https://thosgood.com/Tim Hosgood's blog2022-04-25T00:00:00ZTim Hosgoodhttps://thosgood.comtag:thosgood.com,2022-04-25:/blog/2022/04/25/some-research-questions-from-my-notesbooks/Some research questions from my notebooks2022-04-25T00:00:00Z2022-04-25T00:00:00Z<p>One thing that the past few years have taught me is that I am not
good at doing maths all by myself. In fact, I would go as far as to say
I am completely useless and unmotivated. I do much better when I have
co-authors to give me deadlines and friends to talk to, but, for obvious
reasons, the past two years have not been good for this. Not really the
ideal time for first postdocs, but alas, that’s life.</p>
<p>I recently found an old notebook with some vague questions and
research ideas in it, and then realised that I have had no motivation to
work on any of these alone, so why not put them out there for other
people to see?</p>
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<p>A common fear amongst early-career mathematicians seems to be getting
scooped: having somebody solve the questions you were working on before
you do, maybe even by having read your work and using it as a launchpad.
This is very understandable, since we live in a horrible academic world
where publishing is everything, and trying to publish some results that
you’ve arrived at <em>after</em> somebody else, no matter how shortly or
how differently, suddenly becomes a <em>lot</em> harder, which makes
getting a job next year <em>much</em> more difficult, and… Because of
this, I’ve always been a bit cautious in writing what I’m working on.
But, in the context of my life, the world around me, and some other
things, I realise that these fears aren’t really very well founded for
me right now. There’s no point in me worrying about other people using
my ideas to solve problems before I do if <em>I’m not even working on
applying my ideas myself</em>. Not only that, it’s not like I’m sitting
on some goldmine of potential paths to solutions of big long-standing
open problems — I just have a few small ideas about small things, none
of them are particularly profound or ingenious (in fact, now that I read
back what I’ve written, I see that they’re basically all just “<em>what
if you did X, but with the Čech nerve instead?</em>”). If anything, I
shouldn’t be worried about people scooping my ideas, but the opposite:
trying to get anybody to listen at all!</p>
<p>So I figured I might as well just open up some of<a href="#fn1"
class="footnote-ref" id="fnref1" role="doc-noteref"><sup>1</sup></a> my
“research” in the hopes that one of you might see one of the questions I
have and think “oh, that’s interesting, I wonder if Tim would like to
talk about this sometime” (to which I would almost certainly (time and
obligations permitting) answer, “yes, that would be very lovely”).</p>
<h1 id="some-simplicial-things">Some simplicial things</h1>
<p><em>(These questions are about using the Čech nerve, and sometimes
the notion of (pre)sheaves on the Čech nerve — cf. e.g. Definition 2.2.1
in “<a href="https://arxiv.org/abs/2003.10023">Simplicial Chern-Weil
theory for coherent analytic sheaves, Part I</a>”.)</em></p>
<h2 id="barycentric-global-sections-always-exist">Barycentric global
sections always exist</h2>
<p>Take a presheaf <span class="math inline">\mathscr{F}</span> (of
<span class="math inline">\mathbb{R}</span>-modules, say) on some space
<span class="math inline">X</span>, and pick a bunch of local sections
<span class="math inline">\{s_i\in\Gamma(U_i,\mathscr{F})\}_{i\in
I}</span>, where <span class="math inline">\{U_i\}_{i\in I}</span> is an
open cover of <span class="math inline">X</span>. Since we only have a
presheaf, there’s no reason for these local sections to glue to give a
global section. <em>But</em> if we pull back our presheaf along the Čech
nerve then we can consider the “barycentric global section” <span
class="math inline">\sum_{j=0}^p t_j s_j</span> on any <span
class="math inline">U_{\alpha_0\ldots\alpha_p}</span>. This is like the
“uniform average” of all the sections: if we fibre integrate then we’d
get e.g. <span class="math inline">\frac1{p+1}\sum_{i=0}^p
s_{\alpha_j}</span> on each <span
class="math inline">U_{\alpha_0\ldots\alpha_p}</span>.</p>
<p>Is this ever useful at all, as a sort of “second best thing” for a
global section that you know doesn’t exist?</p>
<h2 id="čech-nerves-of-things">Čech nerves of things</h2>
<p>Consider some complex-analytic “thing” (i.e. manifold, space,
whatever) <span class="math inline">X</span> that is locally algebraic
(whatever that might mean). If we take the Čech nerve then is the
resulting simplicial “thing” an <em>algebraic</em> simplicial “thing”
<span class="math inline">\widetilde{X}</span>? In the case where <span
class="math inline">X</span> is <em>not</em> algebraic, can we measure
how far away it is from being so by looking at the simplicial “thing”
<span class="math inline">\widetilde{X}</span>?</p>
<p>A simpler, similar question: is the Čech nerve of an affine cover of
an (algebraic) scheme an “affine object” in the category of simplicial
schemes?</p>
<p><em>(This last question should be something already well known, but I
just don’t know the answer myself.)</em></p>
<h1 id="some-analytic-geometry">Some analytic geometry</h1>
<p><em>(These questions are about taking things that we know how to do
in the smooth or the algebraic world, and trying to do them in the
complex-analytic world.)</em></p>
<h2 id="chernweil-for-stacks">Chern–Weil for stacks</h2>
<p>One way of cheekily summarising some of the results from <a
href="https://tel.archives-ouvertes.fr/tel-02882140">my PhD thesis</a>
would be that “sheaves on the Čech nerve sometimes allow you to apply
smooth methods to complex-analytic things”, and the application of this
that I considered was Chern–Weil theory via the Atiyah exact sequence.
So is it possible to extend some results concerning Chern–Weil theory
via the Atiyah exact sequence on differentiable stacks to the
complex-analytic case?</p>
<p><em>(This question was prompted by seeing two papers on the arXiv by
Indranil Biswas, Saikat Chatterjee, Praphulla Koushik, and Frank
Neumann: <a
href="https://arxiv.org/abs/2012.08442"><code>2012.08442</code></a> and
<a
href="https://arxiv.org/abs/2012.08447"><code>2012.08447</code></a>.)</em></p>
<h2 id="holomorphic-deligne-cohomology">Holomorphic Deligne
cohomology</h2>
<p>This is one that I’ve been thinking about ever since the middle of my
PhD, and it was actually the original problem that I’d hoped to solve
(but that turned out to be much more difficult than we’d first thought).
Deligne cohomology in the <em>smooth</em> setting is really well
understood — for example, Urs Schreiber has written SO much about this
(and all very very lovely, albeit nearly entirely far over my head)
under the name of <em>differential cohomology</em>. But at some point
quite early on there is a partition of unity argument, which means that
it fails in the holomorphic case. Indeed, there are lots of little
worked examples you can do that show that Deligne cohomology in the
complex-analytic world really is quite different.</p>
<p>One “simple” concrete problem is the following: given a holomorphic
vector bundle (not even an arbitrary coherent analytic sheaf!), write
down <em>Čech representatives</em> for its Chern classes <em>in Deligne
cohomology</em>. Just the first part (Čech representatives) was done<a
href="#fn2" class="footnote-ref" id="fnref2"
role="doc-noteref"><sup>2</sup></a> in <a
href="http://wrap.warwick.ac.uk/40592/">Green’s 1980 PhD thesis</a>;
just the second part (holomorphic Deligne cohomology) was done in <a
href="http://jgrivaux.perso.math.cnrs.fr/articles/Chern.pdf">Grivaux’s
2009 thesis</a>. There is a (beautiful and very good) paper by Brylinski
and McLaughlin (“<a
href="https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-178/issue-1/%C4%8Cech-cocycles-for-characteristic-classes/cmp/1104286562.pdf">Cech
Cocycles for Characteristic Classes</a>”) that seems to give an answer,
but only gives a proof in the smooth setting, and when I sat down (many
<em>many</em> times) and tried to work through it myself, the degrees in
the holomorphic setting seemed to be exactly the wrong ones that you
would get from trying to treat smooth Deligne cohomology like
holomorphic Deligne cohomology (i.e. bidegree <span
class="math inline">(2p,0)</span> instead of <span
class="math inline">(p,p)</span>, if I remember correctly). It is also
one of those answers where they say “ok, here are all the ingredients
you need, so just put them together”, but then don’t just write down the
explicit representatives (something that always frustrates me: if
putting the pieces together is hard, then say it’s hard; if it’s easy,
then why not just do it?!).</p>
<p>To reduce this problem down to the very simplest form: given a
rank-<span class="math inline">2</span> holomorphic vector bundle
defined by transition functions <span
class="math inline">g_{\alpha\beta}</span>, write down a Čech cocycle
<em>in terms of the <span
class="math inline">g_{\alpha\beta}</span></em> representing the second
Chern class in <em>Deligne cohomology</em>.</p>
<p><em>(For example, we know how to do this in de Rham cohomology: you
can take <span
class="math inline">\operatorname{tr}(\omega_{\alpha\beta}g_{\alpha\beta}\omega_{\beta\gamma}g_{\alpha\beta}^{-1})</span>
or, equivalently (via some algebraic manipulations), <span
class="math inline">\operatorname{tr}(\omega_{\alpha\beta}(\omega_{\alpha\gamma}-\omega_{\alpha\beta}))</span>,
where <span
class="math inline">\omega_{\alpha\beta}=\operatorname{d}\log
g_{\alpha\beta}=g_{\alpha\beta}^{-1}dg_{\alpha\beta}</span>, and you can
then extend this to give a closed element of the Čech-de Rham
bicomplex.)</em></p>
<h1 id="some-analytic-sheaves">Some analytic sheaves</h1>
<p><em>(More complex-analytic geometry, this time dealing with
holomorphic vector bundles and their generalisations.)</em></p>
<h2 id="complexes-of-coherent-sheaves">Complexes of coherent
sheaves</h2>
<p>If you’re dealing with sheaves in algebraic geometry, then you might
care about whether or not they’re coherent. Given that we like to think
of complexes of sheaves instead of just single sheaves (e.g. we like
working in the derived category), we are interested in the derived
category <span class="math inline">D^\mathrm{b}\mathsf{Coh}(X)</span> of
(bounded) complexes of coherent sheaves. But there’s another category
which arises quite often in practice, namely the category <span
class="math inline">D^\mathrm{b}_{\mathsf{Coh}}(\mathsf{Sh}(X))</span>
of complexes of sheaves that are not necessarily coherent, but whose
(internal, i.e. “kernel of the differential of the complex modulo the
image of the differential”) cohomology consists of coherent sheaves. The
latter sounds like it should be more general, but in nice algebraic
cases the two are actually equivalent! This follows from a result in SGA
6:</p>
<ul>
<li><strong>SGA 6, II, Corollaire 2.2.2.1.</strong> If <span
class="math inline">X</span> is a Noetherian scheme, then the canonical
fully faithful functor <span
class="math inline">D^\mathrm{b}(\mathsf{Coh}(X))\hookrightarrow
D(\mathsf{Sh}(X))</span> identifies the codomain with the full
subcategory <span
class="math inline">D^\mathrm{b}_{\mathsf{Coh}}(\mathsf{Sh}(X))</span>
of the domain.</li>
</ul>
<p>Now there’s another related result, which follows from applying the
above corollary to a specific case:</p>
<ul>
<li><strong>SGA 6, I, Exemples 5.11 (+ II, Corollaire 2.2.2.1).</strong>
If <span class="math inline">X</span> is a smooth scheme, then there is
a canonical equivalence of triangulated categories <span
class="math inline">\mathsf{Perf}(X)\xrightarrow{\sim}D^{\mathrm{b}}(\mathsf{Coh}(X))</span>.</li>
</ul>
<p>This says that, for <em>smooth</em> schemes, pseudo-coherence is
equivalent to perfectness (being <em>locally</em> resolved by locally
free sheaves).</p>
<p>Now, the analogue of this second statement is still true in the
analytic case: coherent analytic sheaves always have <em>local</em>
locally free resolutions. But the first statement is a corollary to
Proposition 2.2.2, which uses the fact that “every quasi-coherent module
is the filtrant colimit of its coherent submodules”, and this is
<em>not</em> true in the analytic case (in fact, this raises a question
I’ll talk about after this one).</p>
<p>So are these two categories, “complexes of coherent sheaves” and
“complexes of sheaves with coherent cohomology” equivalent in the
analytic setting? In the specific case where <span
class="math inline">X</span> is a smooth compact analytic
<em>surface</em>, yes! This is Corollary 5.2.2 of Bondal and Van den
Bergh’s “<a href="https://arxiv.org/abs/math/0204218">Generators and
representability of functors in commutative and noncommutative
geometry</a>”. But in higher dimensions, it is (as far as I can tell)
still an open question.</p>
<p>I don’t have any particular insight into this problem, except that I
think (yet again) that sheaves on the nerve might have something to say
about this. Furthermore, I haven’t read the details of Bondal and Van
den Bergh’s proofs, nor those in SGA 6, so probably the following is
obvious (or even tautological, actually), but it’s intriguing that these
hypotheses (<span class="math inline">X</span> is smooth, compact, and a
surface) are exactly those found in Schuster’s “<a
href="https://www.degruyter.com/document/doi/10.1515/crll.1982.337.159/html">Locally
free resolutions of coherent sheaves on surfaces</a>” which shows that,
under these hypothesis, coherent analytic sheaves can be
<em>globally</em> resolved by locally free sheaves (the <em>resolution
property</em>).</p>
<h2 id="the-resolution-property">The resolution property</h2>
<p>Whether or not coherent analytic sheaves can be <em>globally</em>
resolved by locally free sheaves (instead of just <em>locally</em>
resolved) is controlled by the so-called <em>resolution property</em>,
mentioned above. We say that something (e.g. a stack) <em>has the
resolution property</em> if every coherent sheaf admits a surjection
from a locally free sheaf — what happens if we change this to “… a
surjection from a locally free sheaf <em>on the nerve</em>”? This could
be useful for Riemann–Roch for Artin stacks: these have the resolution
property when they are quotients of quasi-projective schemes by
reductive groups, but not in general (and so this assumption appears in
Toën’s Riemann–Roch paper). It could also be useful for a formal GAGA
theorem, cf. Geraschenko and Zureick-Brown’s “<a
href="https://arxiv.org/abs/1208.2882">Formal GAGA for good moduli
spaces</a>”.</p>
<h2 id="analogies-between-algebraic-and-analytic-geometry">Analogies
between algebraic and analytic geometry</h2>
<p>Here I’m just going to refer you to a previous blog post of mine: “<a
href="https://thosgood.com/blog/2021/09/24/some-questions-about-analytic-geometry.html">Some
questions about complex-analytic geometry</a>”. The main one here is
“<em>how should we define the notion of quasi-coherence for an analytic
sheaf?</em>”, followed immediately by “<em>why did we pick this
definition, and not the others?</em>”. One possible answer is “read
Scholze and Clausen’s lecture notes on condensed mathematics, Conrad’s
paper on relative ampleness in rigid geometry, and Eschmeier and
Putinar’s book <em>Spectral Decompositions and Analytic Sheaves</em>,
and see if you can put all the pieces together”.</p>
<p>Just for the sake of it, here are some possible contenders for the
definition of quasi-coherence for an analytic sheaf <span
class="math inline">\mathscr{F}</span> of <span
class="math inline">\mathcal{O}_X</span>-modules:</p>
<ol type="1">
<li>being <em>of local presentation</em>, i.e. for all <span
class="math inline">x\in X</span> there exists an open <span
class="math inline">U\subseteq X</span> on which there is an exact
sequence <span class="math display">\mathcal{O}_X^{\oplus I}|U \to
