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On the semisimplicity of pure sheaves. (English) Zbl 0918.14008

Summary: We obtain a criterion for a pure sheaf to be semisimple. As a corollary, we prove the following:
Let \(X_0\) and \(S_0\) be two schemes over a finite field \(\mathbb{F}_q\), and let \(f_0: X_0\rightarrow S_0\) be a proper smooth morphism. Assume \(S_0\) is normal and geometrically connected, and assume there exists a closed point \(s\) in \(S_0\) such that the Frobenius automorphism \(F_s\) acts semisimply on \(H^i(X_{\bar s}, {\overline{\mathbb{Q}_l}})\), where \(X_{\bar s}\) is the geometric fiber of \(f_0\) at \(s\) (this last assumption is unnecessary if the semisimplicity conjecture is true). Then \(R^if_{0\ast} {\overline{\mathbb{Q}_l}}\) is a semisimple sheaf on \(S_0\). This verifies a conjecture of Grothendieck and Serre provided the semisimplicity conjecture holds. As an application, we prove that the Galois representations of function fields associated to the \(l\)-adic cohomologies of \(K3\) surfaces are semisimple. We also get a theorem of Zarkhin about the semisimplicity of the Galois representations of function fields arising from abelian varieties.
The proof relies heavily on P. Deligne’s work on Weil conjectures [cf. Publ. Math., Inst. Hautes Étud. Sci. 52, 137-252 (1980; Zbl 0456.14014)].

MSC:

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14G15 Finite ground fields in algebraic geometry
14F20 Étale and other Grothendieck topologies and (co)homologies
11S20 Galois theory

Citations:

Zbl 0456.14014
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