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Motives, numerical equivalence, and semi-simplicity. (English) Zbl 0762.14003

Let \(k\) be a field and let \({\mathcal V}_ k\) denote the category of smooth projective varieties over \(k\). Also, let \(F\) be a field of characteristic zero, traditionally \(F=\mathbb{Q}\). One can define several adequate equivalence relations on the space of \(F\)-linear algebraic cycles on objects of \({\mathcal V}_ k\), e.g. rational, algebraic, homological and numerical equivalence. Here ‘adequate’ means good behaviour under pull- back, push-forward and intersection. These are ordered: rational equivalence is the finest and numerical equivalence is the coarsest adequate equivalence relation. A corollary of Grothendieck’s standard conjectures on algebraic cycles would be the coincidence of homological and numerical equivalence. These conjectures are at the basis of the theory of motives. For two objects \(X,Y\) in \({\mathcal V}_ k\) a correspondence between \(X\) and \(Y\) is defined as a cycle on \(X\times Y\) modulo an adequate equivalence relation. So, different choices of equivalence relations give rise to different correspondences.
Correspondences can be composed in a natural way. The category of motives (over \(F\)), \({\mathcal M}_ k\), has as objects triples \((X,p,m)\) with \(X\in{\mathcal O}b({\mathcal V}_ k)\), \(p\) a projector, i.e. a cycle on \(X\times X\) modulo one of the aforementioned equivalence relations and satisfying \(p^ 2=p\), and \(m\) is an integer. A morphism in \({\mathcal M}_ k\) from \((X,p,m)\) to \((Y,q,n)\) is the composition of \(p\), a correspondence between \(X\) and \(Y\) of codimension \(\dim(X)-m+n\), and the projector \(q\). For different choices of equivalence relations one obtains different categories of motives, e.g. for rational equivalence one gets the category of Chow motives \({\mathcal M}_ k^{rat}\), and for numerical (or homological?) equivalence one gets Grothendieck’s original category of motives \({\mathcal M}_ k^{num}\). These categories \({\mathcal M}_ k\) are all \(F\)-linear, pseudo-abelian. However, one would like to have a more refined structure on the \({\mathcal M}_ k\), to make them (semi-simple) abelian or even Tannakian, i.e. equivalent to the category of finite dimensional representations of some (pro-)algebraic group scheme. It has been believed for a long time that, to reach such results on \({\mathcal M}_ k\), Grothendieck’s conjectures are needed. In this paper the surprising result on \({\mathcal M}_ k\), without any recourse to Grothendieck’s conjectures, is proven:
Theorem: The following properties are equivalent:
(i) \({\mathcal M}_ k\) is a semi-simple abelian category.
(ii) The group of \(\text{codim}(X)\) self-correspondences of \(X\in{\mathcal O}b({\mathcal V}_ k)\) is a finite-dimensional semi-simple \(F\)-algebra for any smooth projective variety \(X\).
(iii) The equivalence relation on cycles used to define \({\mathcal M}_ k\) is numerical equivalence.
In relation to Grothendieck’s conjectures one obtains a corollary: If the Künneth components of the diagonal (with respect to some fixed Weil cohomology) are algebraic for every \(X\) in \({\mathcal V}_ k\), then \({\mathcal M}_ k^{num}\) is a semi-simple \(F\)-linear Tannakian category.
The condition of the corollary holds e.g. for \(k\subset\overline\mathbb{F}_ q\).

MSC:

14A20 Generalizations (algebraic spaces, stacks)
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References:

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