\mathcal{O}_X^{\oplus J}|U \to \mathscr{F}|U \to 0</span></li>
<li>being the filtrant colimit of its coherent subsheaves</li>
<li>being <em>Fréchet quasi-coherent</em>, or, equivalently, admitting a
global “topologically free” resolution</li>
<li>something analogous to arising from the right Kan extension of
something like the pseudofunctor <span
class="math inline">\mathsf{CRing}\to\mathsf{Cat}</span> defined by
<span class="math inline">R\mapsto\mathsf{Mod}_R</span> (very vague, I
know).</li>
</ol>
<p>I know that there are <a
href="https://math.stackexchange.com/questions/2840594/gaga-and-quasicoherent-sheaf/2841087#2841087">examples
of 1. that do not satisfy 2.</a>, but I actually don’t know how the
other notions interact at all.</p>
<section class="footnotes footnotes-end-of-document"
role="doc-endnotes">
<hr />
<ol>
<li id="fn1" role="doc-endnote"><p>Apart from one paper which I’m
currently working on with a co-author, because that’s not just my story
to share. But this will hopefully hit the arXiv… within a year? I dunno,
I’ve been saying that about this specific paper for the past two years,
so we’ll see what actually happens.<a href="#fnref1"
class="footnote-back" role="doc-backlink">↩︎</a></p></li>
<li id="fn2" role="doc-endnote"><p>Well, the method was explained, but
the actual computations were basically left as an exercise to the reader
— I was one such reader, and I wrote them down in <em>my</em> PhD
thesis.<a href="#fnref2" class="footnote-back"
role="doc-backlink">↩︎</a></p></li>
</ol>
</section>
tag:thosgood.com,2022-04-08:/blog/2022/04/08/diagrammatic-equations-and-multiphysics/Diagrammatic equations and multiphysics2022-04-08T00:00:00Z2022-04-08T00:00:00Z<p>Just a very short post (you don’t even need to click “Continue
reading” if you’re looking at this on my blog archive) — I finally
managed to do some more maths (but only because I had some very hard
working and very good coauthors), and I’ve written about it on the Topos
blog as a two-part series: <a
href="https://topos.site/blog/2022/04/diagrammatic-equations-and-multiphysics-part-1/">Part
1</a> and <a
href="https://topos.site/blog/2022/04/diagrammatic-equations-and-multiphysics-part-2/">Part
2</a>.</p>
<p>I would love to write more, but after having written these I’m afraid
I’m all blogged out for the minute, so ciao for now!</p>
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tag:thosgood.com,2022-02-11:/blog/2022/02/11/graded-commutative-and-graded-and-commutative/Graded commutative rings and graded and commutative rings2022-02-11T00:00:00Z2022-02-11T00:00:00Z<p>One of the many reasons that teaching is fun is because you get to
look back at things that you haven’t seen in a while and try to
understand them in light of what you’ve learnt in the meantime. This
means that you sometimes have the unexpected joy of having to teach
something that always used to confuse you, but that now seems so much
more straightforward! I experienced this last year when teaching an
algebraic topology course: I remember being super lost when it came to
the graded ring structure of cohomology and getting very annoyed at
Hatcher’s book; now I look back and realise that it’s really neat! This
post has a slightly different intended audience than normal: I’m just
gonna assume that you know a bit about rings in the first half; the
second half is aimed for somebody who’s a reasonable way through a first
course on algebraic topology (e.g. knows what the cup product in
cohomology is).</p>
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<h1 id="graded-rings">Graded rings</h1>
<p>Our starting point is the omnipresent <em>polynomial ring</em>: given
a ring <span class="math inline">S</span>, we define <span
class="math inline">S[x]</span> to be the set of polynomials in one
variable (namely <span class="math inline">x</span>) with coefficients
in <span class="math inline">S</span>, so elements of <span
class="math inline">S[x]</span> are of the form <span
class="math display">
f = s_n x^n + s_{n-1} x^{n-1} + \ldots + s_1 x + s_0
</span> where the <span class="math inline">s_i</span> are elements of
<span class="math inline">R</span>. (More generally, we have the
<em>polynomial ring in <span class="math inline">n</span>
indeterminates</em>: <span
class="math inline">S[x_1,x_2,\ldots,x_n]</span>, which is entirely
analogous, but with more than one variable; this is important, but we
don’t need to worry about it too much for now).</p>
<p>These rings are <em>really</em> nice, in lots of ways that we won’t
talk about today, but one <em>particularly</em> nice thing that they
have is a <em>grading</em>, given by the <em>degree</em>. Recall that
the degree of a <em>monomial</em> is defined to be the “total power”,
i.e. <span class="math display">
\deg(x^n) = n
</span> (we say <em>total</em> power to deal with the case of multiple
variables, e.g. <span class="math inline">\deg(x^my^n)=m+n</span>). The
degree behaves nicely under multiplication <span class="math display">
\deg(x^m\cdot x^n) = m+n
</span> and we have a <em>unique</em> way of writing <em>any</em>
polynomial <span class="math inline">f\in S[x]</span> as a sum of
monomials, basically by definition of what it means to be a polynomial:
<span class="math display">
f = \sum_{i=0}^n r_i x^i.
</span></p>
<p>So let’s do what we always do and generalise this to an abstract
structure!</p>
<div class="rmenv" title="Definition">
<p>A ring <span class="math inline">R</span> is a <em>graded ring</em>
if there exist abelian groups <span
class="math inline">(R_d)_{d\in\mathbb{N}}</span>, where <span
class="math inline">R_d\subseteq R</span> for all <span
class="math inline">d\in\mathbb{N}</span>, such that</p>
<ol type="i">
<li><span class="math inline">R\cong\bigoplus_{d=0}^\infty
R_d</span>;</li>
<li><span class="math inline">R_d R_e\subseteq R_{d+e}</span>.</li>
</ol>
<p>Given <span class="math inline">r\in R\setminus\{0\}</span>, if there
exists some <span class="math inline">d\in\mathbb{N}</span> such that
<span class="math inline">r\in R_d</span>, then we say that <span
class="math inline">r</span> is <em>homogeneous of degree <span
class="math inline">d</span></em>, and we write <span
class="math inline">|r|=d</span>.</p>
</div>
<p>The prototypical example is the thing we started with: the polynomial
ring <span class="math inline">R=S[x]</span> is a graded ring with <span
class="math inline">R_d\coloneqq\{sx^d \mid s\in S\}</span>; the more
general polynomial ring <span
class="math inline">R=S[x_1,\ldots,x_n]</span> in <span
class="math inline">n</span> variables is a graded ring with <span
class="math inline">R_d\coloneqq\{s x_1^{m_1}x_2^{m_2}\cdots
x_n^{m_n}\mid s\in S, m_1+m_2+\ldots+m_n=d\}</span>.</p>
<h1 id="commutativity">Commutativity</h1>
<p>Now let’s talk about something confusing: what is a <em>graded
commutative ring</em>? Or should we say <em>commutative graded
ring</em>? Or should these two things be different?</p>
<p>Well, it makes sense that a graded commutative ring would just be a
commutative ring that is graded, i.e. we parse it as “graded
(commutative ring)”. Annoyingly, however, this is <em>not</em> the way
that most algebraists or geometers will parse this! If you want to talk
about commutative rings that are graded, then your best bet is really to
just say “a commutative ring that is graded”, but if you want to be
snappier, then I would advise that you say <em>commutative graded
ring</em>. Why am I making such a point out of this? What do people mean
when they actually say “graded commutative ring” then?</p>
<p>The answer lies in “bracketing” the adjectives in a different way,
namely: “(graded commutative) ring”. But this just prompts the question:
what does it mean for a ring to be “graded commutative”?</p>
<div class="rmenv" title="Definition">
<p>A ring <span class="math inline">R</span> is <em>graded
commutative</em> if <span class="math inline">R</span> is a
<em>graded</em> ring <span class="math inline">R=\bigoplus_{d=0}^\infty
R_d</span> such that <span class="math display">
rs = (-1)^{|r||s|}sr
</span> for all homogeneous elements <span class="math inline">r,s\in
R</span>.</p>
</div>
<p>So this is a slightly odd definition: graded commutativity is like
commutativity, but with a possible minus sign, depending on the degree
of the (homogeneous) elements.<a href="#fn1" class="footnote-ref"
id="fnref1" role="doc-noteref"><sup>1</sup></a> To understand this
better, let’s turn back to our good old friend <span
class="math inline">S[x]</span>. If <span class="math inline">S</span>
is commutative, then <span class="math inline">S[x]</span> is clearly
commutative, but is it graded commutative? Shockingly, no! Indeed, we
are asking if the following equality holds <span class="math display">
x^2 = x\cdot x \overset{?}{=} (-1)^{|x||x|} x\cdot x = -x^2.
</span> We see that this only happens in two cases:</p>
<ol type="1">
<li>if <span class="math inline">2=0</span>; or</li>
<li>if <span class="math inline">x^2=0</span>.</li>
</ol>
<p>The first one can happen (if <span class="math inline">S</span> is a
field of characteristic <span class="math inline">2</span>, for example,
e.g. <span class="math inline">\mathbb{Z}/2\mathbb{Z}</span>), but the
second one cannot, by the very definition of <span
class="math inline">S[x]</span>. <em>But</em> this second case does
happen in the <em>ring of dual numbers</em> <span
class="math inline">S[x]/(x^2)</span>.</p>
<p>Going back to the polynomial ring <span
class="math inline">S[x]</span>, we could also do something entirely
different and <em>define</em> <span class="math inline">x</span> to be
of <em>degree <span class="math inline">2</span></em>. That is, we are
giving <span class="math inline">S[x]</span> a <em>different</em> graded
ring structure: <span
class="math inline">S[x]\cong\bigoplus_{d=0}^\infty R_d</span> where
<span class="math display">
R_d =
\begin{cases}
\{sx^d \mid s\in S\} &\text{if }d\text{ is even;}
\\0 &\text{if }d\text{ is odd.}
\end{cases}
</span> (After a little bit of thought, you can see that this is “the
same as” simply looking at <span class="math inline">S[y^2]</span>,
where <span class="math inline">y</span> is an indeterminate of degree
<span class="math inline">1</span> again). Then we have that <span
class="math display">
(-1)^{|x||x|} x\cdot x = x^2
</span> and so this graded ring <em>is</em> graded commutative.</p>
<p>A nice little summary table would be helpful right about now (and,
just in this table, we’re going to make the assumption that
<strong><span class="math inline">S</span> is commutative</strong>).</p>
<table>
<thead>
<tr class="header">
<th style="text-align: center;">Graded ring</th>
<th style="text-align: center;">Commutative?</th>
<th style="text-align: center;">Graded commutative?</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: center;"><span
class="math inline">S[x]</span></td>
<td style="text-align: center;">✅</td>
<td style="text-align: center;">❌ (<strong>unless</strong> <span
class="math inline">2=0</span> in <span
class="math inline">S</span>)</td>
</tr>
<tr class="even">
<td style="text-align: center;"><span
class="math inline">S[x]/(x^2)</span></td>
<td style="text-align: center;">✅</td>
<td style="text-align: center;">✅</td>
</tr>
<tr class="odd">
<td style="text-align: center;"><span
class="math inline">S[x^2]</span></td>
<td style="text-align: center;">✅</td>
<td style="text-align: center;">✅</td>
</tr>
</tbody>
</table>
<p>Let’s look at one more example that we’re going to need when it comes
to doing some algebraic topology: the <em>exterior algebra</em> <span
class="math inline">\Lambda_S[\alpha_1,\ldots,\alpha_n]</span> of a ring
<span class="math inline">S</span> can be defined as the graded ring
<span class="math display">
\Lambda_S[\alpha_1,\ldots,\alpha_n] \coloneqq
S[\alpha_1,\ldots,\alpha_n]/(\alpha_i^2,\alpha_i\alpha_j+\alpha_j\alpha_i)_{1\leq
i,j\leq n}
</span> where each <span class="math inline">\alpha_i</span> is of
degree <span class="math inline">1</span> (although, again, we can
modify this if we want to). Note that <span
class="math inline">\Lambda_S[\alpha]\cong S[\alpha]/(\alpha^2)</span>,
i.e. <em>the exterior algebra of <span class="math inline">S</span> in
one variable is exactly the ring of dual numbers</em>.</p>
<p>The exterior algebra is commutative (if <span
class="math inline">S</span> is), but is it graded commutative? Well
it’s enough to check the condition on the generators <span
class="math inline">\alpha_i</span>, but we see that <span
class="math display">
\alpha_i\alpha_j
= -\alpha_j\alpha_i
= (-1)^{|\alpha_i||\alpha_j|}\alpha_j\alpha_i
</span> where the first equality is exactly by the definition of <span
class="math inline">\Lambda_S[\alpha_1,\ldots,\alpha_n]</span> (and note
that, if <span class="math inline">i=j</span>, then everything is zero,
and we definitely have that <span class="math inline">0=0</span>).</p>
<p>Back to our table:</p>
<table>
<thead>
<tr class="header">
<th style="text-align: center;">Graded ring</th>
<th style="text-align: center;">Commutative?</th>
<th style="text-align: center;">Graded commutative?</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: center;"><span
class="math inline">S[x]</span></td>
<td style="text-align: center;">✅</td>
<td style="text-align: center;">❌ (<strong>unless</strong> <span
class="math inline">2=0</span> in <span
class="math inline">S</span>)</td>
</tr>
<tr class="even">
<td style="text-align: center;"><span
class="math inline">S[x^2]</span></td>
<td style="text-align: center;">✅</td>
<td style="text-align: center;">✅</td>
</tr>
<tr class="odd">
<td style="text-align: center;"><span
class="math inline">\Lambda_S[\alpha_1,\ldots,\alpha_n]</span></td>
<td style="text-align: center;">❌ (unless <span
class="math inline">n=1</span>)</td>
<td style="text-align: center;">✅</td>
</tr>
</tbody>
</table>
<p>(we’ve removed the row for <span
class="math inline">S[x]/(x^2)</span>, since this is just a specific
example of the exterior algebra <span
class="math inline">\Lambda_S[\alpha_1,\ldots,\alpha_n]</span> where
<span class="math inline">n=1</span>.)</p>
<h1 id="freeness">Freeness</h1>
<p>Unrelated to the property of (graded) commutativity is that of being
<em>free</em>. For <em>commutative</em> rings, being free basically
means being isomorphic to a polynomial ring in finitely many variables.
So <span class="math inline">S[x]</span> is free (as a commutative
ring), as is <span class="math inline">S[x^2]</span>, but <span
class="math inline">S[x]/(x^2)</span> is <em>not</em> free (as a
commutative ring) — the latter has a nilpotent element (i.e. some <span
class="math inline">r</span> such that <span
class="math inline">r^2=0</span>, namely <span
class="math inline">r=x</span>) and free commutative rings never have
nilpotent elements, so it cannot be isomorphic to a free commutative
ring.<a href="#fn2" class="footnote-ref" id="fnref2"
role="doc-noteref"><sup>2</sup></a></p>
<p>But note that I’ve been very careful to say “free <em>as a
commutative ring</em>”, and this is important: <span
class="math inline">\Lambda_S[\alpha]</span> is <strong>not</strong> a
free <em>commutative ring</em>, but it <strong>is</strong> a free
<em>graded commutative ring</em>. What do I mean by this? I mean that,
if we just use the fact that <span
class="math inline">\Lambda_S[\alpha]</span> is a graded commutative
ring, <em>forgetting all about how it’s actually defined</em>, then we
can recover the fact that <span class="math inline">\alpha_i^2=0</span>
and <span class="math inline">\alpha_i\alpha_j=-\alpha_j\alpha_i</span>
<em>without having to ask for it</em> (…most of the time).</p>
<p>That is, if we know that <span
class="math inline">\Lambda_S[\alpha_1,\ldots,\alpha_n]</span> is graded
commutative, then we know that <span class="math display">
\alpha_i\cdot\alpha_i =
(-1)^{|\alpha_i||\alpha_i|}\alpha_i\cdot\alpha_i =
-\alpha_i\cdot\alpha_i
</span> which rearranges to give <span class="math display">
2\alpha_i^2=0.
