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An extension criterion for functors defined on smooth schemes. (Un critère d’extension des foncteurs définis sur les schémas lisses.) (French) Zbl 1075.14012

Let $X$ be an algebraic variety over a field of characteristic zero. Then, by H. Hironaka’s famous theorem on the resolution of singularities, there exists a birational and proper morphism $f:\stackrel{˜}{X}\to X$ from a smooth variety $\stackrel{˜}{X}$ to $X$. In addition, if $Y$ is a closed subvariety of $X$ then there is a resolution $f:\stackrel{˜}{X}\to X$ such that $\stackrel{˜}{Y}:={f}^{-1}\left(Y\right)$ is a normal crossing divisor and $f$ is an isomorphism of the union of $Y$ and the singular locus of $X$. This fundamental theorem, together with Hironaka’s more precise versions of it, has found numerous applications to the study of cohomology theories for algebraic varieties, among them being Grothendieck’s algebraic de Rham cohomology and the Hodge-Deligne cohomology theory.

In the paper under review, the authors continue their work begun in an earlier treatise of theirs published about 15 years ago [cf.: F.Guillén, V. Navarro Aznar, P. Pascual-Gainza and F. Puerta, Hyperrésolutions cubiques et descente cohomologique (Lect. Notes Math. 1335, Springer Verlag, Berlin) (1988; Zbl 0638.00011)]. Starting from the particular precise version of Hironaka’s resolution theorem obtained back then, they prove an extension criterion for a functor defined on the category of separated schemes of finite type which are smooth over the field $k$.

Roughly speaking, this criterion asserts that a functor admitting the usual exact sequence for blowing-up maps can be extended to the whole category of separated schemes of finite type over $k$.

More precisely, for a field $k$ of characteristic zero, denote by $\underline{\text{Sch}}\left(k\right)$ the category of separated schemes of finite type over $k$ and by $\underline{\text{Reg}}\left(k\right)$ the subcategory of smooth schemes. Let $𝒜$ be an abelian category, ${C}^{b}\left(𝒜\right)$ the category of bounded complexes in $𝒜$ and ${D}^{b}\left(𝒜\right)$ the associated derived category. The main theorem of the present paper states that any contra-variant functor $G:\underline{\text{Reg}}\left(k\right)\to {C}^{b}\left(𝒜\right)$ satisfying certain (quasi-)isomorphism conditions with respect to disjoint unions and blowing-up maps, which are here labeled by (F1) and (F2), possesses a unique extension to the category $\underline{\text{Sch}}\left(k\right)$ such that a prescribed cohomological descent property holds.

This general theorem can be applied, as the authors point out in the course of their paper, to several known cohomology functors. For example, the classical singular cohomology, the Beilinson cohomology, the various Weil cohomology theories, the Bloch-Ogus cohomology, but also some non-abelian cohomology theories as well as the known complex-analytic cohomology theories can be interpreted in this general context, in particular with a view toward the treatment of those problems related to questions of cohomological descent (à la P. Deligne). Among the numerous applications of their main theorem, the authors prove the existence of two functors extending Grothendieck’s theory of motives to the category of schemes $\underline{\text{Sch}}\left(k\right)$. These functors are then used to give an alternative proof of the former Serre conjecture on the independence of the virtual motive of an algebraic variety [cf.: J.-P. Serre, in: Journées arithmétiques, Exp. Congr., Luminy/Fr. 1989, Astérisque 198–200, 333–349 (1991; Zbl 0759.14002)], thereby complementing the first proof by H. Gillet and C. Soulé from another point of view.

No doubt, the article under review represents a highly important contribution toward the theory of cohomological descent in its general setting, together with just as significant concrete applications to current research topics.

##### MSC:
 14E15 Global theory and resolution of singularities 14F43 Other algebro-geometric (co)homologies 14F42 Motivic cohomology; motivic homotopy theory 18E30 Derived categories, triangulated categories 18E10 Exact categories, abelian categories