Hopf Algebras and Galois Theory

A-map A*-module structure Ab(A abelian category abelian group abelian presheaves adjointness admissible Hopf subalgebra antipode apply arising B₁ C-map category of commutative coalgebra cocommutative coequalizers coflat Galois G-object cogroup cohomology commutative diagram commutative R-algebras commutative ring completing the proof Corollary 10.4 defined Definition 7.3 diagram below commutes direct summand easy computation equalizer diagram exists Ext A*,U extension faithfully flat faithfully flat R-module finite Hopf algebra finite products finitely generated projective forgetful functor formula full subcategory Galois A-object Galois group guarantees H₁ immediate consequence inverse left A*-module module Morita context morphism natural isomorphism object obtain one-to-one preserves products presheaves Proposition R-submodule Remark rendering the diagram right A*-module right ideal routine computation RZ-object Section Seiten Sets,X sheaf sheaves surjective terminal object Theorem 8.2 theory tripleable unit map unlabeled arrow denoting whence X X G Yoneda Lemma Z-graded