</span> But then, if we can <em>divide by <span
class="math inline">2</span></em> (i.e. if <span
class="math inline">2</span> is an invertible element) in our ring <span
class="math inline">S</span>, then we see that <span
class="math inline">\alpha^2</span> <em>must be</em> equal to zero,
automatically! (The other relation that we need to impose, that <span
class="math inline">\alpha_i\alpha_j=-\alpha_j\alpha_i</span> is exactly
the definition of being graded commutative (since all our <span
class="math inline">\alpha_i</span> are of degree <span
class="math inline">1</span>), so we don’t even need to worry about
enforcing that one separately!)</p>
<p>Of course, I’m going to put this in the table (and I’ll write
“gc-ring” to mean “graded commutative ring”).</p>
<table>
<colgroup>
<col style="width: 18%" />
<col style="width: 19%" />
<col style="width: 31%" />
<col style="width: 31%" />
</colgroup>
<thead>
<tr class="header">
<th style="text-align: center;">Graded ring</th>
<th style="text-align: center;">Commutative?</th>
<th style="text-align: center;">Graded commutative?</th>
<th style="text-align: center;">Free (as a gc-ring)</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: center;"><span
class="math inline">S[x]</span></td>
<td style="text-align: center;">✅</td>
<td style="text-align: center;">❌ (<strong>unless</strong> <span
class="math inline">2=0</span> in <span
class="math inline">S</span>)</td>
<td style="text-align: center;">✅</td>
</tr>
<tr class="even">
<td style="text-align: center;"><span
class="math inline">S[x^d]</span></td>
<td style="text-align: center;">✅</td>
<td style="text-align: center;">❌ (<strong>unless</strong> <span
class="math inline">2</span> divides <span
class="math inline">d</span>)</td>
<td style="text-align: center;">✅</td>
</tr>
<tr class="odd">
<td style="text-align: center;"><span
class="math inline">\Lambda_S[\alpha_1,\ldots,\alpha_n]</span></td>
<td style="text-align: center;">❌ (unless <span
class="math inline">n=1</span>)</td>
<td style="text-align: center;">✅</td>
<td style="text-align: center;">❌ (<strong>unless</strong> <span
class="math inline">2</span> is invertible in <span
class="math inline">S</span>)</td>
</tr>
</tbody>
</table>
<p>By now you have probably noticed that all of the difficulties and
subtleties come from how the number <span class="math inline">2</span>
behaves (except for the exterior algebra only being commutative when in
one variable, but this isn’t a “graded” property, so we’ll ignore that
one), and this is a very common thing to happen.<a href="#fn3"
class="footnote-ref" id="fnref3" role="doc-noteref"><sup>3</sup></a></p>
<h1 id="circles-and-spheres">Circles and spheres</h1>
<p>Before diving into the applications, we just need one more abstract
remark about graded rings: the tensor product <span
class="math inline">R\otimes S</span> of two graded rings has a natural
graded ring structure by setting <span class="math display">
\deg(r\otimes s) \coloneqq \deg(r)+\deg(s)
</span> and defining the multiplication to have a sign that “makes
things work nicely”: <span class="math display">
(r\otimes s)(r'\otimes s') \coloneqq
(-1)^{|r'||s|}(rr')\otimes(ss')
</span> (note that the exponent uses the two “inner” terms, <span
class="math inline">r'</span> and <span
class="math inline">s</span>, <strong>not</strong> <span
class="math inline">r</span> and <span class="math inline">s</span>, nor
<span class="math inline">r'</span> and <span
class="math inline">s'</span> (nor <span
class="math inline">r</span> and <span
class="math inline">s'</span>, for that matter)).</p>
<p>What does this all have to do with algebraic topology? Hopefully
you’ve already seen the ring structure on cohomology (of a manifold),
given by the cup product. In fact, the cup product lets us assemble the
cohomology groups into a <em>commutative graded ring</em> (this is
indeed one justification for there being a minus sign in the definition
of the cup product: it ensures graded commutativity!).</p>
<p><em>In what follows, we say “space” to mean “CW complex that is nice
enough to satisfy whatever hypotheses the theorem in question might
need”.</em></p>
<p>Here’s a fun fact that we won’t prove:</p>
<div class="itenv" title="Lemma">
<p>Let <span class="math inline">X</span> and <span
class="math inline">Y</span> be spaces. If <span
class="math inline">Y</span> is such that <span
class="math inline">\operatorname{H}^\bullet(Y)</span> is <em>free</em>
(as a graded commutative ring) and <em>finitely generated</em>, then
<span class="math display">
\operatorname{H}^\bullet(X\times Y) \cong
\operatorname{H}^\bullet(X)\otimes\operatorname{H}^\bullet(Y)
</span> (where <span class="math inline">\cong</span> means “isomorphic
<em>as graded rings</em>”).</p>
</div>
<p>Using this, we can look at two examples.</p>
<div class="rmenv" title="Example 1">
<p>We know that <span class="math display">
\operatorname{H}^n(S^1) =
\begin{cases}
\mathbb{Z} &\text{if }n=0,1\text{;}
\\0 &\text{otherwise.}
\end{cases}
</span> If we denote by <span class="math inline">\alpha</span> the
generator of <span class="math inline">\operatorname{H}^1(S^1)</span>,
then <span class="math display">
\alpha^2 \coloneqq \alpha\smile\alpha
</span> is an element of <span
class="math inline">\operatorname{H}^2(S^1)</span>, but this group is
zero, and so it must be the case that <span
class="math inline">\alpha^2=0</span>. Thus <span class="math display">
\operatorname{H}^\bullet(S^1) \cong \mathbb{Z}\langle1\rangle \oplus
\mathbb{Z}\langle\alpha\rangle/(\alpha^2)
= \Lambda_{\mathbb{Z}}[\alpha]
</span> (where here the angled brackets mean “the abelian group
generated by these elements”).</p>
<p>But we know that <span
class="math inline">\Lambda_{\mathbb{Z}}[\alpha]</span> is free as a
graded commutative ring, and it has only one generator so it’s also
finitely generated; we can apply the Lemma above to get that <span
class="math display">
\operatorname{H}^\bullet(S^1\times S^1)
\cong
\operatorname{H}^\bullet(S^1)\otimes\operatorname{H}^\bullet(S^1)
</span> but this is exactly <span
class="math inline">\Lambda_{\mathbb{Z}}[\alpha_1,\alpha_2]</span>
(where <span class="math inline">\alpha_i</span> is the generator of the
first cohomology of the <span class="math inline">i</span>-th copy of
<span class="math inline">S^1</span>).</p>
<p>Finally, recall that <span
class="math inline">\Lambda_{\mathbb{Z}}[\alpha_1,\alpha_2]</span> is
graded commutative but <em>not</em> commutative (i.e. <span
class="math inline">\alpha_1\alpha_2=-\alpha_2\alpha_1</span>).</p>
</div>
<div class="rmenv" title="Example 2">
<p>We know that <span class="math display">
\operatorname{H}^n(S^2) =
\begin{cases}
\mathbb{Z} &\text{if }n=0,2\text{;}
\\0 &\text{otherwise.}
\end{cases}
</span> If we denote by <span class="math inline">\beta</span> the
generator of <span class="math inline">\operatorname{H}^2(S^2)</span>,
then <span class="math display">
\beta^2 \coloneqq \beta\smile\beta
</span> is an element of <span
class="math inline">\operatorname{H}^4(S^2)</span>, but this group is
zero, and so it must be the case that <span
class="math inline">\beta^2=0</span>. Thus <span class="math display">
\operatorname{H}^\bullet(S^2) \cong \mathbb{Z}\langle1\rangle \oplus
\mathbb{Z}\langle\beta\rangle/(\beta^2)
= \Lambda_{\mathbb{Z}}[\beta]
</span> where <em><span class="math inline">\beta</span> is of degree
<span class="math inline">2</span></em>.</p>
<p>As in the previous example, the Lemma gives us that <span
class="math display">
\operatorname{H}^\bullet(S^2\times S^2)
\cong
\operatorname{H}^\bullet(S^2)\otimes\operatorname{H}^\bullet(S^2)
</span> which is exactly <span
class="math inline">\Lambda_{\mathbb{Z}}[\beta_1,\beta_2]</span>, but
<em>where <span class="math inline">\beta</span> is of degree <span
class="math inline">2</span></em> (we’re repeating this because it’s
important).</p>
<p>Now, <span
class="math inline">\Lambda_{\mathbb{Z}}[\beta_1,\beta_2]</span> is
graded commutative (since it’s a cohomology ring, and recall that these
are always graded commutative, by the sign in the definition of the cup
product), but we see that <span class="math display">
\beta_1\beta_2
= (-1)^{|\beta_1||\beta_2|} \beta_2\beta_1
= (-1)^{4}\beta_2\beta_1
= \beta_2\beta_1
</span> and so this cohomology ring is actually commutative!</p>
</div>
<div class="itenv" title="Corollary">
<p>The cohomology rings of <span class="math inline">S^1\times
S^1</span> and <span class="math inline">S^2\times S^2</span> are not
isomorphic (because one is commutative and the other is not).</p>
</div>
<h1 id="the-künneth-theorem">The Künneth theorem</h1>
<p>What happens to our useful lemma if the cohomology ring <span
class="math inline">\operatorname{H}^\bullet(Y)</span> is finitely
generated but <em>not</em> free? That is (by the classification of
finite groups), what if it has <em>torsion</em>? The <em>Künneth
theorem</em> tells us that this torsion is exactly the “correction term”
needed to fix the Lemma.</p>
<div class="itenv" title="Corollary (to the Künneth theorem)">
<p>Let <span class="math inline">X</span> and <span
class="math inline">Y</span> be spaces. If <span
class="math inline">Y</span> is such that <span
class="math inline">\operatorname{H}^\bullet(Y)</span> is finitely
generated, then <span class="math display">
\operatorname{H}^\bullet(X\times Y)
\cong (\operatorname{H}^\bullet(X)\otimes\operatorname{H}^\bullet(Y))
\oplus
\operatorname{Tor}_{\bullet+1}(\operatorname{H}^\bullet(X),\operatorname{H}^\bullet(Y))
</span> (note that this <span
class="math inline">\operatorname{Tor}</span> term is zero if <span
class="math inline">\operatorname{H}^\bullet(Y)</span> is free<a
href="#fn4" class="footnote-ref" id="fnref4"
role="doc-noteref"><sup>4</sup></a>, so this really is a generalisation
of our previous lemma).</p>
</div>
<div class="itenv" title="Example">
<p>We know that <span class="math display">
\operatorname{H}^n(\mathbb{RP}^2) =
\begin{cases}
\mathbb{Z} &\text{if }n=0\text{;}
\\\mathbb{Z}/2\mathbb{Z} &\text{if }n=2\text{;}
\\0 &\text{otherwise.}
\end{cases}
</span> Doing some <span class="math inline">\operatorname{Tor}</span>
calculations (use the fact that <span
class="math inline">\mathbb{Z}</span> is free and thus flat (or even
just projective) to see that <span
class="math inline">\operatorname{Tor}(\mathbb{Z},-)=\operatorname{Tor}(-,\mathbb{Z})=0</span>;
use the fact that <span
class="math inline">\operatorname{Tor}(A,\mathbb{Z}/m\mathbb{Z})\cong\{a\in
A\mid ma=0\}</span> to see that <span
class="math inline">\operatorname{Tor}(\mathbb{Z}/2\mathbb{Z},\mathbb{Z}/2\mathbb{Z})\cong\mathbb{Z}/2\mathbb{Z}</span>),
the above Corollary gives us that <span class="math display">
\operatorname{H}^n(\mathbb{RP}^2\times\mathbb{RP}^2) =
\begin{cases}
\mathbb{Z} &\text{if }n=0\text{;}
\\(\mathbb{Z}/2\mathbb{Z})^2 &\text{if }n=2\text{;}
\\\mathbb{Z}/2\mathbb{Z} &\text{if }n=3{;}
\\\mathbb{Z}/2\mathbb{Z} &\text{if }n=4{;}
\\0 &\text{otherwise.}
\end{cases}
</span> We’ve written the case <span class="math inline">n=3</span> on a
separate line because this is exactly the “correction term” given by the
Künneth theorem, i.e. the previous Lemma would have given us everything
else but would have said that <span
class="math inline">\operatorname{H}^3=0</span>; Künneth tells us
otherwise.</p>
</div>
<div class="rmenv" title="Exercise">
<p>Calculate <span
class="math inline">\operatorname{H}^\bullet(\mathbb{RP}^2\times\mathbb{RP}^2;\mathbb{Q}/\mathbb{Z})</span>.</p>
<p><em>Hint: use the Universal Coefficient Theorem, or the long exact
sequence associated to the short exact sequence <span
class="math inline">0\to\mathbb{Z}\hookrightarrow\mathbb{Q}\twoheadrightarrow\mathbb{Q}/\mathbb{Z}\to0</span>,
and recall that <span
class="math inline">\operatorname{Tor}(A,\mathbb{Q}/\mathbb{Z})\cong\operatorname{tors}(A)</span>,
the torsion part of <span class="math inline">A</span>.</em></p>
</div>
<h1 id="some-useful-cohomology-rings">Some useful cohomology rings</h1>
<p>Real projective spaces have pretty nice cohomology rings, but they
differ in presentation depending on the parity of the dimension:</p>
<ul>
<li><span
class="math inline">\operatorname{H}^\bullet(\mathbb{RP}^{2n})\cong\mathbb{Z}[\beta]/(2\beta,\beta^{n+1})</span>,
where <span class="math inline">|\beta|=2</span>;</li>
<li><span
class="math inline">\operatorname{H}^\bullet(\mathbb{RP}^{2n+1})\cong\mathbb{Z}[\beta,\varepsilon]/(2\beta,\beta^{n+1},\epsilon^2,\beta\epsilon)</span>,
where <span class="math inline">|\beta|=2</span> and <span
class="math inline">|\varepsilon|=2n+1</span> (here <span
class="math inline">\epsilon</span> is the generator of <span
class="math inline">\operatorname{H}^{2n+1}</span>).</li>
</ul>
<p>What’s nice is that, if we work with <span
class="math inline">\mathbb{Z}/2\mathbb{Z}</span> coefficients (instead
of <span class="math inline">\mathbb{Z}</span>), then we can write both
cases together as one thing:</p>
<ul>
<li><span
class="math inline">\operatorname{H}^\bullet(\mathbb{RP}^n;\mathbb{Z}/2\mathbb{Z})\cong(\mathbb{Z}/2\mathbb{Z})[\alpha]/(\alpha^{n+1})</span>,
where <span class="math inline">|\alpha|=1</span>.</li>
</ul>
<p>Given that <span class="math inline">\mathbb{C}</span> is <span
class="math inline">2</span>-dimensional over <span
class="math inline">\mathbb{R}</span>, complex projective space behaves
much nicer, since we don’t need to worry about parity:</p>
<ul>
<li><span
class="math inline">\operatorname{H}^\bullet(\mathbb{CP}^n)\cong\mathbb{Z}[\beta]/(\beta^{n+1})</span>,
where <span class="math inline">|\beta|=2</span>.</li>
</ul>
<p>Finally, if you know how to define infinite dimensional projective
spaces, then it’s a very lovely cheeky little fact that the cohomology
rings are just given by “taking the limit <span
class="math inline">n\to\infty</span>”, i.e.</p>
<ul>
<li><span
class="math inline">\operatorname{H}^\bullet(\mathbb{RP}^\infty)\cong\mathbb{Z}[\alpha]</span>,
where <span class="math inline">|\alpha|=1</span>;</li>
<li><span
class="math inline">\operatorname{H}^\bullet(\mathbb{CP}^\infty)\cong\mathbb{Z}[\beta]</span>,
where <span class="math inline">|\beta|=2</span>.</li>
</ul>
<p>Nice!</p>
<section class="footnotes footnotes-end-of-document"
role="doc-endnotes">
<hr />
<ol>
<li id="fn1" role="doc-endnote"><p>Note that this condition extends to
give some condition on the multiplication of arbitrary elements of <span
class="math inline">R</span>, since every element can be written (in a
unique way) as a sum of homogeneous elements, by the fact that <span
