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

Research paper by Alexander Goncharov, Guangyu Zhu

Indexed on: 03 Mar '18Published on: 09 Feb '18Published in: Selecta Mathematica



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.