Endomorphism algebras over weak (α, β)-Yetter–Drinfeld modules

Let H be a weak Hopf algebra with a bijective antipode, α, β ∈ Aut weak Hopf (H) and M a finite-dimensional weak (α, β)-Yetter–Drinfeld module. Then in this paper we prove that the endomorphism algebras End Hs(M) and End Ht(M) op endowed with certain structures become algebras in H𝒲𝒴𝒟H and we also study the isomorphic relations between different endomorphism algebras. We prove that End Hs(M) endowed with certain structures becomes an H-Azumaya algebra.

GALOIS CORRESPONDENCES FOR PARTIAL GALOIS AZUMAYA EXTENSIONS

Let α be a partial action, having globalization, of a finite group G on a unital ring R. Let Rα denote the subring of the α-invariant elements of R and CR(Rα) the centralizer of Rα in R. In this paper we will show that there are one-to-one correspondences among sets of suitable separable subalgebras of R, Rα and CR(Rα). In particular, we extend to the setting of partial group actions similar results due to DeMeyer [Some notes on the general Galois theory of rings, Osaka J. Math.2 (1965) 117–127], and Alfaro and Szeto [On Galois extensions of an Azumaya algebra, Commun. Algebra25 (1997) 1873–1882].

A Gersten-Witt complex for hermitian Witt groups of coherent algebras over schemes II: Involution of the second kind

Let X be a regular noetherian scheme of finite Krull dimension with involution σ and an Azumaya algebra over X with involution τ of the second kind with respect to σ. We construct a hermitian and a skew-hermitian Gersten-Witt complex for (, τ) and show that these complexes are exact if X = Spec R is the spectrum of a regular semilocal ring R of geometric type, such that R is a quadratic étale extension of the fixed ring of σ.

CENTRALIZERS AND INDUCTION

Given a ring homomorphism B → A, consider its centralizer R = AB, bimodule endomorphism ring S = End BAB and sub-tensor-square ring T = (A ⊗ BA)B. Nonassociative tensoring by the cyclic modules RT or SR leads to an equivalence of categories inverse to the functors of induction of restricted A-modules or restricted coinduction of B-modules in case A | B is separable, H-separable, split or left depth two (D2). If RT or SR are projective, this property characterizes separability or splitness for a ring extension. Only in the case of H-separability is RT a progenerator, which takes the place of the key module AAe for an Azumaya algebra A. In addition, we characterize left D2 extensions in terms of the module TR, and show that the centralizer of a depth two extension is a normal subring in the sense of Rieffel as well as pre-braided commutative. For example, the notion of normality yields a version for Hopf subalgebras of the fact that normal subgroups have normal centralizers, and yields a special case of a conjecture that D2 Hopf subalgebras are normal.

On Azumaya algebras with a finite automorphism group

LetBbe a ring with 1,Cthe center ofB, andGa finite automorphism group ofB. It is shown that ifBis an Azumaya algebra such thatB=⊕∑g∈GJgwhereJg={b∈B|bx=g(x)b for all x∈B}, then there exist orthogonal central idempotents{fi∈C|i=1,2,…,m for some integer m}and subgroupsHiofGsuch thatB=(⊕∑i=1mBfi)⊕DwhereBfiis a central Galois algebra with Galois groupHi|Bfi≅Hifor eachi=1,2,…,mandDis contained inC.