class="math inline">R\cong\bigoplus_{d=0}^\infty R_d</span>.<a
href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
<li id="fn2" role="doc-endnote"><p>This is very similar to the fact that
a group homomorphism must send an element of order <span
class="math inline">n</span> to an element of order <span
class="math inline">m</span> <em>such that <span
class="math inline">m</span> divides <span
class="math inline">n</span></em>.<a href="#fnref2"
class="footnote-back" role="doc-backlink">↩︎</a></p></li>
<li id="fn3" role="doc-endnote"><p>Both number theorists and algebraic
geometers get very tired of having to write “let <span
class="math inline">k</span> be a field of characteristic not equal to
<span class="math inline">2</span>”, since <span
class="math inline">2</span> “behaves badly” for them; some algebraic
topologists, on the other hand, get very excited when you say “calculate
(co)homology with coefficients in a field of characteristic <span
class="math inline">2</span>”, since <span class="math inline">2</span>
“behaves nicely” for them. Basically, <span class="math inline">2</span>
is a tricky number.<a href="#fnref3" class="footnote-back"
role="doc-backlink">↩︎</a></p></li>
<li id="fn4" role="doc-endnote"><p>Or if we work over a field instead of
over <span class="math inline">\mathbb{Z}</span>…<a href="#fnref4"
class="footnote-back" role="doc-backlink">↩︎</a></p></li>
</ol>
</section>
tag:thosgood.com,2022-02-05:/blog/2022/02/05/every-bundle-is-flat-from-infinity-pov/Every principal bundle is flat, in the infinity world2022-02-05T00:00:00Z2022-02-05T00:00:00Z<p>Earlier today, Mahmoud Zeinalian explained something to me that
Dennis Sullivan once explained to him, and it’s been sitting in my brain
ever since then. In an attempt to empty out my thoughts, and also
preserve what little understanding I currently believe to have of the
story, I thought I’d write a little blog post about it. It’s going to
move quite quickly, because I don’t want to spend time developing the
prerequisites — the main purpose is for this to jog my brain two weeks
down the line when I forget all the details!</p>
<!-- more -->
<p>The rather “provocative” statement that I’m going to try to justify
today is the following:</p>
<blockquote>
<p><em>Every principal bundle is flat, from the <span
class="math inline">(\infty,1)</span>-point-of-view.</em></p>
</blockquote>
<p>Let’s give some setup and explain why, first of all, this should
sound rather bizarre, if not completely incorrect.</p>
<h1 id="the-setup">The setup</h1>
<p>Pick some principal <span class="math inline">G</span>-bundle <span
class="math inline">\pi\colon E\to B</span>, endowed with a connection
<span class="math inline">\nabla</span>. We’re going to look at what
sort of information we can associated to singular cells in the base
space <span class="math inline">B</span>, using the things that we have
available to us.</p>
<p>We know that the fibre <span
class="math inline">E_b=\pi^{-1}(b)</span> of <span
class="math inline">E</span> at any point <span class="math inline">b\in
B</span> “looks like” the group <span class="math inline">G</span>, so,
using the axiom of choice if we have to (or whatever really — we’re
playing fast and loose here!), pick an element in each fibre to be the
identity element of <span class="math inline">G</span>. That is,
<em>very</em> discontinuously think of all the fibres as copies of <span
class="math inline">G</span>.</p>
<p>Now let’s look at what a <span class="math inline">1</span>-simplex
(i.e. a line) in <span class="math inline">B</span> gives us. Say the
line goes from the point <span class="math inline">b</span> to the point
<span class="math inline">b'</span>. We get a <span
class="math inline">G</span>-equivariant map <span
class="math inline">\Gamma_b^{b'}\colon E_b\to E_{b'}</span> by
using the parallel transport offered to us from the connection <span
class="math inline">\nabla</span>. But, using the fact that we have
already identified our fibres with <span class="math inline">G</span>,
and a nice technical lemma about Lie groups (saying (roughly) that, if a
map commutes with all left actions, then it must be given by right
action, i.e. by an element of the Lie group itself), we see that this is
exactly the data of an element of <span class="math inline">G</span>,
say <span class="math inline">g_{b,b'}</span>.</p>
<p>The next step is to think about what happens with <span
class="math inline">2</span>-simplices (i.e. triangles) in <span
class="math inline">B</span>. If we call the three vertices <span
class="math inline">0</span>, <span class="math inline">1</span>, and
<span class="math inline">2</span>, and label the edge between <span
class="math inline">i</span> and <span class="math inline">j</span> by
<span class="math inline">ij</span>, then, by the above, the three edges
will give us three elements <span
class="math inline">g_{01},g_{12},g_{02}\in G</span>. But here is where
things “break”: it is <em>not</em> necessarily the case that <span
class="math inline">g_{12}g_{01}=g_{02}</span>; this identity that the
notation suggestively implies is only true if <span
class="math inline">\nabla</span> is flat! (Recall: flatness of the
connections corresponds to path-independence of the associated parallel
transport).</p>
<p>So we’ve come to the conclusion that a bundle is flat if and only if…
it is flat. Great!</p>
<p>So how about this claim that <em>every</em> bundle is flat “from the
<span class="math inline">(\infty,1)</span>-point-of-view”?</p>
<h1 id="some-cubical-wizardry">Some cubical wizardry</h1>
<p>Trying to be ever so slightly more precise, we can formalise what we
were doing in the above construction: for <span
class="math inline">p=1,2</span>, to each singular <span
class="math inline">p</span>-chain (i.e. each <span
class="math inline">p</span>-simplex) in <span
class="math inline">B</span> we were associating a <span
class="math inline">(p-1)</span>-cube with values in <span
class="math inline">G</span>. That is, were were constructing an element
of the singular cochain complex on <span class="math inline">B</span>
with values in “cubical chains with values in <span
class="math inline">G</span>”, which we could write as <span
class="math display">
\varphi_\nabla \in
\mathrm{C}^{\bullet,\mathrm{sing}}(B,\mathrm{C}_{\bullet-1}^\mathrm{cube}(B,G)).
</span> The fact that our connection is not flat is reflected in the
fact that this element does not satisfy the Maurer–Cartan equation:
<span class="math display">
\mathrm{d}\varphi_\nabla+\varphi_\nabla^2 \neq 0
</span> where defining the product turns out to be really easy, because
cubes, <em>unlike simplices</em>, actually satisfy “the product of two
cubes is a cube”.</p>
<p>So here’s the question: we’ve constructed <span
class="math inline">\varphi_\nabla</span> to be concentrated only in two
(low) degrees; can we “extend” it to give an element that <em>does</em>
satisfy Maurer–Cartan? The answer is <strong>YES</strong>, and this is
exactly what we mean by our opening claim that “every bundle is flat in
the <span class="math inline">(\infty,1)</span>-sense”. I won’t spell
out the details on how <em>exactly</em> this works (since I’ve already
been very informal up until now anyway), but I’ll show you some very
lovely pictures, and then afterwards tell you where you can go to read
about this in proper detail.</p>
<p>Thinking about what we need to “construct” in order to extend our
element, we can guess that we need some sort of “homotopy” between <span
class="math inline">g_{12}g_{01}</span> and <span
class="math inline">g_{02}</span>. But how exactly can we do this? And,
further, how can we do it in a cubical way?</p>
<p>Well we know that every path in <span class="math inline">B</span>
gives us an element of <span class="math inline">G</span>, and we know
that the path <span class="math inline">02</span> gives us <span
class="math inline">g_{02}</span>, and the composite path <span
class="math inline">01;12</span> gives us the element <span
class="math inline">g_{12}g_{01}</span>, so let’s draw a bunch of
intermediate paths! We can do this by starting at <span
class="math inline">0</span>, heading along the line <span
class="math inline">01</span> <em>for some time <span
class="math inline">0\leq t_0\leq1</span></em>, and then heading towards
the point <span class="math inline">2</span> for the rest of the time
<span class="math inline">t_0-1</span>. When <span
class="math inline">t_0=0</span> we recover exactly the line <span
class="math inline">02</span>, and when <span
class="math inline">t_0=1</span> we recover exactly the composite <span
class="math inline">01;12</span>; for intermediate values, we get
something like in the (very hastily drawn) picture below.</p>
<figure>
<img
src="/assets/post-images/2022-02-05-every-bundle-is-flat-from-infinity-pov-paths-in-delta-2.png"
alt="Paths in the 2-simplex." />
<figcaption aria-hidden="true">Paths in the <span
class="math inline">2</span>-simplex.</figcaption>
</figure>
<p>What’s wonderful is that, when we look at doing something similar for
the <span class="math inline">3</span>-simplex, we see that all our
paths are parametrised by variables <span class="math inline">0\leq
t_0,t_1\leq 1</span>, i.e. by the <span
class="math inline">2</span>-cube!</p>
<figure>
<img
src="/assets/post-images/2022-02-05-every-bundle-is-flat-from-infinity-pov-paths-in-delta-3.png"
alt="Paths in the 3-simplex." />
<figcaption aria-hidden="true">Paths in the <span
class="math inline">3</span>-simplex.</figcaption>
</figure>
<h1 id="behind-the-scenes-and-some-history">Behind the scenes, and some
history</h1>
<p>This whole story is really an explicit example of the <em>universal
twisting cochain construction</em> for the path space bundle. What do I
mean by that? Well, given any based topological space <span
class="math inline">(X,x_0)</span>, we have the (based) path space
bundle <span class="math inline">\mathcal{P}_{x_0}X\hookrightarrow
X</span> given by sending a path to its endpoint; the fibre at the point
<span class="math inline">x_0</span> is exactly the (based) loop space.
Although this bundle doesn’t have a connection, we can use the homotopy
lifting property to get something akin to parallel transport. We can
make a nice simplification too: instead of looking at singular chains in
<span class="math inline">X</span>, we can look at singular chains in
<span class="math inline">X</span> whose <span
class="math inline">0</span>-cells are all sent to the point <span
class="math inline">x_0</span>; somehow this carries the same
“homological information” as the usual singular chain complex. Then a
<span class="math inline">1</span>-simplex is exactly a loop based at
<span class="math inline">x_0</span>, which is exactly a <span
class="math inline">0</span>-simplex in the loop space — this
corresponds to the same “dimension drop” that we saw in the example
above (where e.g. a line was sent to an element of <span
class="math inline">G</span>).</p>
<p>You know what? At this point, I’m just going to give up trying to
explain things any better. Hopefully you’re confused enough by my
meandering and poorly-written exposé that you want to read some proper
references. This construction was described abstractly in J.F. Adams’
“On the Cobar Construction”<a href="#fn1" class="footnote-ref"
id="fnref1" role="doc-noteref"><sup>1</sup></a>; the concrete
description (using these paths parametrised by the <span
class="math inline">(p-1)</span>-cube) is found in Edgar H. Brown, Jr.’s
“Twisted Tensor Products, I”<a href="#fn2" class="footnote-ref"
id="fnref2" role="doc-noteref"><sup>2</sup></a>.</p>
<p>Enjoy!</p>
<section class="footnotes footnotes-end-of-document"
role="doc-endnotes">
<hr />
<ol>
<li id="fn1" role="doc-endnote"><p>J. F. Adams, “On the Cobar
Construction”. <em>PNAS</em> <strong>42</strong> (1956), 409–412. <a
href="https://www.jstor.org/stable/89694">JSTOR:89694</a>.<a
href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
<li id="fn2" role="doc-endnote"><p>Edgar H. Brown, Jr., “Twisted Tensor
Products, I”. <em>Annals of Mathematics</em> <strong>69</strong> (1959),
223–246. <a
href="https://doi.org/10.2307/1970101">DOI:10.2307/1970101</a>.<a
href="#fnref2" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
</ol>
</section>
tag:thosgood.com,2022-01-23:/blog/2022/01/23/translations-part-3/Translations2022-01-23T00:00:00Z2022-01-23T00:00:00Z<p>I haven’t blogged about it in a while, but I’ve been working on just
making <a href="https://thosgood.com/translations">my translations</a> a
bit better, both in terms of content and accessibility. Let’s have a
look at what I’ve done, shall we?</p>
<!-- more -->
<p>The main change is that <em>nearly</em> all of the translations are
now viewable as web versions, <em>as well as</em> PDFs. This hopefully
makes them much more accessible to people who use screen readers, but is
also nice even for people who don’t: you can change the font, the font
size, switch to dark mode, use the table of contents which scrolls down
the page with you, fit the whole page onto your phone, etc. etc. I also
took this opportunity to refresh the CSS a bit, so hopefully things look
slick and nice now.</p>
<figure>
<img src="/assets/post-images/2022-01-23-new-css.png"
title="New style for web versions of translations"
alt="New style for web versions of translations: the buttons in the top left are for toggling the table of contents, changing font options, and viewing the PDF version. Thanks Bookdown!" />
<figcaption aria-hidden="true">New style for web versions of
translations: the buttons in the top left are for toggling the table of
contents, changing font options, and viewing the PDF version. Thanks <a
href="https://bookdown.org/">Bookdown</a>!</figcaption>
</figure>
<p>There are some translations which I probably won’t switch over to
HTML, simply because they have commutative diagrams which cannot be
rendered in the <a
href="https://ctan.org/pkg/amscd?lang=en"><code>CD</code>
environment</a> (which is now supported by KaTeX), and I just haven’t
gotten around to sorting out image generation for these web versions
yet.</p>
<p>I’ve also finished some new translations since I last blogged. Here
are the articles:</p>
<ul>
<li>M Balazard, E Saias, M Yor. “Notes sur la fonction ζ de Riemann, 2”.
<em>Adv. in Math.</em> <strong>143</strong> (1999), 284–287. <a
href="https://labs.thosgood.com/translations/AIM-143-1999-284.html">HTML</a></li>
<li>P Deligne. “Variétés abéliennes ordinaires sur un corps fini”.
