On the semisimplicity of pure sheaves

Author:
Lei Fu

Journal:
Proc. Amer. Math. Soc. **127** (1999), 2529-2533

MSC (1991):
Primary 14F20, 14G15

DOI:
https://doi.org/10.1090/S0002-9939-99-05414-3

Published electronically:
May 19, 1999

MathSciNet review:
1676348

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We obtain a criteria for a pure sheaf to be semisimple. As a corollary, we prove the following: Let and be two schemes over a finite field , and let be a proper smooth morphism. Assume is normal and geometrically connected, and assume there exists a closed point in such that the Frobenius automorphism acts semisimply on , where is the geometric fiber of at (this last assumption is unnecessary if the semisimplicity conjecture is true). Then is a semisimple sheaf on . 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 -adic cohomologies of surfaces are semisimple. We also get a theorem of Zarhin about the semisimplicity of the Galois representations of function fields arising from abelian varieties. The proof relies heavily on Deligne's work on Weil conjectures.

**[D1]**P. Deligne,*La Conjecture de Weil, II*, Publ. Math. IHES 52 (1980), 137-252. MR**83c:14017****[D2]**P. Deligne,*La Conjecture de Weil pour les Surfaces ,*Inventiones Math. 15 (1972), 206-226. MR**45:5137****[PS]**I. I. Piatetski-Shapiro and I. R. Shafarevich,*The Arithmetic of surfaces*, Proc. Steklov Inst. Math. 132 (1975), 45-57. MR**49:302****[T]**J. Tate,*Algebraic Cycles and the Poles of Zeta Functions*, Arithmetic Algebraic Geometry, Harper and Row, New York (1965), 93-110. MR**37: 1371****[W]**A. Weil, Courbes Algébriques et Variétés Abéliennes, Hermann, Paris (1971).**[Z]**Y. Zarhin,*Endomorphisms of Abelian Varieties over Fields of Finite Characteristic*, Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), No. 2, 272-277. MR**51:8114**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
14F20,
14G15

Retrieve articles in all journals with MSC (1991): 14F20, 14G15

Additional Information

**Lei Fu**

Affiliation:
Institute of Mathematics, Nankai University, Tianjin, People’s Republic of China

Email:
leifu@sun.nankai.edu.cn

DOI:
https://doi.org/10.1090/S0002-9939-99-05414-3

Received by editor(s):
August 11, 1997

Published electronically:
May 19, 1999

Communicated by:
Ron Donagi

Article copyright:
© Copyright 1999
American Mathematical Society