Involutions on sheaves of endomorphisms of modules over ringed spaces

Abstract

The study of Azumaya algebras over schemes has had a comparatively formidable reputation in algebraic geometry over the past decades. In this thesis, we provide in the sheaf-theoretic setting counterparts of results pertaining to involutions of the first kind on algebras of endomorphisms of faithfully projective -modules, where is a commutative ring. More precisely, let be a locally finitely presented module over an affine scheme X, and let be an involution of the first kind on . Then, there exists an invertible module over the ringed space such that . Moreover, given a vector sheaf of finite rank on a locally ringed space and involution of the first kind on and an invertible -module such that , then σ locally will depend on an invertible section of .