<em>Inv. Math.</em> <strong>8</strong> (1969), 238–243. <a
href="https://labs.thosgood.com/translations/IM-8-1969-238.html">HTML</a></li>
<li>P Deligne. “Théorie de Hodge I”. <em>Actes du Congrès intern.
math.</em> <strong>1</strong> (1970), 425–430. <a
href="https://labs.thosgood.com/translations/ACIM-1-1970-425.html">HTML</a></li>
<li>Y Diers. “Catégories Multialgébriques”. <em>Archiv der Math.</em>
<strong>34</strong> (1980), 193–209. <a
href="https://labs.thosgood.com/translations/ADM-34-1980-193.pdf">PDF</a></li>
<li>A Grothendieck. “Résumé des résultats essentiels dans la théorie des
produits tensoriels topologiques et des espaces nucléaires”. <em>Annales
de l’Institut Fourier</em> <strong>4</strong> (1952), 73–112. <a
href="https://labs.thosgood.com/translations/AIF-4-1952-73.html">HTML</a></li>
</ul>
<p>And here are the seminars:</p>
<ul>
<li>H Cartan. “Les travaux de Koszul, I, II, and III”. <em>Séminaire
Bourbaki</em> <strong>1</strong> (1952), Talks no. 1, 8, and 12. <a
href="https://labs.thosgood.com/translations/SB-1-1%2B8%2B12.html">HTML</a></li>
<li>A Douady. “Variétés et espaces mixtes; Déformations régulières;
Obstruction primaire à la déformation”. <em>Séminaire Henri Cartan</em>
<strong>13(1)</strong> (1960/61), Talks no. 2, 3, and 4. <a
href="https://labs.thosgood.com/translations/SHC-13(1)-2+3+4.html">HTML</a></li>
<li>Grothendieck, A. “Technique de descente et théorèmes d’existence en
géométrie algébrique, I, II, and III”. <em>Séminaire Bourbaki</em>
<strong>12</strong> and <strong>13</strong> (1959/60 and 1960/61), Talks
no. 190, 195, and 212. <a href="https://thosgood.com/fga">HTML</a></li>
</ul>
<p>You might notice that this last one is Grothendieck’s FGA, and I’ve
already talked a lot about how much progress has been made (although,
admittedly, not for a while, due to general life chaos) about the <a
href="https://github.com/ryankeleti/ega">EGA translation</a>, so the
natural question to ask is <em>what about SGA?</em> Although I said I
wasn’t going to work on this until I’d at least finished the FGA
translation (of which there are only two and a half chapters left), I
ended up needing to read bits of SGA 6 for my research anyway, so I just
gave in and translated some: the homepage of the SGA translation is <a
href="https://thosgood.com/sga/">thosgood.com/sga</a> (to accompany <a
href="https://thosgood.com/fga">thosgood.com/fga</a>), and here you can
see what I’ve done so far (just the first section of SGA 1, and a little
bit of the introduction from SGA 6), but <em>do not expect any updates
to this any time soon</em> — I really have to finish up some old
projects before I get started on any news ones! (Although, as always,
you can always contribute anything yourself if the fancy ever takes you:
<a
href="https://github.com/thosgood/sga">github.com/thosgood/sga</a>.)</p>
<p>Anyways, that’s enough for now. Belated happy new year to you
all!</p>
tag:thosgood.com,2021-12-22:/blog/2021/12/22/blog-comments/Comments on blog posts2021-12-22T00:00:00Z2021-12-22T00:00:00Z<p>This blog now has support for comments! I haven’t had a chance to
properly test things yet, and there are still some kinks left to iron
out, but please do use this as an excuse to browse back through old
posts and say nice things.</p>
<!-- more -->
<p>Using the lovely <a
href="https://github.com/souramoo/commentoplusplus#how-is-this-different-to-the-original-commento">Commento++</a>,
I’ve now enabled comments at the bottom of blog posts. Some things to
point out:</p>
<ul>
<li>I haven’t yet made LaTeX rendering work, but it’s next on my to-do
list. For now, feel free to write stuff surrounded by <code>$</code> or
<code>$$</code> (or wrap it in <code>\(\)</code> or <code>\[\]</code>),
and at some point in the future this should actually render
properly.</li>
<li>I also haven’t yet made it so that people can post comments via
their Twitter/GitHub/whatever accounts, but I also don’t expect that
this will really be something that people particularly want to do? I
assume I get around six readers per month on this blog, so there’s not
particularly high demand for anything…</li>
<li>If you want to comment a lot, you can create an account (using the
“Login” button) which will let you set a profile picture and link to
your website, but this isn’t necessary! You can always just leave a
name, or even comment anonymously.</li>
<li>Things might generally be a bit buggy, but if that’s the case, then
please be patient, and also just
<a href="mailto:tim.hosgood@posteo.net">let me know what went
wrong</a>.</li>
</ul>
<p>Hopefully see you all in the comments soon!</p>
tag:thosgood.com,2021-11-29:/blog/2021/11/29/lie-algebra-cohomology-part-3/The basics of Lie algebra cohomology2021-11-29T00:00:00Z2021-11-29T00:00:00Z<p>We’ll end this series (for now) by talking about two things that we
really should talk about: Lie algebra <em>homology</em>, and
<em>relative</em> Lie algebra (co)homology. (We will work over a field
<span class="math inline">k</span>, but really we are thinking only of
the case <span class="math inline">k=\mathbb{C}</span>).</p>
<!-- more -->
<h1 id="lie-algebra-homology">Lie algebra homology</h1>
<p>Just as Lie algebra cohomology is constructed to be the derived
functor of the functor of invariants, Lie algebra homology is
constructed to be the derived functor of the functor of
<em>coinvariants</em>: <span class="math display">
\begin{aligned}
\operatorname{Coinv}_\mathfrak{g}(M) = M_\mathfrak{g}&\coloneqq
M/\mathfrak{g}M
\\\operatorname{H}_\bullet(\mathfrak{g},M) &\coloneqq
\operatorname{Tor}_\bullet^{U(\mathfrak{g})}(k,M).
\end{aligned}
</span> We need to justify why this <span
class="math inline">\operatorname{Tor}</span> is indeed the derived
functor of <span class="math inline">\operatorname{Coinv}</span>, since
the latter doesn’t seem to have anything to do with tensor products a
priori. To do this, we need to make a small definition (which we should
really have made back when we constructed the projective resolution
<span
class="math inline">U(\mathfrak{g})\otimes_k\wedge^p\mathfrak{g}\twoheadrightarrow
k^\mathrm{triv}</span>).</p>
<div class="rmenv" title="Definition">
<p>Write <span class="math inline">i\colon\mathfrak{g}\to
U(\mathfrak{g})</span> to mean the composite map <span
class="math inline">\mathfrak{g}\hookrightarrow
T(\mathfrak{g})\twoheadrightarrow U(\mathfrak{g})</span>. The <span
class="math inline">k</span>-algebra homomorphism <span
class="math inline">\varepsilon\colon U(\mathfrak{g})\to k</span>
defined by sending <span class="math inline">i(\mathfrak{g})</span> to
zero is called the <em>augmentation map</em>, and we call its kernel
<span class="math inline">\mathfrak{J}</span> the <em>augmentation
ideal</em>.</p>
</div>
<div class="itenv" title="Lemma">
<p>The augmentation ideal <span class="math inline">\mathfrak{J}</span>
is generated (as a <span
class="math inline">U(\mathfrak{g})</span>-module) by the image <span
class="math inline">i(\mathfrak{g})</span>. Further, <span
class="math inline">k\cong
U(\mathfrak{g})/\mathfrak{J}=U(\mathfrak{g})_\mathfrak{g}</span>.</p>
</div>
<p>With this, we can see why the derived functor of the coinvariants
functor is calculated by <span
class="math inline">\operatorname{Tor}</span>, since <span
class="math display">
\begin{aligned}
k\otimes_{U(\mathfrak{g})}M
&\cong U(\mathfrak{g})/\mathfrak{J}_{U(\mathfrak{g})}M
\\&\cong M/\mathfrak{J}M
\\&\cong M/\mathfrak{g}M
= M_\mathfrak{g}.
\end{aligned}
</span></p>
<p>As we did for cohomology, we can construct a chain complex that
computes the Lie algebra homology: we define <span class="math display">
\wedge_p = \wedge_p(\mathfrak{g},M) \coloneqq
\wedge^p(\mathfrak{g})\otimes_k M
</span> (so that <span class="math inline">\wedge_0=M</span>), with
differential <span
class="math inline">\partial\colon\wedge_p\to\wedge_{p-1}</span> given
by <span class="math display">
\begin{aligned}
&\partial_p(x_1\wedge\ldots\wedge x_p\otimes m) =
\\\,&\sum_{1\leqslant i<j\leqslant p} (-1)^{i+j}
[x_i,x_j]\wedge
x_1\wedge\ldots\wedge\widehat{x_i}\wedge\ldots\wedge\widehat{x_j}\wedge\ldots\wedge
x_p \otimes m
\\+&\sum_{i=1}^p (-1)^i
x_1\wedge\ldots\wedge\widehat{x_i}\wedge\ldots\wedge x_p \otimes (x_im).
\end{aligned}
</span> The claim (which we won’t prove) is then that <span
class="math display">
\operatorname{H}_p(\mathfrak{g},M) \cong
\operatorname{Z}_p(\mathfrak{g},M)/\operatorname{B}_p(\mathfrak{g},M).
</span></p>
<h1 id="relative-lie-algebra-cohomology-some-properties">Relative Lie
algebra (co)homology: some properties</h1>
<p>Again, as is usual for (co)homology theories, we wish to have some
notion of the (co)homology <em>of a pair</em>. So let <span
class="math inline">(\mathfrak{g},\mathfrak{h})</span> be a Lie algebra
pair (i.e. <span class="math inline">\mathfrak{h}</span> and <span
class="math inline">\mathfrak{g}</span> are both Lie algebras, with
<span class="math inline">\mathfrak{h}\hookrightarrow\mathfrak{g}</span>
a subalgebra), and <span class="math inline">M</span> a <span
class="math inline">\mathfrak{g}</span>-module. It should be the case
that, however we define the (co)homology of this pair (with coefficients
in <span class="math inline">M</span>), we at the very least recover the
non-relative (co)homology of <span
class="math inline">\mathfrak{g}</span> when <span
class="math inline">\mathfrak{h}</span> is zero, i.e. we should
definitely have that <span class="math display">
\operatorname{H}^\bullet(\mathfrak{g},0;M) \cong
\operatorname{H}^\bullet(\mathfrak{g};M).
</span> But before giving the definition, it’s maybe also helpful to see
another theorem which will hold, which tells us about the case of the
relative cohomology of a <em>reductive</em> subalgebra.</p>
<div class="rmenv" title="Definition">
<p>Let <span class="math inline">(\mathfrak{g},\mathfrak{h})</span> be a
Lie algebra pair. We say that it is <em>reductive</em> if the adjoint
Lie algebra representation of <span
class="math inline">\mathfrak{h}</span> on <span
class="math inline">\mathfrak{g}</span> is completely reducible.</p>
</div>
<div class="itenv" title="Theorem">
<p>Let <span class="math inline">(\mathfrak{g},\mathfrak{h})</span> be a
Lie algebra pair, and <span class="math inline">M</span> a <span
class="math inline">\mathfrak{g}</span>-module. Assume that</p>
<ul>
<li>everything (i.e. <span class="math inline">\mathfrak{h}</span>,
<span class="math inline">\mathfrak{g}</span>, and <span
class="math inline">M</span>) is finite dimensional (I’m not actually
sure that this is necessary, but I’m just being safe here);</li>
<li><span class="math inline">(\mathfrak{g},\mathfrak{h})</span> is a
reductive pair; and</li>
<li>we can write <span class="math inline">\mathfrak{g}</span> as a
semi-direct product <span
class="math inline">\mathfrak{g}=\mathfrak{h}\ltimes\mathfrak{a}</span>
(where <span class="math inline">\mathfrak{a}</span> is thus, in
particular, a <em>Lie ideal</em> of <span
class="math inline">\mathfrak{g}</span>, i.e. a vector subspace such
that <span
class="math inline">[\mathfrak{a},\mathfrak{g}]\subseteq\mathfrak{a}</span>).</li>
</ul>
<p>Then <span class="math display">
\operatorname{H}^\bullet(\mathfrak{a};M)^\mathfrak{g}\simeq
\operatorname{H}^\bullet(\mathfrak{a};M)^\mathfrak{h}\simeq
\operatorname{H}^\bullet(\mathfrak{g},\mathfrak{h};M)
</span> (where the superscript denotes the space of invariants).</p>
</div>
<h1 id="relative-lie-algebra-cohomology-some-definitions">Relative Lie
algebra (co)homology: some definitions</h1>
<p>As before, we will first give a complex whose internal cohomology
computes relative Lie algebra cohomology, and then construct a
projective resolution of <span
class="math inline">k^\mathrm{triv}</span> so that we can show how to
recover the derived functor definition.</p>
<div class="rmenv" title="Definition">
<p>The <em>relative Chevalley–Eilenberg complex <span
class="math inline">\operatorname{CE}^\bullet(\mathfrak{g},\mathfrak{h};M)</span></em>
is defined by <span class="math display">
\operatorname{CE}^p(\mathfrak{g},\mathfrak{h};M) \coloneqq \big(
(\wedge^p(\mathfrak{g}/\mathfrak{h})^*)\otimes_k M \big)^\mathfrak{h}
</span> and the differential is given by the duals of the Lie bracket
and the action of <span class="math inline">\mathfrak{g}</span> on <span
class="math inline">M</span>, i.e. <span class="math display">
d \coloneqq \rho^* + [-,-]^*.
</span></p>
</div>
<p>By definition, if <span class="math inline">\mathfrak{h}=0</span>,
then we recover exactly the Chevalley–Eilenberg complex of <span
class="math inline">\mathfrak{g}</span>, which is one thing that we
wanted. (To see this, we use the general fact that we have an
isomorphism of the form <span
class="math inline">\operatorname{Hom}_k(A,V)\cong A^*\otimes_k
V</span>).</p>
<p>As for relative homology, we define the complex <span
class="math display">
\wedge_p = \wedge_p(\mathfrak{g},\mathfrak{h};M) \coloneqq
\frac{\wedge^p(\mathfrak{g}/\mathfrak{h})\otimes_k
M}{\mathfrak{h}\cdot(\wedge^p(\mathfrak{g}/\mathfrak{h})\otimes_k M)}
</span> where <span class="math inline">\mathfrak{h}\cdot V</span>, for
an <span class="math inline">\mathfrak{h}</span>-module <span
class="math inline">V</span>, denotes the span of elements of the form
<span class="math inline">x\cdot v</span> for <span
class="math inline">x\in\mathfrak{h}</span> and <span
class="math inline">v\in V</span>.</p>
<p>So far, we have not really been making any assumptions about whether
or not <span class="math inline">M</span> is finite dimensional. It
turns out that we don’t have to, but we do need to impose at least some
sort of condition relating to its dimension, whence the following
definition.</p>
<div class="rmenv" title="Definition">
<p>We say that a <span class="math inline">\mathfrak{g}</span>-module is
<em>finitely semisimple</em> if it is the sum of its finite-dimensional
irreducible <span
class="math inline">\mathfrak{g}</span>-submodules.</p>
</div>
<p>Being finitely semisimple is preserved under quotients, taking
submodules, and tensor products.</p>
<div class="itenv" title="Lemma">
<p>Let <span class="math inline">(\mathfrak{g},\mathfrak{h})</span> be a
Lie algebra pair such that <span class="math inline">\mathfrak{g}</span>
is finitely semisimple under the adjoint action of <span
class="math inline">\mathfrak{h}</span>. Write <span
class="math inline">\mathcal{G}\coloneqq U(\mathfrak{g})</span>, and
<span class="math inline">\mathcal{H}\coloneqq U(\mathfrak{h})</span>.
Then the complex <span class="math display">
\ldots \to D_2 \to D_1 \to D_0
</span> is a <span
class="math inline">(\mathcal{G},\mathcal{H})</span>-projective
resolution of <span class="math inline">k^\mathrm{triv}</span>, where
<span class="math display">
D_p \coloneqq
\mathcal{G}\otimes_\mathcal{H}\wedge^p(\mathfrak{g}/\mathfrak{h}).
