The Gauss-Manin connection and Tannaka duality

Abstract

If f : X → Spec(K) is a smooth, geometrically connected variety defined over a field of characteristic 0, K ⊃ k is a field extension, and x ∈ X(K) is a rational point, one considers three Tannaka categories C(X/K), C(X/k), C(K/k) of flat connections with compatible fiber functors. The objects of C(X/K) are bundles (i.e., locally free coherent modules of finite type) with relative flat connections ((V,∇/K),∇/K : V → ΩX/K ⊗OX V), the ones of C(X/k) are bundles with flat absolute connections ((V,∇), ∇ : V → ΩX/k ⊗OX V), and the ones of C(K/k) are K-vector spaces with flat connections ((V,∇), ∇ : V → ΩK/k⊗KV). The morphisms are the flat morphisms and the fiber functor has values in the category of finite dimensional vector spaces VecK over K, defined by the restriction of V to x for C(X/K), C(X/k) and by V for C(K/k). Then C(X/K) is a neutral Tannaka category, and Tannaka duality [2,Theorem 2.11] yields the existence of a proalgebraic group schemeG(X/K) overK, so that C(X/K) becomes equivalent to the representation category Repf(G(X/K)) on finite dimensional K-vector spaces. The two other Tannaka categories C(X/k), C(K/k) are not necessarily neutral. We assume that they are defined over k, which is to say that k is the endomorphism ring EndC(K/k)((K, dK/k)) of the unit object,which in this case is the same as the subfield of K of flat sections. Equivalently, this is saying that k is algebraically closed in K. Then Tannaka duality [3, theoreme 1.12] yields the existence of groupoid schemes G(X/k), G(K/k) over k acting on Spec(K) ×k Spec(K), so that, in the groupoid sense, C(X/k) (resp., C(K/k)) becomes equivalent to the representation category Repf(K :