Relative Hermitian Morita theory. I: Morita equivalences of algebras with involution
The aim is to develop a hermitian Morita theory for arbitrary involutive Grothendieck categories. By the Gabriel-Popescu Theorem this is equivalent to the relative Hermitian Morita theory for algebras with involution, where ``relative'' means ``with respect to a hereditary torsion theory, or equivalently, to an idempotent kernel functor''. More precisely, one considers the notion of a torsion triple \((A,\alpha,\sigma)\) where \(A\) is an algebra over a commutative ring with identity, \(\alpha\) is an involution of \(A\), and \(\sigma\) is an idempotent kernel functor in \(A\)-Mod. Then one introduces the notion of a relative hermitian Morita equivalence by means of so-called involutive relatively invertible bimodules, and finally one proves hermitian versions of Morita I and Morita II. For example, the hermitian Morita I shows that any relative hermitian Morita equivalence produces an equivalence between the corresponding categories of so-called \(\sigma\)-sesquilinear left \(A\)- modules. The paper contains no example, as well as no application of the theory developed here.
involutive relatively invertible bimodules
\(\sigma\)-sesquilinear left \(A\)-modules
scientific article; zbMATH DE number 554201