Involutions on sheaves of endomorphisms of locally finitely presented \(\mathscr{O}_X\)-modules
type: Item
instance of: scholarly article
title: Involutions on sheaves of endomorphisms of locally finitely presented \(\mathscr{O}_X\)-modules
zbMATH Open document ID: 1486.16020
author: Richard Ng'ambi
author: Patrice Ntumba
published in: Mediterranean Journal of Mathematics
publication date: 2022-03-21T00:00:00Z
review text: The authors deal with local rings \(R\) and Azumaya \(R\)-algebras \(A\) of finite rank.\par As the first main result, they construct some correspondence between the set of all \(R\)-linear involutions of End\(_R(A)\) (in the case of sheaves, \(O_{\mathrm{Spec}(R)}\)-linear involutions of \(\widetilde{\mathrm{End}_R(A)}\)) and the set of all nonsingular bilinear forms, which are either symmetric or skew-symmetric.\par As the second main result, they show that every locally projective quasi-coherent \(O_X\)-module (where \(X\) is a scheme) of constant rank \(2\) is a commutative \(O_X\)-algebra, endowed with a unique standard involution.\par The last (also the longest and most technical) part of the paper is dedicated to the study of involutions of the first kind on the algebra End\(_{O_X}(E)\), where \((X, O_X)\) is a locally ringed space and and \(E\) is a sheaf (more exactly, either a locally finitely presented \(O_X\)-module or a vector sheaf of finite rank on \(X\)). The authors give a formula for presenting every involution using some sections of another sheaf (an invertible \(O_X\)-module, which is connected with the sheaf \(E\) through some specific isomorphisms).
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 14F22
Mathematics Subject Classification ID: 14F06
Mathematics Subject Classification ID: 18F20
zbMATH DE Number: 7493754
zbMATH Keywords: Azumaya \(R\)-algebra
zbMATH Keywords: Brauer equivalence
zbMATH Keywords: \(R\)-progenerator
zbMATH Keywords: \(O_X\)-module
zbMATH Keywords: involution
zbMATH Keywords: Morita equivalence.
reviewed by: Mart Abel
MaRDI profile type: Publication
full work available at URL: s00009-021-01924-0
OpenAlex ID: W4220749661
cites work: Q5562621
cites work: Algebraic geometry and commutative algebra
cites work: Q5565097
cites work: Q3277493
cites work: Q2967735
cites work: Separable Algebras
cites work: Q5322842
cites work: Categories and Sheaves
cites work: Q3997794
cites work: Q4210545
cites work: Théorie de la descente et algèbres d'Azumaya
cites work: Q4143433
cites work: Q5421346
cites work: Q4393464
cites work: On the commutativity of the Clifford and ``extension of scalars'' functors
cites work: Critères de platitude et de projectivité. Techniques de ''platification'' d'un module. (Criterial of flatness and projectivity. Technics of ''flatification of a module.)
cites work: Separable algebras over commutative rings
DOI: 10.1007/S00009-021-01924-0
Recommended article: On invariants and endomorphism rings of certain local cohomology modules
Recommended article: On the Endomorphism Rings of Local Cohomology Modules
Recommended article: A NOTE ON ENDOMORPHISMS OF LOCAL COHOMOLOGY MODULES
Recommended article: On endomorphism rings of local cohomology modules
Recommended article: On endomorphism rings and dimensions of local cohomology modules
Recommended article: On the structure of the endomorphism ring of a certain local cohomology module
Recommended article: Locally finitely presented categories of sheaves
Recommended article: An invariant for finitely presented \(\mathbb{C} G\)-modules
Recommended article: On the radical of endomorphism rings of local modules
Recommended article: On the finiteness properties of extension and torsion functors of local cohomology modules
description: scientific article; zbMATH DE number 7493754