Involutions on sheaves of endomorphisms of locally finitely presented \(\mathscr{O}_X\)-modules (Q2117369)

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).