# 1 Duality theorems for coherent algebraic sheaves

Grothendieck, A.
“Théorème de dualité pour les faisceaux algébriques cohérents”.
*Séminaire Bourbaki* **9** (1956–57), Talk no. 149.

The results that follow, inspired by Serre’s “theorem of algebraic duality”, were discovered in the winter of 1955 and the winter of 1956. They can be established very simply, thanks to reasonably elementary results on the cohomology of projective spaces [21]. and an intensive use of Cartan–Eilenberg’s homological algebra, in the form given in [7].

## 1.1 \operatorname{Ext} of sheaves of modules

*(cf. [7], chap. 3 and 4)*

Let *groups* of *sheaf* of germs of homomorphisms from

where

We evidently have

For every

From this, we obtain “*boundary homomorphisms*”, as well as a short exact sequence, which we will not write.

If

(given by the boundary homomorphisms of the spectral sequence).
In particular, we have a canonical isomorphism

To use these results, we need to know how to explicitly describe the

that uniquely extend to homomorphisms of cohomological functors (in

If

Let

*Proof*. The proof is standard: we consider the bicomplex

To finish, we note that the two *cohomological bifunctors*, covariant in

## 1.2 The composition law in \operatorname{Ext}

The results of this section are due, independently, to Cartier and Yoneda;
see a talk by Cartier [2] for more details.
Let

which we write as

Now suppose that

In the case where the category in question is the category

Finally, recall that we also have a multiplicative structure between functors

## 1.3 Results concerning local cohomology

Let

where the tensor and exterior products are taken over the ring

such that the following conditions are satisfied:

\varphi_{x_1,\ldots,x_p} depends on the system of thex_i\in{\mathfrak{J}} in an alternatingA -multilinear way;\varphi_{x_1,\ldots,x_p} is zero when any of thex_i is in{\mathfrak{J}}^2 .

In fact, ii. follows from i., since

To define the *free* augmented complex, with augmentation module

But we immediately note that, in the maximal dimension

By composing the homomorphisms in (3.3) and (3.4) we obtain the homomorphisms in (3.2) that we wanted to define. The verification of i. is tedious, but does not present any difficulties.

Let

(The essential point, from which all others follow, is the acyclicity of

With

*Proof*. Indeed,

In particular, there is a distinguished element in *fundamental class* of the ideal

Let

Equation (3.6 bis) can serve as the *definition* of

Let

*Proof*. Since the question is local, we can assume that *dévissage* leads to the case where

Denote by

## 1.4 Cohomology class associated to a subvariety

In all that follows,

Let

If

where

*Proof*. The formula in (4.1) is an immediate consequence of the spectral sequence from Proposition 1, as well as Proposition 5;
by the formula in (3.8), we can write

Setting, in particular,

Now suppose, for simplicity, that *non-singular*, so that *fundamental section* of the sheaf *cohomology class of Y in X*;
it is induced by the section

We define a *non-singular cycle* of dimension

Let *intersect transversally* if every component of

If

where the product on the right-hand side is the cup product:

This last hypothesis is used only to be able to conclude that every coherent algebraic sheaf on

*Proof*. To prove Theorem 1, we can assume that

The pairings in the two columns on the right are induced by the pairing of complexes of sheaves

(where the product on the left-hand side is that from the last column of (4.5)).
This formula in (4.6), which is of a purely local nature, can easily be proven by taking

(a formula which holds true if

a formula which holds true if

Let

*Proof*. Let

For all *fundamental class* of

To have a satisfying theory, we must define *if Z and Z' are two algebraically-equivalent cycles, then P_X(Z)=P_X(Z')*
(a claim which does not seem to follow from the above, even if

*[Comp.]*
As I pointed out in my conference at the international Congress of Mathematicians in 1958^{2}, the questions raised here are now completely resolved.