</span></p>
</div>
<div class="itenv" title="Corollary">
<p>Under the same assumptions, the complex <span
class="math inline">D_\bullet\otimes_k M</span> is a <span
class="math inline">(\mathcal{G},\mathcal{H})</span>-projective
resolution of <span class="math inline">M</span>.</p>
</div>
<p>Using this resolution, we can show the following:</p>
<div class="itenv" title="Lemma">
<p>Let <span class="math inline">(\mathfrak{g},\mathfrak{h})</span> be a
Lie algebra pair such that <span class="math inline">\mathfrak{g}</span>
is finitely semisimple under the adjoint action of <span
class="math inline">\mathfrak{h}</span>. Then <span
class="math display">
\operatorname{H}_\bullet(\mathfrak{g},\mathfrak{h};M) \simeq
\operatorname{Tor}_\bullet^{(\mathcal{G},\mathcal{H})}(M^t,k) \simeq
\operatorname{Tor}_\bullet^{(\mathcal{G},\mathcal{H})}(k,M)
</span> (where <span class="math inline">M^t</span> is the right <span
class="math inline">\mathfrak{g}</span>-module with the same underlying
space as <span class="math inline">M</span>, but with <span
class="math inline">m\cdot x\coloneqq -xm</span> for <span
class="math inline">m\in M</span> and <span
class="math inline">x\in\mathfrak{g}</span>), and <span
class="math display">
\operatorname{H}^\bullet(\mathfrak{g},\mathfrak{h};M) \simeq
\operatorname{Ext}_{(\mathcal{G},\mathcal{H})}^\bullet(k,M).
</span></p>
</div>
<p>A “fun” exercise is then to prove that, for any Lie algebra pair
<span class="math inline">(\mathfrak{g},\mathfrak{h})</span> and any
<span class="math inline">\mathfrak{g}</span>-module <span
class="math inline">M</span>, <span class="math display">
\operatorname{H}^\bullet(\mathfrak{g},\mathfrak{h};M^*) \simeq
\operatorname{H}_\bullet(\mathfrak{g},\mathfrak{h};M)^*.
</span></p>
tag:thosgood.com,2021-11-26:/blog/2021/11/25/left-adjoints-lenses-and-localisation/Left adjoints, lenses, and localisation2021-11-26T00:00:00Z2021-11-26T00:00:00Z<p>This is really just a cross-post announcement: I wrote <a
href="https://topos.site/blog/2021/11/left-adjoints-lenses-and-localisation/">a
post over on the Topos blog</a>, but it’s something I’ve been thinking
about a lot, so I wanted to share a link to it here as well. It’s
basically the result of me, knowing a bit about derived categories and
model categories, trying to digest this lovely bit of Australian
category theory by <a href="https://bryceclarke.github.io/">Bryce
Clarke</a> concerning <a
href="https://arxiv.org/abs/2009.06835v1">internal lenses</a>. In fact,
it’s really a fermented and distilled version of an old blog post from
here, namely <a
href="https://thosgood.com/blog/2019/07/14/cauchy-completion-and-profunctors.html"><em>Cauchy
completion and profunctors</em></a>.</p>
<!-- more -->
<p>I won’t copy the contents of the entire post here, but I will just
repeat the question that I ask at the end (and will refer anybody
interested in more context over to <a
href="https://topos.site/blog/2021/11/left-adjoints-lenses-and-localisation/">my
post</a>).</p>
<blockquote>
<p>Is there some setting in which I could say something like (maybe
swapping “projective” for “injective”) the following: cofunctors are the
morphisms in some “derived” category; weak equivalences are
bijective-on-objects functors; fibrations are discrete opfibrations; the
category does not have enough projectives; lenses correspond (via
looking at their domain) to K-projective objects.</p>
</blockquote>
tag:thosgood.com,2021-11-24:/blog/2021/11/24/lie-algebra-cohomology-part-2/The basics of Lie algebra cohomology2021-11-24T00:00:00Z2021-11-24T00:00:00Z<p>Continuing on from last time, let’s now take a look at some actual
computational methods for Lie algebra cohomology, as well as some
applications and important results. We’ll study the cohomology of
semisimple Lie algebras, finite dimensional nilpotent Lie algebras, and
then take a little detour to talk about the Borel–Weil–Bott theorem.</p>
<!-- more -->
<h1 id="decompositions-for-computation">Decompositions for
computation</h1>
<p>Explicitly calculating the Lie algebra cohomology for some specific
<span class="math inline">\mathfrak{g}</span> and <span
class="math inline">M</span> can be difficult, but there are some useful
results (as is normally the case for (co)homology) that tell us how to
split <span
class="math inline">\operatorname{H}^\bullet(\mathfrak{g},M)</span> up
into smaller parts if we know some sort of decomposition of <span
class="math inline">\mathfrak{g}</span> or of <span
class="math inline">M</span>. The idea is then to apply these
decomposition results until we can reduce down to the case where
e.g. <span class="math inline">\mathfrak{g}</span> is semisimple and
<span class="math inline">M</span> is finite dimensional (since then we
have very strict vanishing results, as we will see in the next
section).</p>
<p>We first consider the case where we know how to decompose <span
class="math inline">M</span>.</p>
<div class="itenv" title="Lemma (Long exact sequence in cohomology)">
<p>Let <span class="math display">
0\to M'\to M\to M''\to0
</span> be a short exact sequence of <span
class="math inline">\mathfrak{g}</span>-modules. Then <span
class="math display">
\ldots \to \operatorname{H}^{i-1}(\mathfrak{g},M'') \to
\operatorname{H}^i(\mathfrak{g},M') \to
\operatorname{H}^i(\mathfrak{g},M) \to
\operatorname{H}^i(\mathfrak{g},M'') \to \ldots
</span> is a long exact sequence.</p>
</div>
<p>This is exactly the cohomology long exact sequence associated to a
short exact sequence that we always expect any nice cohomology theory to
give us.</p>
<p>We also have the following useful result for when we know how to
decompose <span class="math inline">\mathfrak{g}</span>.</p>
<div class="itenv" title="Theorem (Hochschild–Serre spectral sequence)">
<p>Let <span class="math display">
0\to\mathfrak{g}'\to\mathfrak{g}\to\mathfrak{g}''\to0
</span> be a short exact sequence of Lie algebras, and let <span
class="math inline">M</span> be a <span
class="math inline">\mathfrak{g}</span>-module. Then there exists a
spectral sequence <span class="math inline">E_r^{p,q}</span> such
that</p>
<ol type="i">
<li><span
class="math inline">E_1^{p,q}\cong\operatorname{H}^p(\mathfrak{g}',\operatorname{CE}^q(\mathfrak{g}'',M))</span></li>
<li><span
class="math inline">E_2^{p,q}\cong\operatorname{H}^p(\mathfrak{g}'',\operatorname{CE}^q(\mathfrak{g}',M))</span></li>
</ol>
<p>and such that <span
class="math inline">E_r^{p,q}\Rightarrow\operatorname{H}^\bullet(\mathfrak{g},M)</span>.</p>
</div>
<p>One final result, which is useful for studying the <a
href="https://en.wikipedia.org/wiki/Virasoro_algebra">Virasoro
algebra</a> (arising as the unique <span
class="math inline">1</span>-dimensional central extension of the Lie
algebra of vector fields on the circle), pertains to the case where
<span class="math inline">\mathfrak{g}</span> has a <em>grading
element</em>.</p>
<div class="rmenv" title="Definition">
<p>If there exists an element <span
class="math inline">e_0\in\mathfrak{g}</span> such that <span
class="math display">
\mathfrak{g}\cong \bigoplus_{n\in\mathbb{Z}}\mathfrak{g}_n
</span> where <span class="math display">
\mathfrak{g}_n \coloneqq \{x\in\mathfrak{g}\mid [e_0,x]=nx\}
</span> then we call <span class="math inline">e_0</span> a <em>grading
element</em> of <span class="math inline">\mathfrak{g}</span>.</p>
</div>
<div class="itenv" title="Theorem">
<p>Suppose that <span class="math inline">\mathfrak{g}</span> has a
grading element <span class="math inline">e_0</span>. Then all the
cochain spaces <span
class="math inline">\operatorname{CE}^p(\mathfrak{g},M)</span> are
graded, and the inclusion <span
class="math inline">\operatorname{CE}_0^\bullet(\mathfrak{g},M)\hookrightarrow\operatorname{CE}^\bullet(\mathfrak{g},M)</span>
induces an isomorphism in cohomology.</p>
</div>
<h1 id="semisimple-and-nilpotent-lie-algebras">Semisimple and nilpotent
Lie algebras</h1>
<p>As mentioned above, the case where <span
class="math inline">\mathfrak{g}</span> is semisimple (over <span
class="math inline">\mathbb{C}</span>) and <span
class="math inline">M</span> finite dimensional gives particularly nice
vanishing results.</p>
<div class="itenv" title="Lemma (Whitehead's Lemmas)">
<p>Let <span class="math inline">\mathfrak{g}</span> be a complex
semisimple Lie algebra, and <span class="math inline">M</span> a finite
dimensional <span class="math inline">\mathfrak{g}</span>-module.
Then</p>
<ol type="i">
<li><span
class="math inline">\operatorname{H}^1(\mathfrak{g},M)=0</span>,</li>
<li><span
class="math inline">\operatorname{H}^2(\mathfrak{g},M)=0</span>.</li>
</ol>
</div>
<div class="itenv" title="Corollary (Weyl's Theorem)">
<p>Let <span class="math inline">\mathfrak{g}</span> be a semisimple Lie
algebra over a field of characteristic <span
class="math inline">0</span>. Then every finite dimensional <span
class="math inline">\mathfrak{g}</span>-module is semisimple.</p>
</div>
<p><em>Proof.</em> For a contradiction, suppose that <span
class="math inline">M_1</span> is not a direct sum of simple modules,
and take it to be of smallest dimension (so <span
class="math inline">M_1</span> is not necessarily unique, but we just
pick any non-semisimple module of smallest dimension!). In particular,
<span class="math inline">M_1</span> is not simple, and thus contains a
proper submodule <span class="math inline">M_0</span>. This submodule
<span class="math inline">M_0</span> must be a direct sum of simple
modules, otherwise it would contradict the minimality of <span
class="math inline">M_1</span>; similarly, <span
class="math inline">M_2\coloneqq M_1/M_0</span> must also be a direct
sum of simple modules. This means that the short exact sequence <span
class="math inline">0\to M_0\to M_1\to M_2\to0</span> is not split
(otherwise the semisimplicity of <span class="math inline">M_0</span>
and <span class="math inline">M_2</span> would imply the semisimplicity
of <span class="math inline">M_1</span>), and so this gives a non-zero
extension class in <span
class="math inline">\operatorname{Ext}_{U(\mathfrak{g})}^1(M_2,M_0)</span>.
But then <span
class="math inline">\operatorname{H}^1(\mathfrak{g},\operatorname{Hom}_k(M_2,M_0))\cong\operatorname{Ext}_{U(\mathfrak{g})}^1(M_2,M_0)</span>
is non-zero, which contradicts Whitehead’s first lemma.</p>
<p>Whitehead’s second lemma also gives us a classical result: <em>Levi’s
Theorem</em>, which states that every Lie algebra is a split extension
of a semisimple algebra by a solvable algebra. This follows from the
fact that <span
class="math inline">\operatorname{H}^2(\mathfrak{g},M)</span> classifies
extensions of <span class="math inline">\mathfrak{g}</span> by <span
class="math inline">M</span> with some specified action of <span
class="math inline">\mathfrak{g}</span> on <span
class="math inline">M</span>, but we won’t delve into this here.</p>
<p>The next obvious question, following Whitehead’s lemmas, is <em>“what
about the cohomology in degree <span class="math inline">0</span> and
degrees <span class="math inline">\geqslant 3</span>?”</em>. Happily,
there are some nice results about this. Firstly, if we can apply
Whitehead’s Lemmas, then the vanishing of <span
class="math inline">\operatorname{H}^0</span> ensures the vanishing of
<em>all</em> cohomology, <em>as long as <span
class="math inline">M</span> is semisimple</em>:</p>
<div class="itenv" title="Theorem">
<p>Let <span class="math inline">\mathfrak{g}</span> be a complex
semisimple Lie algebra, and <span class="math inline">M</span> a finite
dimensional <span class="math inline">\mathfrak{g}</span>-module (so
that Whitehead’s Lemmas apply). If <span class="math inline">M</span>
(which must be semisimple, by Weyl’s theorem) is such that <span
class="math inline">M^\mathfrak{g}=0</span>, then <span
class="math display">
\operatorname{H}^i(\mathfrak{g},M)=0
</span> for all <span class="math inline">i\geqslant 0</span>.</p>
</div>
<p>This theorem holds more generally, for reductive Lie algebras over
fields of characteristic zero, and finite-dimensional semisimple
modules. Indeed, Whitehead’s lemmas <em>also</em> hold over arbitrary
fields of characteristic zero, since semisimplicity is preserved by
field extensions, as is cohomology, and so we can always pass to the
algebraic closure.</p>
<p>In general, however, it is not the case that <span
class="math inline">\operatorname{H}^3</span> vanishes; indeed, we have
the following result:</p>
<div class="itenv" title="Theorem">
<p>Let <span class="math inline">\mathfrak{g}</span> be a complex
semisimple Lie algebra, and <span class="math inline">M</span> a (not
necessarily finite dimensional) <span
class="math inline">\mathfrak{g}</span>-module. Then <span
class="math display">
\operatorname{H}^3(\mathfrak{g},M)\cong\mathbb{C}^s
</span> where <span class="math inline">s</span> is the number of simple
factors of <span class="math inline">M</span> contained in <span
class="math inline">\mathfrak{g}</span>.</p>
</div>
<p>(We can actually give a more constructive statement than the above:
for each simple factor with Killing form <span
class="math inline">\langle-,-\rangle</span>, the <span
class="math inline">3</span>-cocycle <span
class="math inline">\langle[-,-],-\rangle</span> generates the
corresponding factor in <span
class="math inline">\mathbb{C}^s</span>.)</p>
<p>As mentioned in the previous post, we are often interested in the
case where <span class="math inline">M=k</span>, and we can say
something about this in the complex case as well, namely that the
cohomology vanishes in all <em>even</em> degrees:</p>
<div class="itenv" title="Theorem">
<p>Let <span class="math inline">\mathfrak{g}</span> be a complex
semisimple Lie algebra. Then <span
class="math inline">\operatorname{H}^\bullet(\mathfrak{g},\mathbb{C})</span>
is an exterior algebra in odd generators.</p>
</div>
<p>Contrary to the case of semisimple Lie algebras (which have “not
much” cohomology), nilpotent Lie algebras have “a lot of” cohomology,
even when finite dimensional, as evidenced by the following theorem:</p>
<div class="itenv" title="Theorem">
<p>Let <span class="math inline">\mathfrak{g}</span> be an <span
class="math inline">n</span>-dimensional nilpotent Lie algebra. Then
<span class="math display">
\dim\operatorname{H}^i(\mathfrak{g},k)\geqslant 2
</span> for all <span
class="math inline">i\in\{1,\ldots,n-1\}</span>.</p>
</div>
<p>In fact, this suggests a conjecture:</p>
<div class="itenv"
title="Conjecture (Halperin's total rank conjecture)">
<p>Let <span class="math inline">\mathfrak{g}</span> be an <span
class="math inline">n</span>-dimensional nilpotent Lie algebra. Then
<span class="math display">
\sum_{i\in\mathbb{N}}\dim\operatorname{H}^i(\mathfrak{g},k)\geqslant
2^{\dim Z(\mathfrak{g})}
</span> where <span class="math inline">Z(\mathfrak{g})</span> is the
centre of <span class="math inline">\mathfrak{g}</span>.</p>
</div>
<h1 id="kostant-and-borelweilbott">Kostant and Borel–Weil–Bott</h1>
<p>As I mentioned in the first post in this series, the real context for
me studying all this stuff is to better understand Kac–Moody algebras.
