Zoran Skoda global Hopf-Galois extension

(be careful, I use $g$ and $G$ here, unrelated)

Recall that a functor $L^*$ possessing a right adjoint $L_*$ is a localization iff the counit of the adjunction is iso and that a functor is affine if it has a right adjoint which is faithful and has its own right adjoint (thus $L^*\dashv L_{*}\dashv L^!$). If the functors are additive among abelian categories $L^*$ is said to be flat (coflat, biflat) if the inverse image part $L^*$ (resp. direct image part $L_*$, resp. both) are exact. A family of flat functors is a cover if it is conservative.

Let an adjoint functor $g_* : A\to \mathcal{M} : g^*$ together with a finite cover $\{Q^*_\lambda : A\to A_\lambda\}_\lambda$ together of $A$ by biflat affine localization functors define a structure of a noncommutative scheme over $\mathcal{M}$ in the sense of Rosenberg. In addition to the stated, being a noncommutative scheme means that $Q_{\lambda}^*\circ g^*$ is affine for every $\lambda$, and that $g_*$ is exact and faithful.

Let now $H$ be a Hopf $k$-algebra over a commutative unital ring $k$. Let $\mathcal{M} = k-Mod$. Let an additive comonad $G$ on $A$ be given, which is strongly compatible with each of the localizations $Q^*_\lambda$. That implies that there is am induced comonad $G^\lambda$ on $A_\lambda\cong A^{Q_\lambda}$ lifting $G$. Let $E_\lambda = (Q_\lambda^* \circ g^*)(k)$; by affiness, $E_\lambda$ is then a $k$-algebra. Suppose, $E_\lambda$ is a faithfully flat $H$-Hopf-Galois extension with coinvariants $U_\lambda = E_\lambda^{co H}$; and $A_\lambda \cong E_\lambda-Mod$ (we can take that they are equal). We suppose that $G^\lambda: {}_{E_\lambda}\mathcal{M}^H$ is given with an isomorphism with the comonad for Hopf modules (compatible left $E_\lambda$-modules, right $H$-comodules) $\otimes H$. We say that these data form a global Hopf-Galois extension structure on the noncommutative scheme given above.

Theorem. $g_*^G = g_*\circ U^G$ and localizations $(Q^G_\lambda)^*$ induced by the compatible localizations $Q^*$ induce a structure of noncommutative scheme on $A^G$ over $k-Mod$ and the algebra of coinvariants $E_\lambda^{co H}$ is isomorphic to $(Q^{G_\lambda*}\circ g^{G*})(k)$ (as a $k$-module, and this isomorphism gives it a structure of a $k$-algebra, making it a progenerator in $(A_\lambda)^{G_\lambda}$).

Naturalness: If $A\to B$ is any other affine biflat strongly compatible localization such that $B$ is the category of relative Hopf modules over the appropriate pullback of $k$ and such that the localization factors via one of the elements of the initial compatible cover, then the inverse image functor of the other part of the factorization is induced by an inclusion of the algebras of coinvariants.

(We say that $A^G$ together with this structure of noncommutative scheme over $\mathcal{M}$ is a noncommutative quotient scheme of $A$ with respect to the globalized action of a Hopf algebra $H$.)

Main elements of the proof: $g^*\circ U^G$ is faithful because $g^*$ is and the forgetful functor from EM category is; it is exact because $g^*$ is exact and $U^{G_\lambda}$ is for all $\lambda$, for the latter use the fact that locally we have modules over the coinvariants. For affiness of the composition of $(g^G_\lambda)^*$ with the induced localizations one in fact proves and uses that this functor is isomorphic with the extension of scalars from $k$ to $E_\lambda^{co H}$. For this one of course uses the Schneider’s equivalence of categories in the left module-right comodule version (no need for bijective antipode assumption). In this proof, it is convenient to think of the Schneider’s equivalence as the equivalence of the EM categories, which is induced by the isomorphism of $k$-linear comonads $M\mapsto E_\lambda\otimes_{U_\lambda}M$ and $M\mapsto M\otimes_k H$ in the category of left $E_\lambda$-modules; the latter isomorphism of course canonically extends the $k$-linear isomorphism of left $E_\lambda$-modules $E_\lambda\otimes_{U_\lambda} E_\lambda\cong E_\lambda\otimes_k H$ given by the Hopf-Galois condition.

For induced localizations one uses the compatibility, as well as for the right adjoint (not difficult); the exactness of both is also easy.

For affiness of the induced localizations themselves one uses the Gabi’s result of existence of the second adjoint (the faithfulness is easy). For conservativeness of the cover one just uses that the forgetful functor from $G$-comodules is faithful. For the naturalness one uses the construction via coinvariants an the naturalness of the Schneider’s equivalence of categories with respect to the inclusions of Hopf-Galois extensions for a fixed Hopf $k$-algebra $H$.