Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 0919.16005
Grandjean, F.; Vitale, E.M.
Morita equivalence for regular algebras. (English)
[J] Cah. Topologie Géom. Différ. Catég. 39, No.2, 137-153 (1998). ISSN 0008-0004

Let $R$ be a commutative ring with identity and $A$ be an $R$-algebra, not necessarily with identity. A left $A$-module $M$ is called regular if the canonical map $A\otimes_A M\to M$ is an isomorphism. An $R$-algebra $A$ is said to be regular if it is regular as left (and hence also as right) $A$-module. An $A$-$B$-bimodule is regular if it is regular both as a left $A$- and a right $B$-module. By observing the simple but crucial fact that the category of regular modules over a regular algebra is a colocalization of the category of all modules, the authors are able to generalize the Eilenberg-Watts characterization of coproduct-preserving right exact functors to regular colimit-preserving functors between categories of regular modules over regular algebras. This result, Proposition 1.6, is used to develop a satisfactory Morita theory for regular algebras. As a result, the authors show that the classifying category of the bicategory of regular $R$-algebras and regular bimodules is compact closed. This provides an embedding of the Brauer group of $R$ in the Abelian group of Morita equivalence classes of invertible regular algebras as it is shown in Proposition 2.8.
[Anh Pham Ngoc (Budapest)]

MSC 2000:: *16D90 Module categories (assoc. rings and algebras)
16D20 Bimodules (assoc. rings and algebras)

Keywords: Picard groups; categories of regular modules; regular algebras; coproduct-preserving right exact functors; colimit-preserving functors; Morita theory; bicategories; regular bimodules; Brauer groups; Morita equivalences

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Article

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]