The reader will find more information on the duality of coherent sheaves in *loco citato*, p. 112–115, as well as in EGA III.2, and in SGA (1962).
A more systematic treatment can be found in a later chapter of EGA (chapter IX in the provisional plan).

## 1.5 The duality theorem

In this section,

The fundamental class

(The proof of this will be given later on).
With the above theorem, we can thus identify

Taking Theorem 2 into account, this pairing defines a homomorphism

This homomorphism is functorial in

The homomorphism in (5.2) is an isomorphism.

In particular, we recover the following result of Serre:

Let ^{3}

*Proof*. It suffices to apply Theorem 3 and Corollary 1 of Proposition 1.

Theorem 2 and Theorem 3 will follow from the following claim:

The homomorphism

(where

We will show that (D) implies Theorem 2.
Let

Its transpose can be identified with the homomorphism

induced by the homomorphism between the

Since (5.3) is an isomorphism, so too is (5.4), and thus so too is (5.5).
Since

It remains only to prove the statement of (D), which will follow in a purely formal way from some elementary facts summarised in the following lemmas.
Here we suppose that

The statement of (D) is true if

*Proof*. This lemma can be proved by a direct calculation.
The explicit calculation of the

Every coherent algebraic sheaf

*Proof*. This follows from the fact that

Let

*Proof*. The first claim follows from the explicit calculations mentioned above;
for the second, we note that we have an isomorphism

Combining the previous two lemmas, we find:

Let

Let

*Proof*. This follows immediately from the spectral sequence in Proposition 1 applied to

We now prove (D) in the case where

Now suppose that

Taking

We can prove that the isomorphism in (5.6) is exactly (5.2 bis) with

## 1.6 The duality theorem for singular varieties

Let

where we set^{4}

Then consider the pairings defined by the composition of the

which are compatible with the boundary maps (generalising (5.2)).
We can prove that, for

For any given integer

The functorial homomorphism in (6.5) is an isomorphism for

n-k\leqslant p\leqslant n .\operatorname{H}^p(X,{\mathscr{O}}_X(-m)) = 0 forn-k\leqslant p<n and large enoughm .The functor

\operatorname{H}^p(X,{\mathscr{F}}) on the category of coherent algebraic sheaves onX is coeffaceable forn-k\leqslant p<n .E^i({\mathscr{O}}_X) = \underline{\operatorname{Ext}}_{{\mathscr{O}}_\mathbf{P}}^{r-n+i}({\mathscr{O}}_X,\omega_\mathbf{P}^r) = 0 for0<i\leqslant k .

*Proof*. i.

Let

where we set

*Proof*. Indeed, by the duality theorem for

For

Recall that the

## 1.7 Poincaré duality

Let

We have seen in §4 that the effect of

If

This formula is equivalent to

*Proof*. To prove Theorem 4, we consider, by Theorem 3, the transpose of the homomorphism

These relations (which could have been given in §4) can be stated, and are indeed true, for arbitrary non-singular varieties, with the second, for example, following from the commutativity of the following diagram of canonical endomorphisms:

We thus obtain an exact equivalent of the formalism of Poincaré duality for compact oriented varieties.
In particular, Theorem 4 allows us to determine the cohomology class associated to the diagonal of *Lefschetz formula*:

Let

The restriction on

## 1.8 Generalisation of the duality theorem

Let

where

We then see that this spectral sequence itself does not depend on the chosen resolutions, and its abutment is the left-hand side of (8.1).
We can easily define the coboundary maps relative to miscellaneous arguments

We define, on the system of functors *composition* (generalising that which was described in §2):

that satisfies the evident properties of associativity, compatibility with the functorial homomorphisms and the coboundary homomorphisms, and spectral sequences.
Similarly, we have symmetry operations, whose explicit descriptions we leave to the reader.
We further have a *contraction* operation every time one of the arguments

Furthermore, if some argument

Now let

If

*Proof*. This follows in a purely formal way from the corollary of Theorem 3.
In fact, it easily follows from this corollary that, if *locally-free* coherent algebraic sheaves, then the hypercohomology of

