nLab Schneider's theorem

Redirected from "Schneider's descent theorem".

Idea

The Schneider’s descent theorem is a noncommutative affine analogue of the descent along a torsor where the structure group is also replaced by a Hopf algebra.

The theorem

Preliminaries

Let HH be a Hopf algebra with comultiplication Δ\Delta and E=(E,ρ E:EEH)E = (E,\rho_E:E\to E\otimes H) a right HH-comodule algebra. Consider the subalgebra U=E coHEU = E^{co H}\subset E of coinvariants. Consider the category E H{}_E\mathcal{M}^H of relative Hopf modules and the category U{}_U\mathcal{M} of left UU-modules. There is a pair of adjoint functors between these two categories:

Taking coinvariants extends to a functor () coH: E H U()^{co H}:{}_E\mathcal{M}^H\to {}_U\mathcal{M}. Scalar extension of a UU-module MM, E UME\otimes_U M is canonically a relative Hopf module via coaction e Um(e (0) Um)e (1)e\otimes_U m\mapsto (e_{(0)}\otimes_U m)\otimes e_{(1)} and the map ME UMM\mapsto E\otimes_U M extends to a functor E U: U E HE\otimes_U -:{}_U\mathcal{M}\to {}_E\mathcal{M}^H which is adjoint to () coH()^{co H}. The unit of the adjunction has components η N:N(E UN) coH\eta_N : N\mapsto (E\otimes_U N)^{co H}, n1 Unn\mapsto 1\otimes_U n and the counit is ϵ M:E UM coHM\epsilon_M:E\otimes_U M^{co H}\to M, e Umeme\otimes_U m\mapsto e m.

Statement

In the left-right version, it says that given a Hopf algebra HH and a faithfully flat right Hopf-Galois extension UEU\hookrightarrow E which is faithfully flat from the left, the category of left-right relative Hopf modules E H{}_E\mathcal{M}^H is equivalent to the category of left UU-modules U{}_U \mathcal{M} via the adjoint equivalence described above.

Schneider proves the equivalence for left-left Hopf modules, hence he needs an additional assumption that HH has a bijective antipode.

Modern proofs

Schneider’s original proof is rather involved. Modern proofs begin by rephrasing the category of relative Hopf modules in different terms, either as modules over certain cosimplicial algebra (“coBorel construction” by Lunts-Škoda 2002, see ŠkodaGMJ2009), or as comodules over related comonad GG on the category of EE-modules, or comodules over an EE-coring (Brzezinski 2002). Then the Hopf-Galois condition E UEEHE\otimes_U E\cong E\otimes H directly gives the equivalence of any of the three constructions with the analogue in the descent theory for extension of rings; for example the Sweedler coring E UEE\otimes_U E is isomorphic to the mentioned coring. Hence the categories of comodules are equivalent and the theorem hence boils down to the standard faithfully flat descent (for noncommutative rings) or comonadicity criterium in comonadic version.

Let us describe the comonad GG on the category of left EE-modules, E{}_E\mathcal{M}. As an endofunctor, GG sends a left EE-modules MM to MHM\otimes H with HH-coaction id MΔ:MH(MH)H\id_M\otimes\Delta:M\otimes H\to (M\otimes H)\otimes H and left EE-action,

e(mh)=e (0)me (1)h.e\otimes (m\otimes h) = \sum e_{(0)} m\otimes e_{(1)} h.

(Remark: This action is induced by a distributive law.) The Eilenberg-Moore category of this comonad is equivalent to E H{}_E\mathcal{M}^H.

Literature

A generalization to Hopf algebroids is proven in

Last revised on May 14, 2024 at 16:44:20. See the history of this page for a list of all contributions to it.