Fu, Lei On the semisimplicity of pure sheaves. (English) Zbl 0918.14008 Proc. Am. Math. Soc. 127, No. 9, 2529-2533 (1999). 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)]. Cited in 2 Documents 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 Keywords:semisimplicity of pure sheaves; finite ground field; Frobenius automorphism; semisimplicity of the Galois representations of function fields Citations:Zbl 0456.14014 PDFBibTeX XMLCite \textit{L. Fu}, Proc. Am. Math. Soc. 127, No. 9, 2529--2533 (1999; Zbl 0918.14008) Full Text: DOI