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The category (mathrm{MHT}_{mathbb {Q}}) 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 ({{{mathcal {H}}}}_bullet = oplus _{n=0}^infty {{{mathcal {H}}}}_n) over  {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}}/(2pi i)^n{mathbb {Q}}) over (nu003e0). 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  {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}}, 2pi i {mathbb {Q}})) gives rise to the Hopf algebra (mathcal{A}_bullet ({mathbb {C}})). Another example of a Tate dg-algebra ((Omega _X^bullet , 2pi i{mathbb {Q}})) is provided by the holomorphic de Rham complex  _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.

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