A Sheaf-theoretic Reformulation of the Tate Conjecture

We present a conjecture which unifies several conjectures on motives in characteristic p. Introduction In a talk at the June 1996 Oberwolfach algebraic K-theory conference, I explained that, in view of Voevodsky’s proof of the Milnor conjecture, the Bass conjecture on finite generation of K-groups of regular schemes of finite type over the integers implies the rational Beilinson-Soulé conjecture on vanishing of algebraic K-theory of low weights in characteristic 6= 2. This argument is reproduced in the appendix. The next day, Thomas Geisser explained that, in characteristic > 0, the Tate conjecture on surjectivity of the Ql-adic cycle map for smooth, projective varieties over a finite field, together with the conjecture that rational and numerical equivalences agree for such varieties, also implies the Beilinson-Soulé conjecture. His work is now available in [10]. The present paper stems from an attempt to understand the relationship between these two facts. Let l be a prime number. We present a conjecture (conjecture 8.12) which is equivalent to the conjunction of three well-known conjectures on smooth, projective varieties X over Fp (p 6= l): 1. Tate’s conjecture: the geometric cycle map CH(X)⊗ Ql → H (X,Ql(n)) G (*) is surjective (X = X ×Fp F̄p, G = Gal(F̄p/Fp)). 2. Partial semi-simplicity: the characteristic subspace of H(X,Ql(n)) corresponding to the eigenvalue 1 of Frobenius is semi-simple. 1