Resumen de The Galois group of the category of mixed Hodge–Tate structures

Alexander Goncharov, Guangyu Zhu

• 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∙=⊕∞n=0Hn over Q . Since the category MHTQ has homological dimension one, H∙ is isomorphic to the commutative graded Hopf algebra provided by the tensor algebra of the graded vector space given by the sum of Ext1MHTQ(Q(0),Q(n))=C/(2πi)nQ 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∙ . Namely, let C∗Q:=C∗⊗Q . Set A∙(C):=Q⊕⨁n=1∞C∗Q⊗QC⊗n−1.

  We provide it with a commutative graded Hopf algebra structure, such that H∙=A∙(C) . This implies another construction of the big period map Hn⟶C∗Q⊗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∙(R) . For example, the Tate algebra (C,2πiQ) gives rise to the Hopf algebra A∙(C) . Another example of a Tate dg-algebra (Ω∙X,2πiQ) is provided by the holomorphic de Rham complex Ω∙X of a complex manifold X. The sheaf of Hopf dg-algebras A∙(Ω∙X) describes a dg-model of the derived category of variations of Hodge–Tate structures on X. The cobar complex of A∙(Ω∙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 Bcrys / Bst produces graded/dg Hopf algebras which we relate to the p-adic Hodge theory.