Involutions on sheaves of endomorphisms of locally finitely presented \(\mathscr{O}_X\)-modules. (English) Zbl 1486.16020
The authors deal with local rings \(R\) and Azumaya \(R\)-algebras \(A\) of finite rank.
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.
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.
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).
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.
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.
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).
Reviewer: Mart Abel (Tartu)
MSC:
| 16H05 | Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.) |
| 16W10 | Rings with involution; Lie, Jordan and other nonassociative structures |
| 14F22 | Brauer groups of schemes |
| 14F06 | Sheaves in algebraic geometry |
| 18F20 | Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) |
Keywords:
Azumaya \(R\)-algebra; Brauer equivalence; \(R\)-progenerator; \(O_X\)-module; involution; Morita equivalence.References:
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