Graded associative algebras and Grothendieck Standard Conjectures
This paper is concerned with Grothendieck's standard conjectures
on algebraic cycles, introduced independently by Grothendieck and Bombieri
to explain the Weil conjectures on the zeta-function of algebraic varieties.
We prove that the semisimplicity of the algebra of algebraic
correspondences A(X) of a projective irreducible smooth variety X implies
the standard conjecture of Lefschetz type for X. It was proved by
U. Jannsen that the algebra A(X) is semisimple when the numerical and
homological equivalences of algebraic cycles on X are the same. Thus, with
Jannsen's theorem our result asserts that the standard conjecture of
Lefschetz type follows from Grothendieck's conjecture about the equality of
the numerical and homological equivalences. This was known before only in
the presence of the standard conjecture of Hodge type.
(This paper has appeared in Invent. Math. 128 (1997), 201--206)
O. N. Smirnov
<smirnov@oreo.uottawa.ca>