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Epsilon-strongly graded rings: Azumaya algebras and partial crossed products

  • Dirceu Bagio , Luis Martínez and Héctor Pinedo ORCID logo EMAIL logo
Published/Copyright: June 27, 2023
Forum Mathematicum
From the journal Forum Mathematicum Volume 35 Issue 5

Abstract

Let G be a group, let A = ⊕ g ∈ G A g be an epsilon-strongly graded ring over G, let R := A 1 be the homogeneous component associated with the identity of G, and let 𝙿𝚒𝚌𝚂 ⁢ ( R ) be the Picard semigroup of R. In the first part of this paper, we prove that the isomorphism class [ A g ] is an element of 𝙿𝚒𝚌𝚂 ⁢ ( R ) for all g ∈ G . Moreover, the association g ↦ [ A g ] determines a partial representation of G on 𝙿𝚒𝚌𝚂 ⁢ ( R ) which induces a partial action γ of G on the center Z ⁢ ( R ) of R. Sufficient conditions for A to be an Azumaya R γ -algebra are presented if R is commutative. In the second part, we study when B is a partial crossed product in the following cases: B = M n ⁡ ( A ) is the ring of matrices with entries in A, or B = END A ( M ) = ⊕ l ∈ G Mor A ( M , M ) l is the direct sum of graded endomorphisms of graded left A-modules M with degree l, or B = END A ⁡ ( M ) where M = A ⊗ R N is the induced module of a left R-module N. Assuming that R is semiperfect, we prove that there exists a subring of A which is an epsilon-strongly graded ring over a subgroup of G and it is graded equivalent to a partial crossed product.

Keywords: Epsilon-strongly graded ring; Azumaya algebra; partial crossed product and graded equivalence
MSC 2020: 16W50; 16W22; 16W55; 13A50

Communicated by Siegfried Echterhoff

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Received: 2022-09-05
Revised: 2023-03-16
Published Online: 2023-06-27
Published in Print: 2023-09-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. On the arithmetic of polynomial semidomains
  3. Extra-special Leibniz superalgebras
  4. Restricted iso-minimum condition
  5. On the spectral large sieve inequality for symmetric-squares
  6. Simple 𝔰𝔩(V)-modules which are free over an abelian subalgebra
  7. Epsilon-strongly graded rings: Azumaya algebras and partial crossed products
  8. Non-weight modules over N = 1 Lie superalgebras of Block type
  9. Simpler foundations for the hyperbolic plane
  10. Characterizations of the mixed radial-angular central Campanato space via the commutators of Hardy type
  11. The fourth moment of Dirichlet L-functions along the critical line
  12. Diagonal restriction of Eisenstein series and Kudla–Millson theta lift
  13. Gibbons’ conjecture for quasilinear elliptic equations involving a gradient term
  14. Jantzen filtration of Weyl modules for general linear supergroups
  15. Integral formulas for a class of curvature PDE’s and applications to isoperimetric inequalities and to symmetry problems
https://doi.org/10.1515/forum-2022-0262
Keywords for this article
Epsilon-strongly graded ring; Azumaya algebra; partial crossed product and graded equivalence

Articles in the same Issue

  1. Frontmatter
  2. On the arithmetic of polynomial semidomains
  3. Extra-special Leibniz superalgebras
  4. Restricted iso-minimum condition
  5. On the spectral large sieve inequality for symmetric-squares
  6. Simple 𝔰𝔩(V)-modules which are free over an abelian subalgebra
  7. Epsilon-strongly graded rings: Azumaya algebras and partial crossed products
  8. Non-weight modules over N = 1 Lie superalgebras of Block type
  9. Simpler foundations for the hyperbolic plane
  10. Characterizations of the mixed radial-angular central Campanato space via the commutators of Hardy type
  11. The fourth moment of Dirichlet L-functions along the critical line
  12. Diagonal restriction of Eisenstein series and Kudla–Millson theta lift
  13. Gibbons’ conjecture for quasilinear elliptic equations involving a gradient term
  14. Jantzen filtration of Weyl modules for general linear supergroups
  15. Integral formulas for a class of curvature PDE’s and applications to isoperimetric inequalities and to symmetry problems
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