zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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:X ˜X from a smooth variety X ˜ to X. In addition, if Y is a closed subvariety of X then there is a resolution f:X ˜X such that Y ˜:=f -1 (Y) 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 Sch ̲(k) the category of separated schemes of finite type over k and by Reg ̲(k) the subcategory of smooth schemes. Let 𝒜 be an abelian category, C b (𝒜) the category of bounded complexes in 𝒜 and D b (𝒜) the associated derived category. The main theorem of the present paper states that any contra-variant functor G:Reg ̲(k)C b (𝒜) 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 Sch ̲(k) 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 Sch ̲(k). 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:
14E15Global theory and resolution of singularities
14F43Other algebro-geometric (co)homologies
14F42Motivic cohomology; motivic homotopy theory
18E30Derived categories, triangulated categories
18E10Exact categories, abelian categories