# I. Generalities on finiteness conditions in derived categories

## 0. Introduction

The aim of Exposés I to IV is to develop, with generality suitable for this seminar, the formalism of finiteness conditions in the derived categories of ringed toposes.

As was mentioned in Exp. 0, it is the need to define “Grothendieck groups” that have good variance properties on arbitrary schemes that requires us to generalise and relax the notions of finiteness used up until now.

The classical notion of coherent sheaves on a ringed space *pseudo-coherent* if it satisfies the above condition, i.e. if it is of finite *complex* *pseudo-coherent* if it is

If *finite* left resolution by free modules of finite type, i.e. there exists, locally, an exact sequence
*perfect*.
More generally, we say that a *complex* *perfect* if there exists, locally, a quasi-isomorphism **(TO-DO: check that {\mathscr{L}} isn’t also just perfect in degree \geqslant n)**
Conversely, perfectness can be recovered from pseudo-coherence by an additional regularity condition: a complex is perfect if and only if it is pseudo-coherent and locally of finite tor-dimension (cf. I.5).
We thus recover the fact that, if the local rings are regular, then every coherent sheaf is perfect, and, more generally, that every pseudo-coherent complex with locally bounded cohomology is perfect.

Pseudo-coherence and perfectness are the two fundamental notions of finiteness with which we will work in this seminar. Sections I.1 to I.5 of Exposé I are dedicated to their definition, and to establishing their elementary stability properties. For this, we only use two or three local properties of the category of free modules of finite type with respect to the category of all modules. It also turns out to be practical — and useful, most notably in Exp. II — to axiomatise the situation, by introducing a notion of pseudo-coherence (resp. perfectness) in a fibred category over a site with respect to a fibred subcategory that respects suitable conditions.

Sections I.6 to I.8 of Exposé I generalise a certain number of well known notions for locally free sheaves of finite type (rank, duality, and trace of an endomorphism) to the setting of perfect complexes.

In Exp. II, we examine the problem of “the existence of global resolutions”: under what conditions, on a ringed topos *globally* isomorphic in

Exp. III studies the stability of finiteness conditions under the derived direct image.
To obtain reasonable statements, we need to put the notions of pseudo-coherence and perfectness “into perspective”.
We place ourselves here in the setting of ordinary schemes, which suffices for the seminar, but there is no doubt that we must sooner or later develop an analogous theory for relative schemes or analytic spaces.
Let *pseudo-coherent* (resp. *perfect*) *with respect to p* if we can locally embed (by a closed immersion)

*finiteness theorem*, which affirms (in a slightly more precise way) that, if

*proper*morphism of

S is locally Noetherian;f is projective.

Unfortunately, it seems that the extension to the general case is of the same order of difficulty as the analogous theorem in analytic geometry (Grauert’s “theorem”).
Combining the finiteness theorem with an essentially trivial formula called the *projection formula*, we obtain tractable criteria for the stability of relative perfectness under direct images.
We recover, as a corollary, the “Grauert’s continuity and semi-continuity theorems” (EGA III, 7.6).

Exp. IV translates the results of Exposés I to III into the language of “Grothendieck groups”.
On a ringed space

## 1. Preliminary definitions

### 1.1 Fibred categories with additive (resp. abelian, resp. triangulated) fibres

Let *additive* (resp. *triangulated*) * {\mathcal{S}}-category* (or that

*abelian*

*flat*abelian

Let