Two key notions in this area are that of <em>root decompositions</em>
and <em>highest weight modules</em>, both of which I will now recklessly
assume that we have prior knowledge of (if this isn’t the case, then I
highly recommend <a
href="https://ahilado.wordpress.com/2021/06/12/reductive-groups-part-i-over-algebraically-closed-fields/">Anton
Hilado’s blog</a> as a learning resource).</p>
<p>I’m only going to scratch the surface when talking about these two
theorems, so if you want to learn more then I would turn to <a
href="https://www.math.ias.edu/~lurie/papers/bwb.pdf">Lurie’s notes on
the Borel–Weil-Bott Theorem</a> and <a
href="https://faculty.math.illinois.edu/~mim2/KostantTheorem.pdf">the
UGA VIGRE Algebra Group’s notes on Kostant’s Theorem</a>, both of which
are thorough and include concrete calculations.</p>
<div class="itenv" title="Theorem (Kostant's Theorem)">
<p>Let <span class="math inline">\mathfrak{g}</span> be a semisimple Lie
algebra, and <span
class="math inline">\mathfrak{g}=\mathfrak{n}_+\oplus\mathfrak{h}\oplus\mathfrak{n}_-</span>
a root decomposition. Let <span class="math inline">M</span> be a finite
dimensional, irreducible, highest weight <span
class="math inline">\mathfrak{g}</span>-module of highest weight <span
class="math inline">\lambda</span>.</p>
<p>Then each <span
class="math inline">\operatorname{H}^i(\mathfrak{n}_+,M)</span> splits
(as a <span class="math inline">\mathfrak{g}</span>-module) into a
direct sum of <span class="math inline">1</span>-dimensional modules of
multiplicity <span class="math inline">1</span>. Further, the
corresponding weights are exactly the elements of the form <span
class="math inline">w(\lambda+\rho)-\rho</span>, where <span
class="math inline">w</span> is any element of the Weyl group of length
<span class="math inline">k</span>, and <span
class="math inline">\rho</span> denotes the half sum of the positive
roots.</p>
</div>
<p>An nice application of this theorem is that it gives an algebraic
proof of the <a
href="https://en.wikipedia.org/wiki/Borel%E2%80%93Weil%E2%80%93Bott_theorem"><em>Borel–Weil–Bott
Theorem</em></a>. We won’t state (let alone prove) the theorem here, but
simply content ourselves with saying that it somehow relates so-called
<em>dominant weights</em> of semisimple complex Lie algebras with the
sheaf cohomology of certain line bundles. Or, to quote from <a
href="https://en.wikipedia.org/wiki/Borel%E2%80%93Weil%E2%80%93Bott_theorem#Borel%E2%80%93Weil_theorem">the
Wikipedia article</a>,</p>
<blockquote>
<p>The Borel–Weil theorem provides a concrete model for irreducible
representations of compact Lie groups and irreducible holomorphic
representations of complex semisimple Lie groups. These representations
are realized in the spaces of global sections of holomorphic line
bundles on the flag manifold of the group. The Borel–Weil–Bott theorem
is its generalization to higher cohomology spaces.</p>
</blockquote>
<p>To further add to the sense of mystery, we will vaguely discuss an
example. Let <span
class="math inline">G=\operatorname{SL}_2(\mathbb{C})</span>, which has
the Borel subgroup consisting of upper-triangular matrices (of
determinant <span class="math inline">1</span>); the flag variety <span
class="math inline">G/B</span> is the Riemann sphere <span
class="math inline">\mathbb{P}_\mathbb{C}^1</span>, and we write its
coordinates as <span class="math inline">x</span> and <span
class="math inline">y</span>.. The integral weights of <span
class="math inline">G</span> are exactly the integers <span
class="math inline">n\in\mathbb{Z}</span>, with <em>dominant</em>
weights corresponding to <em>non-negative</em> integers. Here are some
observations that we can make, which happen to be those coming from the
Borel–Weil Theorem:</p>
<ul>
<li>There exists a line bundle <span
class="math inline">L_n\coloneqq\mathscr{O}(n)</span> on <span
class="math inline">G/B</span> for each integral weight <span
class="math inline">n</span>.</li>
<li>The line bundle <span class="math inline">L_n</span> has global
sections if and only if <span class="math inline">n</span> is a dominant
weight.</li>
<li>When <span class="math inline">n</span> is a dominant weight
(i.e. when <span class="math inline">n\geqslant 0</span>), the space
<span class="math inline">\Gamma(L_n)</span> of global sections of <span
class="math inline">L_n</span> is exactly the <span
class="math inline">(n+1)</span>-dimensional space of homogeneous
polynomials of degree <span class="math inline">n</span> in <span
class="math inline">\mathbb{C}[x,y]</span>.</li>
<li>The global sections of <span class="math inline">L_n</span> (when
<span class="math inline">n</span> is a dominant weight) form an
irreducible representation of <span class="math inline">G</span> on
<span class="math inline">\mathbb{C}[x,y]</span>, which is further a
highest weight representation of highest weight <span
class="math inline">n</span>, with weight vectors <span
class="math inline">x^iy^{n-i}</span> (for <span
class="math inline">0\leqslant i\leqslant n</span>) of weight <span
class="math inline">2i-n</span>, and highest weight vector <span
class="math inline">x^n</span>.</li>
</ul>
<p>The Borel–Weil–Bott Theorem further explains the representation
theory of <span
class="math inline">\mathfrak{g}=\operatorname{\mathfrak{sl}}_2(\mathbb{C})</span>,
in that <span
class="math inline">\Gamma(L_1)=\Gamma(\mathscr{O}(1))</span>
corresponds to the standard representation, and <span
class="math inline">\Gamma(L_n)=\Gamma(\mathscr{O}(n))</span>
corresponds to the <span class="math inline">n</span>-th symmetric power
of the standard representation. Explicitly, we can even write down a
description of this standard action of the Lie algebra on <span
class="math inline">\mathbb{C}[x,y]</span> as follows: take the usual
generators <span
class="math inline">X=\left(\begin{smallmatrix}0&1\\0&0\end{smallmatrix}\right)</span>,
<span
class="math inline">Y=\left(\begin{smallmatrix}0&0\\1&0\end{smallmatrix}\right)</span>,
and <span
class="math inline">H=\left(\begin{smallmatrix}1&0\\0&-1\end{smallmatrix}\right)</span>
of <span
class="math inline">\operatorname{\mathfrak{sl}}_2(\mathbb{C})</span>;
then <span class="math display">
\begin{aligned}
X &\leadsto x\frac{\mathrm{d}}{\mathrm{d}y}
\\Y &\leadsto y\frac{\mathrm{d}}{\mathrm{d}x}
\\H &\leadsto x\frac{\mathrm{d}}{\mathrm{d}x} -
y\frac{\mathrm{d}}{\mathrm{d}y}
\end{aligned}
</span> (which could (but won’t, today) lead us very neatly into a
discussion on the <a
href="https://en.wikipedia.org/wiki/Jordan_map">Jordan–Schwinger
map</a>, and how we can understand quantum angular momentum via quantum
harmonic oscillators).</p>
tag:thosgood.com,2021-11-22:/blog/2021/11/22/lie-algebra-cohomology-part-1/The basics of Lie algebra cohomology2021-11-22T00:00:00Z2021-11-22T00:00:00Z<p>As part of a reading group I’m in on Kac–Moody groups (something I
know nothing about), I had to give a talk introducing the basics of Lie
algebra cohomology (something I know very little about), along with some
sort of motivation and intuition, and some worked examples. Since I’ve
written these notes anyway, I figured I might as well put them up on
this blog, and then, when I inevitably forget all I ever once knew, I
can refer back to here.</p>
<!-- more -->
<p>The main two references that I consulted were Danny Cryster’s <a
href="http://myslu.stlawu.edu/~dcrytser/EnvelopingAlgebrasNotes.pdf">“Notes
on Enveloping Algebras”</a> and Friedrich Wagemann’s <a
href="https://www.math.sciences.univ-nantes.fr/~wagemann/LAlecture.pdf">“Introduction
to Lie algebra cohomology with a view towards BRST cohomology”</a>.
Honestly, you’d maybe be better off just going and reading those two
papers to be honest, since all I’m going to add here is some background
on group cohomology, and then splice together the relevant parts of the
two linked papers. Anyway, let’s get on with it.</p>
<h1 id="first-motivation-group-cohomology">First motivation: group
cohomology</h1>
<p>Let <span class="math inline">G</span> be a group. Unravelling
definitions, we can see that a <span
class="math inline">\mathbb{Z}[G]</span>-module is exactly the same
thing as an abelian group endowed with an action of <span
class="math inline">G</span>. This equivalence gives us another way of
thinking about <em>invariant elements</em>, i.e. elements of the group
<span class="math inline">\operatorname{Inv}_G(A)\coloneqq\{a\in A\mid
ga=a \text{ for all } g\in G\}</span>.</p>
<div class="itenv" title="Lemma">
<p><span class="math inline">\operatorname{Inv}_G(A) \cong
\operatorname{Hom}_{\mathbb{Z}[G]}(\mathbb{Z}^\mathrm{triv},A)</span>.</p>
</div>
<p><em>Proof.</em> Given <span
class="math inline">\varphi\colon\mathbb{Z}^\mathrm{triv}\to A</span>,
we obtain invariant elements, since <span
class="math inline">\varphi(n)=\varphi(gn)=g\varphi(n)</span> for all
<span class="math inline">n\in\mathbb{Z}</span>; conversely, given <span
class="math inline">a\in\operatorname{Inv}_G(A)</span>, we can define
<span class="math inline">\varphi_a\colon1\mapsto a</span>.</p>
<p>This leads us to consider the <em>functor of invariants</em> <span
class="math display">
\operatorname{Inv}\coloneqq\operatorname{Hom}_{\mathbb{Z}[G]}(\mathbb{Z}^\mathrm{triv},-)\colon
\mathsf{Mod}_{\mathbb{Z}[G]}\to\mathsf{Mod}_{\mathbb{Z}}.
</span> But, as always, since this is a functor between categories of
modules, we are interested in <em>deriving it</em>, and this is exactly
the definition of group cohomology: <span class="math display">
\operatorname{H}^n(G,A) \coloneqq
\operatorname{Ext}_{\mathbb{Z}[G]}^\bullet(\mathbb{Z}^\mathrm{triv},-).
</span></p>
<p>To actually compute this, then, it suffices to find a projective
resolution of <span class="math inline">\mathbb{Z}^\mathrm{triv}</span>
as a <span class="math inline">\mathbb{Z}[G]</span>-module, and there is
a canonical way of doing this, via the so-called <em>bar
construction</em>.</p>
<p>We define <span class="math inline">B_0</span> to be the <span
class="math inline">\mathbb{Z}[G]</span>-module <span
class="math inline">\mathbb{Z}[G]</span>; for <span
class="math inline">n\geqslant 1</span>, we define <span
class="math inline">B_n</span> to be the free <span
class="math inline">\mathbb{Z}[G]</span>-module on the set <span
class="math inline">G^{\times(n+1)}</span>, and write its elements as
symbols of the form <span class="math inline">(g_0|\ldots|g_n)</span>
for <span class="math inline">g_i\in G</span>. Then we define <span
class="math inline">d_0\colon B_0\to\mathbb{Z}^\mathrm{triv}</span> by
<span class="math display">
d_0\left(\sum_i m_i g_i\right) = \sum_i m_i
</span> and <span class="math inline">d_n\colon B_n \to B_{n-1}</span>
by <span class="math display">
\begin{aligned}
d_n(g_0|\ldots|g_n) =
\,\,&g_0(g_1|\ldots|g_n)
\\+&\sum_{i=0}^{n-1} (-1)^i (g_0|\ldots|g_{i-1}|g_i
g_{i+1}|g_{i+2}|\ldots|g_n)
\\\pm&(g_0|\ldots|g_n).
\end{aligned}
</span></p>
<p>Something that we won’t discuss is that this complicated equation is
actually somehow “natural”, in that the bar construction is a “nice
thing” that “makes sense”. For more on this (since it would require a
large detour here), I highly recommend Emily Riehl’s beautiful (and
freely-available) book <a
href="https://math.jhu.edu/~eriehl/cathtpy.pdf"><em>Categorical homotopy
theory</em></a>.</p>
<p>We can understand low-dimensional group cohomology quite
explicitly:</p>
<ul>
<li><span
class="math inline">\operatorname{H}^0(G,A)=A^G\coloneqq\operatorname{Inv}_G(A)</span>.</li>
<li><span class="math inline">\operatorname{H}^1(G,A)</span> is given by
the quotient <span class="math display">
\{\varphi\colon\underline{G}\to\underline{A} \mid
\varphi(gh)=\varphi(g)+g\varphi(h)\}\Big/\{\varphi=\varphi_a\colon
g\mapsto ag-a\}
</span> where we write <span class="math inline">\underline{G}</span>
to denote the underlying <em>set</em> of a group (note that this reduces
to <span class="math inline">\operatorname{Hom}(G,A)</span> in the case
where <span class="math inline">A</span> has the trivial <span
class="math inline">G</span>-action).</li>
<li><span class="math inline">\operatorname{H}^2(G,A)</span> classifies
isomorphism classes of extensions of <span class="math inline">G</span>
by <span class="math inline">A</span> such that… some condition
involving the action of <span class="math inline">G</span> on <span
class="math inline">A</span> holds (note here the degree shift: <span
class="math inline">\operatorname{H}^2(G,A)=\operatorname{Ext}_{\mathbb{Z}[G]}^2(G,A)</span>
is (some subset of) <span
class="math inline">\operatorname{Ext}_\mathbb{Z}^1(G,A)</span>; the
<span class="math inline">\operatorname{Ext}</span> are in different
degrees, but this is because they are taken in different
categories).</li>
</ul>
<h1 id="generalising-to-lie-algebras">Generalising to Lie algebras</h1>
<p>When we define Lie algebra cohomology, we’re going to do something
entirely analogous to what we did above for group cohomology. Indeed,
Lie algebras “are just” infinitesimal Lie groups, which “are just”
particularly nice groups, so we can imagine carrying forward the
definition for groups, making some suitable adjustments where necessary.
One thing to note is that, when we do the bar construction (which we
will do later on), the differential will look more complicated,
resembling some sort of total differential of a double complex, with the
second part looking like some sort of “Čech differential”. This is
because, whereas groups have “one structure” (the group operation), Lie
algebras have “two” (the group operation <em>and</em> the Lie bracket).
