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

Abstract

The category $\mathrm{MHT}_{\mathbb {Q}}$ of mixed Hodge–Tate structures over ${\mathbb {Q}}$ 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 ${{{\mathcal {H}}}}_\bullet = \oplus _{n=0}^\infty {{{\mathcal {H}}}}_n$ over ${\mathbb {Q}}$. Since the category $\mathrm{MHT}_{\mathbb {Q}}$ has homological dimension one, ${{{\mathcal {H}}}}_\bullet$ is isomorphic to the commutative graded Hopf algebra provided by the tensor algebra of the graded vector space given by the sum of $\mathrm{Ext}_{\mathrm{MHT}_{\mathbb {Q}}}^1({\mathbb {Q}}(0), {\mathbb {Q}}(n)) = {\mathbb {C}}/(2\pi i)^n{\mathbb {Q}}$ 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 ${{{\mathcal {H}}}}_\bullet$. Namely, let ${\mathbb {C}}^*_{\mathbb {Q}}:={\mathbb {C}}^* \otimes {\mathbb {Q}}$. Set

$$\begin{aligned} {{{\mathcal {A}}}}_\bullet ({\mathbb {C}}):= {\mathbb {Q}}\oplus \bigoplus _{n=1}^\infty {\mathbb {C}}_{\mathbb {Q}}^* \otimes _{\mathbb {Q}}{\mathbb {C}}^{\otimes n-1}. \end{aligned}$$

We provide it with a commutative graded Hopf algebra structure, such that ${{{\mathcal {H}}}}_\bullet = {{{\mathcal {A}}}}_\bullet ({\mathbb {C}})$. This implies another construction of the big period map ${{{\mathcal {H}}}}_n \longrightarrow {\mathbb {C}}_{\mathbb {Q}}^* \otimes {\mathbb {C}}$ 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 ${{{\mathcal {A}}}}_\bullet (R)$. For example, the Tate algebra $({\mathbb {C}}, 2\pi i {\mathbb {Q}})$ gives rise to the Hopf algebra $\mathcal{A}_\bullet ({\mathbb {C}})$. Another example of a Tate dg-algebra $(\Omega _X^\bullet , 2\pi i{\mathbb {Q}})$ is provided by the holomorphic de Rham complex $\Omega _X^\bullet$ of a complex manifold X. The sheaf of Hopf dg-algebras ${{{\mathcal {A}}}}_\bullet (\Omega _X^\bullet )$ describes a dg-model of the derived category of variations of Hodge–Tate structures on X. The cobar complex of $\mathcal{A}_\bullet (\Omega _X^\bullet )$ 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 $\mathrm{B}_{\mathrm{crys}}$/$\mathrm{B}_{\mathrm{st}}$ produces graded/dg Hopf algebras which we relate to the p-adic Hodge theory.