—

For the definitions preceding Theorem 6, it was not necessary for

X to be non-singular, since it was not necessary to work with only*finite*resolutions. But, ifX is singular, then we can no long be sure, a priori, that the(\underline{T}_r^s)^p({\mathscr{A}}_1,\ldots;{\mathscr{B}}_1,\ldots) are*coherent*sheaves, since, in the complex of sheaves\underline{\operatorname{Hom}}_{{\mathscr{O}}_X}(\underline{L}({\mathscr{A}}_1)\otimes\ldots,\underline{L}({\mathscr{B}}_1)\otimes\ldots) there will be an infinite number of components of any given total degree.We can easily verify that, in the formulas in (8.1), we can replace

*one*of the\underline{L}({\mathscr{B}}_i) with{\mathscr{B}}_i . Taking Proposition 3 into account, this shows that we have\begin{aligned} \underline{T}_1^{1\bullet}({\mathscr{A}};{\mathscr{B}}) &= \underline{\operatorname{Ext}}_{{\mathscr{O}}_X}^\bullet({\mathscr{A}},{\mathscr{B}}) \\T_1^{1\bullet}({\mathscr{A}};{\mathscr{B}}) &= \operatorname{Ext}_{{\mathscr{O}}_X}^\bullet(X;{\mathscr{A}},{\mathscr{B}}). \end{aligned} \tag{8.8} In particular, takingr=s=1 and{\mathscr{A}}_1={\mathscr{O}}_X in (8.6), we recover Theorem 3. Equation (8.8) also implies thatT_0^{1\bullet}({\mathscr{B}})=\operatorname{H}^\bullet(X,{\mathscr{B}}) , and thatT_1^{0\bullet}({\mathscr{A}})=\operatorname{Ext}_{{\mathscr{O}}_X}^\bullet(X;{\mathscr{A}},{\mathscr{O}}_X) .We see, in (8.1), that the functors

\underline{T}_r^{s\bullet} andT_r^{s\bullet} have, in general, components in positive*and*negative degrees. Using the above remark, we see that, if the dimension ofX isn , then the non-zero components of\underline{T}_r^{s\bullet} are concentrated between degrees-(s-1)n andrn ifs>0 , and between degrees0 andrn ifs=0 ; the non-zero components ofT_r^{s\bullet} are concentrated between degrees-(s-1)n and(r+1)n ifs>0 , and between degrees0 and(r+1)n ifs=0 (and, unless I am mistaken, ifr>0 , between degrees-(s-1)n andrn , resp.0 andrn ).

### References

[2] P. Cartier. “Des groups *Séminaire A. Grothendieck: Algèbre homologique*. **1** (1957), Talk no. 3.

[7] A. Grothendieck. “Sur quelques points d’algèbre homologique.” *Tohoku Math. J.* **9** (1957), 119–221.

[21] J.-P. Serre. “Faisceaux algébriques cohérents.” *Annals of Math.* **61** (1955), 197–278.

[22] J.-P. Serre. “Sur la dimension homologique des anneaux et des modules noethériens,” in: *Proc. Intern. Symp. On Alg. Number Theory [1955, Tokyo et Nikko]*. Science Council of Japan, Tokyo, 1956: pp. 175–189.

Grothendieck, A. “The cohomology theory of abstract algebraic varieties”, in

*Proceedings of the international Congress of Mathematicians [1958, Edinburgh]*, Cambridge University Press (1960), 103–118.↩︎*[Trans.]*This equation is labelled (5.3) in the original copy of the notes, but this seems to be a typo, since a later equation shares the same number, and any references to (5.3) seem to indeed point to the later equation instead of this one.↩︎*[Trans.]*This equation is labelled (6.2) in the original, but this seems to be a typo, since a later equation shares the same number, and any references to (6.2) seem to indeed point to the later equation instead of this one.↩︎