But the story behind everything is exactly the same: <em>Lie algebra
cohomology is the derived functor of the functor of invariants</em>.</p>
<p>Before giving any actual definitions, let’s see what <em>will</em> be
true (for a Lie algebra <span class="math inline">\mathfrak{g}</span>
(over a field <span class="math inline">k</span>) and a <span
class="math inline">\mathfrak{g}</span>-module <span
class="math inline">M</span>) in analogy to our low-dimensional
statements for group cohomology:</p>
<ul>
<li><span
class="math inline">\operatorname{H}^0(\mathfrak{g},M)=M^\mathfrak{g}</span>
is the group of <span class="math inline">\mathfrak{g}</span>-invariants
of <span class="math inline">M</span>.</li>
<li><span class="math inline">\operatorname{H}^1(\mathfrak{g},M)</span>
is given by <a
href="https://en.wikipedia.org/wiki/Lie_algebra#Derivations">derivations</a>
<span class="math inline">\operatorname{Der}(\mathfrak{g},M)</span>
modulo <em>inner</em> derivations <span
class="math inline">\operatorname{PDer}(\mathfrak{g},M)</span>, where
<span class="math display">
\begin{aligned}
\operatorname{Der}(\mathfrak{g},M) &\coloneqq
\{\varphi\in\operatorname{Hom}_k(\mathfrak{g},M) \mid
\varphi([x,y])=xf(y)-yf(x)\}
\\\operatorname{PDer}(\mathfrak{g},M) &\coloneqq
\{\varphi\in\operatorname{Hom}_k(\mathfrak{g},M) \mid
\varphi=\varphi_m\colon x\mapsto xm\}
\end{aligned}
</span> In particular, <span
class="math inline">\operatorname{H}^1(\mathfrak{g},\mathfrak{g})</span>
(where <span class="math inline">\mathfrak{g}</span> acts on itself by
the adjoint action) will be given by <a
href="https://en.wikipedia.org/wiki/Lie_algebra#Derivations"><em>outer</em>
derivations</a> of <span class="math inline">\mathfrak{g}</span>; the
other “canonical” choice for <span class="math inline">M</span>, namely
<span class="math inline">M=k^\mathrm{triv}</span> (the analogy of <span
class="math inline">\mathbb{Z}^\mathrm{triv}</span> from before), gives
<span
class="math inline">\operatorname{H}^1(\mathfrak{g},k^\mathrm{triv})=(\mathfrak{g}/[\mathfrak{g},\mathfrak{g}])^*</span>.</li>
<li><span class="math inline">\operatorname{H}^2(\mathfrak{g},M)</span>
classifies <em>abelian</em> (i.e. those such that the image of <span
class="math inline">M</span> is abelian) <a
href="https://en.wikipedia.org/wiki/Lie_algebra_extension">extensions of
<span class="math inline">\mathfrak{g}</span> by <span
class="math inline">M</span></a> (again, <span
class="math inline">\operatorname{Ext}^2</span> in the category of <span
class="math inline">\mathfrak{g}</span>-modules, but <span
class="math inline">\operatorname{Ext}^1</span> in the category of Lie
algebras).</li>
<li><span class="math inline">\operatorname{H}^3(\mathfrak{g},M)</span>
classifies certain <a
href="https://ncatlab.org/nlab/show/differential+crossed+module"><em>crossed
modules</em></a>.</li>
</ul>
<p>Further, Lie algebra cohomology tells us not only <em>algebraic</em>
information, but also <em>geometric</em> information:</p>
<div class="itenv" title="Cartan's Theorem">
<p>Let <span class="math inline">G</span> be a compact connected Lie
group, and <span class="math inline">\mathfrak{g}</span> its Lie
algebra. Then <span class="math display">
\operatorname{H}_\mathrm{dR}^\bullet(G) \cong
\operatorname{H}^\bullet(\mathfrak{g},\mathbb{R}) \cong
\operatorname{Inv}_G(\wedge^\bullet\mathfrak{g}^*).
</span></p>
</div>
<h1 id="the-universal-enveloping-algebra-and-poincarébirkhoff-witt">The
universal enveloping algebra and Poincaré–Birkhoff-Witt</h1>
<p>One key ingredient in giving a useful definition of Lie algebra
cohomology is the <em>universal enveloping algebra</em> of a Lie
algebra, so let’s talk about this for a bit.</p>
<p>Any (associative) algebra <span class="math inline">A</span> has an
“underlying” Lie algebra <span class="math inline">A_-</span>, whose set
of elements is the same, and is endowed with the Lie bracket defined by
<span class="math display">
[x,y] \coloneqq xy-yx.
</span> This gives us a “forgetful” functor <span class="math display">
\begin{aligned}
\mathsf{Alg} &\to \mathsf{LieAlg}
\\A &\mapsto A_-
\end{aligned}
</span> and so we can ask the natural question, <em>does this functor
have a left adjoint?</em> We will show that the answer to this question
is “yes”, by functorially constructing an algebra <span
class="math inline">U(\mathfrak{g})</span> such that <span
class="math inline">\mathfrak{g}</span> sits inside <span
class="math inline">U(\mathfrak{g})</span>, generating it as an algebra
(<strong>not</strong> as a Lie algebra), in such a way that
representations of <span class="math inline">U(\mathfrak{g})</span>
correspond to representations of <span
class="math inline">\mathfrak{g}</span>.</p>
<div class="rmenv" title="Definition">
<p>The <em>tensor algebra <span
class="math inline">T(\mathfrak{g})</span></em> is the algebra <span
class="math display">
T(\mathfrak{g}) \coloneqq \bigoplus_{n=0}^\infty \mathfrak{g}^{\otimes
n}
</span> (where we take the tensor product of <span
class="math inline">\mathfrak{g}</span> as a <em>vector space</em>).</p>
</div>
<p>Note that, a priori, <span class="math inline">x\otimes y-y\otimes
x\neq[x,y]</span>. In other words, the inclusion <span
class="math inline">\mathfrak{g}\hookrightarrow T(\mathfrak{g})</span>
is <em>not</em> a Lie algebra homomorphism.</p>
<div class="rmenv" title="Definition">
<p>The <em>universal enveloping algebra <span
class="math inline">U(\mathfrak{g})</span></em> is the largest quotient
algebra of <span class="math inline">T(\mathfrak{g})</span> such that
the composite <span class="math display">
\mathfrak{g}\hookrightarrow T(\mathfrak{g}) \twoheadrightarrow
U(\mathfrak{g})
</span> is a Lie algebra homomorphism. Constructively, <span
class="math display">
U(\mathfrak{g}) \coloneqq T(\mathfrak{g})/(x\otimes y-y\otimes
x-[x,y]).
</span></p>
</div>
<p>Note that, if <span class="math inline">\mathfrak{g}</span> is
abelian (i.e. if the bracket is zero), then <span
class="math inline">U(\mathfrak{g})</span> is exactly the symmetric
algebra <span
class="math inline">\operatorname{Sym}(\mathfrak{g})</span>.</p>
<div class="itenv" title="Lemma">
<p>The universal enveloping algebra construction is left adjoint to the
underlying Lie algebra functor, i.e. <span class="math display">
\operatorname{Hom}_\mathsf{Alg}(U(\mathfrak{g}),A) \cong
\operatorname{Hom}_\mathsf{LieAlg}(\mathfrak{g},A_-)
</span> functorially in <span class="math inline">\mathfrak{g}</span>
and <span class="math inline">A</span>.</p>
</div>
<p>Another way of stating this universal property (and, indeed, the one
that is found in a lot of introductory texts) is the following:</p>
<div class="itenv" title="Lemma">
<p>Let <span class="math inline">\varphi\colon\mathfrak{g}\to A_-</span>
be a Lie algebra homomorphism. Then there exists a unique algebra
homomorphism <span class="math inline">\widehat{\varphi}\colon
U(\mathfrak{g})\to A</span> such that <span
class="math inline">\varphi=\widehat{\varphi}\circ h</span>, where <span
class="math inline">h\colon\mathfrak{g}\to U(\mathfrak{g})</span> is the
canonical map.</p>
</div>
<p>In particular, this implies (by taking <span
class="math inline">A=\operatorname{End}(M)</span>) that <em>every
representation <span class="math inline">M</span> of <span
class="math inline">\mathfrak{g}</span> extends uniquely to a
representation of <span
class="math inline">U(\mathfrak{g})</span></em>.</p>
<p>One question we can ask now is if the map <span
class="math inline">h\colon\mathfrak{g}\to U(\mathfrak{g})</span> is
somehow an embedding, and the (positive) answer to this is given by the
following <a
href="https://en.wikipedia.org/wiki/Universal_enveloping_algebra#Poincar%C3%A9%E2%80%93Birkhoff%E2%80%93Witt_theorem">important
theorem</a>.</p>
<div class="itenv" title="Poincaré–Birkhoff–Witt Theorem">
<p>Let <span class="math inline">(x_1,\ldots,x_n)</span> be a vector
space basis for <span class="math inline">\mathfrak{g}</span>. Then the
ordered set of monomials <span class="math display">
(x_1^{i_1}x_2^{i_2}\ldots x_n^{i_n} \mid
(i_1,\ldots,i_n)\in\mathbb{N}^n)
</span> is a vector space basis for <span
class="math inline">U(\mathfrak{g})</span>.</p>
</div>
<h1 id="actual-definitions">Actual definitions</h1>
<p>I’m going to end this post by actually defining the thing that I’ve
apparently been trying to explain this whole time: Lie algebra
cohomology.</p>
<p>Firstly, analogously to how <span
class="math inline">\mathbb{Z}[G]</span>-modules are exactly abelian
groups endowed with a <span class="math inline">G</span>-action, we can
interchangeably talk about <em>representations</em> and <em>modules</em>
of a Lie algebra.</p>
<div class="rmenv" title="Definition">
<p>Let <span class="math inline">\mathfrak{g}</span> be a Lie algebra.
Then a <em><span class="math inline">\mathfrak{g}</span>-module <span
class="math inline">M</span></em> is a vector space endowed with a <span
class="math inline">\mathfrak{g}</span>-action such that <span
class="math display">
[x,y]\cdot m = x\cdot(y\cdot m) - y\cdot(x\cdot m).
</span> Equivalently, we can require that the map <span
class="math display">
\begin{aligned}
\mathfrak{g}&\to \operatorname{End}(\mathfrak{g})
\\x &\mapsto x\cdot(-)
\end{aligned}
</span> induced by the <span
class="math inline">\mathfrak{g}</span>-action be a Lie algebra
homomorphism.</p>
</div>
<p>We mentioned right near the start of this post that Lie algebra
cohomology was going to be defined as the derived functor of the functor
of invariants. Although this is technically true, it’s not really very
obvious with the specific definition that we’re about to give, but we’ll
explain how they are equivalent afterwards.</p>
<div class="rmenv" title="Definition">
<p>The <em>Chevalley–Eilenberg complex <span
class="math inline">\operatorname{CE}^\bullet(\mathfrak{g},M)</span></em>
is defined by <span class="math display">
\operatorname{CE}(\mathfrak{g},M)^p \coloneqq
\operatorname{Hom}_k(\wedge^p\mathfrak{g},M)
</span> (i.e. the set of <span class="math inline">p</span>-linear
alternating maps <span class="math inline">\mathfrak{g}^{\times p}\to
M</span>) with differential <span
class="math inline">d\colon\operatorname{CE}(\mathfrak{g},M)^p\to\operatorname{CE}(\mathfrak{g},M)^{p+1}</span>
given by <span class="math display">
\begin{aligned}
(dc)(x_1,\ldots,x_{p+1}) \coloneqq
\,\,&\sum_{1\leqslant i<j\leqslant p+1}
(-1)^{i+j}c([x_i,x_j],x_1,\ldots,\widehat{x_i},\ldots,\widehat{x_j},\ldots,x_{p+1})
\\+&\sum_{i=1}^{p+1} (-1)^{i+1} x_i
c(x_1,\ldots,\widehat{x_i},\ldots,x_{p+1}).
\end{aligned}
</span> We then define the <em>Lie algebra cohomology of <span
class="math inline">\mathfrak{g}</span></em> (<em>with coefficients in
<span class="math inline">M</span></em>) to be the internal cohomology
of this complex, i.e. <span class="math display">
\operatorname{H}^p(\mathfrak{g},M) \coloneqq
\operatorname{Z}^p(\mathfrak{g},M)/\operatorname{B}^p(\mathfrak{g},M)
</span> (“the kernel of <span class="math inline">d_p</span> modulo the
image of <span class="math inline">d_{p-1}</span>”).</p>
</div>
<p>This Chevalley–Eilenberg complex deserves a whole blog post to
itself, because there’s a beautiful story about how it links to
deformation theory and Maurer–Cartan elements, but to tell this we would
need to pass to the more general theory of <em>differential graded Lie
algebras</em>, which we don’t have time for today, alas.</p>
<p>As a small side note, we don’t assume <span
class="math inline">M</span> to be finite dimensional, but we
<em>do</em> require <span class="math inline">\mathfrak{g}</span> to be
finite dimensional, otherwise the cohomology groups are just “too big”
to be of much use. In the case where <span
class="math inline">\mathfrak{g}</span> <em>is</em> infinite
dimensional, we can still get something useful if we have some sort of
topological structure on <span class="math inline">\mathfrak{g}</span>
and <span class="math inline">M</span>, because we can replace the
tensor product in the definition of <span
class="math inline">\wedge^p\mathfrak{g}</span> with the
<em>topological</em> tensor product <span
class="math inline">\otimes^\pi</span>, resulting in so-called <a
href="https://en.wikipedia.org/wiki/Gelfand%E2%80%93Fuks_cohomology">Gelfand–Fuks
cohomology</a>. But we’re not going to worry about that here!</p>
<p>Now let’s finish up by asking how this definition recovers the idea
of deriving the functor of invariants. As we’ve already mentioned, the
fact that the universal enveloping algebra construction is left adjoint
to the underlying Lie algebra construction implies that
<em>representations of <span class="math inline">\mathfrak{g}</span>
give us representations of <span
class="math inline">U(\mathfrak{g})</span></em>; the
Poincaré–Birkhoff–Witt theorem gives us a converse to this, telling us
that <em>representations of <span
class="math inline">U(\mathfrak{g})</span> give us representations of
<span class="math inline">\mathfrak{g}</span></em> (since <span
class="math inline">\mathfrak{g}</span> embeds into <span
class="math inline">U(\mathfrak{g})</span>). In fact, these two
correspondences are inverse, and so <em>there is a bijective
correspondence between <span
class="math inline">\mathfrak{g}</span>-modules and <span
class="math inline">U(\mathfrak{g})</span>-modules</em>.</p>
<p>Using this, we can prove a result analogous to what we had for group
cohomology, namely that <span class="math display">
M^\mathfrak{g}\cong
\operatorname{Hom}_{U(\mathfrak{g})}(k^\mathrm{triv},M).
</span> So, if we want to show that the derived functor of invariants is
“the same” as the notion Lie algebra cohomology that we defined above,
we need to show that <span class="math display">
\operatorname{Ext}_{U(\mathfrak{g})}^\bullet(k^\mathrm{triv},M) \simeq
\operatorname{H}^\bullet(\operatorname{Hom}_k(\wedge^p\mathfrak{g},M)).
</span> To do this, it would suffice to show that <span
class="math display">
\operatorname{Hom}_k(\wedge^p\mathfrak{g},M) \cong
\operatorname{Hom}_{U(\mathfrak{g})}(P^\bullet,M)
</span> for some projective resolution <span
class="math inline">P^\bullet</span> of <span
class="math inline">k^\mathrm{triv}</span>. We won’t give the details
here, but this is indeed true, and comes from the projective resolution
<span class="math display">
U(\mathfrak{g})\otimes_k\wedge^p\mathfrak{g}\twoheadrightarrow
k^\mathrm{triv}
</span> where we choose some “clever” definition of differential for the
complex <span
class="math inline">U(\mathfrak{g})\otimes_k\wedge^p\mathfrak{g}</span>
(namely that of the <a
href="https://en.wikipedia.org/wiki/Koszul_complex">Koszul complex</a>),
which looks as similar as possible to the differential in the
Chevalley-Eilenberg complex.</p>