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.

In a left-right version, it says that given a Hopf algebra $H$ and a faithfully flat Hopf-Galois extension $U\hookrightarrow E$ (where $E$ is a right $H$-comodule algebra), the category of left-right relative Hopf modules ${}_E\mathcal{M}^H$ is equivalent to the category of left $U$-modules ${}_U \mathcal{M}$. The adjoint equivalence is given canonically…

- Hans-Jürgen Schneider,
*Principal homogeneous spaces for arbitrary Hopf algebras*, Israel J. Math.**72**(1990), no. 1-2, 167–195 MR92a:16047 doi - Z. Škoda,
*Some equivariant constructions in noncommutative algebraic geometry*Georgian Mathematical Journal**16**(2009), No. 1, 183–202, arXiv:0811.4770 MR2011b:14004 - A. Ardizzoni, G. Böhm, C. Menini,
*A Schneider type theorem for Hopf algebroids*, J. Algebra**318**(2007), no. 1, 225–269 MR2008j:16103 doi math.QA/0612633 (arXiv version is unified, corrected);*Corrigendum*, J. Algebra**321**:6 (2009) 1786-1796 MR2010b:16060 doi

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