The Galois group of the category of mixed Hodge-Tate structures

TL;DR

This paper constructs a natural Hopf algebra for mixed Hodge-Tate structures, generalizes it to dg-algebras, and relates it to derived categories, Deligne cohomology, and p-adic Hodge theory.

Contribution

It provides an explicit, natural construction of the Hopf algebra for mixed Hodge-Tate structures and extends this to dg-algebras, connecting to various aspects of Hodge theory.

Findings

Constructed a natural Hopf algebra for mixed Hodge-Tate structures.

Extended the construction to dg-algebras related to complex manifolds.

Connected the dg-Hopf algebra framework to Deligne cohomology and p-adic Hodge theory.

Abstract

The category of rational mixed Hodge-Tate structures is a mixed Tate category. So thanks to the Tannakian formalism, it is equivalent to the category of finite dimensional graded comodules over a graded commutative Hopf algebra H over Q. Since the category has homological dimension 1, the Hopf algebra H is isomorphic to the commutative graded Hopf algebra provided by the tensor algebra of the graded vector space given by the direct sum of the groups C/Q(n) over n>0. However this isomorphism is not natural in any sense, e.g. does not work in families. We give a natural explicit construction of the Hopf algebra H. Generalizing this, we define a Hopf dg-algebra related to any dg-algebra R over a field k, equipped with an invertible line k(1). When R is the sheaf of algebras given by the holomorphic de Rham complex of a complex manifold X, and the line is Q(1), the related Hopf dg-algebra describes a dg-model of the derived category of variations of Hodge-Tate structures on X. Its cobar complex provides a dg-model for the rational Deligne cohomology of X. We consider a variant of our construction which starts from Fontaine's crystalline / semi-stable period rings and produces graded / dg Hopf algebras, which we relate to the p-adic Hodge theory.