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

The category MHTQ of mixed Hodge-Tate structures over Q is a mixed Tate category. Thanks to the Tannakian formalism it is equivalent to the category of graded comodules over a commutative graded Hopf algebra H-center dot = circle plus H-infinity(n=0)n over Q. Since the category MHTQ has homological dimension one, H-center dot is isomorphic to the commutative graded Hopf algebra provided by the tensor algebra of the graded vector space given by the sum of Ext(MHTQ)(1) (Q(0), Q(n)) = C/(2 pi i)(n)Q over n > 0. However this isomorphism is not natural in any sense, e.g. does not exist in families. We give a natural construction of the Hopf algebra H-center dot. Namely, let C*(Q) := C* circle times Q. Set A(center dot)(C) := Q circle plus circle plus C-infinity(n=1)*(Q)circle times(Q) C circle times n-1. We provide it with a commutative graded Hopf algebra structure, such that H-center dot = A(center dot)(C). This implies another construction of the big period map H-n -> C*(Q) circle times C from Goncharov (JAMS 12(2):569-618, 1999. arXiv:alg-geom/9601021, Annales de la Faculte des Sciences de Toulouse XXV(2-3):397-459, 2016. arXiv:1510.07270). Generalizing this, we introduce a notion of a Tate dg-algebra (R, k(1)), and assign to it a Hopf dg-algebra A(center dot)(R). For example, the Tate algebra (C, 2 pi iQ) gives rise to the Hopf algebra A(center dot)(C). Another example of a Tate dg-algebra (Omega(center dot)(X), 2 pi iQ) is provided by the holomorphic de Rham complex Omega(center dot)(X) of a complex manifold X. The sheaf of Hopf dg-algebras A(center dot)(Omega(center dot)(X)) describes a dg-model of the derived category of variations of Hodge-Tate structures on X. The cobar complex of A(center dot)(Omega(center dot)(X)) is a dg-model for the rational Deligne cohomology of X. We consider a variant of our construction which starting from Fontaine's period rings B-crys/B-st produces graded/dg Hopf algebras which we relate to the p-adic Hodge theory.