Graded associative algebras and Grothendieck standard conjectures (Q1359217)

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 \({\mathcal A}^*({\mathcal X}\times {\mathcal X})\) of a projective irreducible smooth variety \({\mathcal X}\) implies the standard conjecture of Lefschetz type for \({\mathcal X}\). It was proved by U. Jannsen that the algebra \({\mathcal A}^*({\mathcal X}\times {\mathcal X})\) is semisimple when the numerical and homological equivalences of algebraic cycles on \({\mathcal X}\) 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.