Decomoposition of module and comodule categories
by
Robert Wisbauer
Düsseldorf

Let A be an associative algebra over a commutative ring R. For an A-module M we denote by \sigma[M] the category of those A-modules which are submodules of M-generated modules. We define a \sigma-decomposition

\sigma[M] = Å \Lambda  \sigma[N\lambda],

for a family { N_\lambda}_\Lambda of modules in \sigma[M], meaning that for every module L in \sigma[M], L=\oplus_\Lambda L_\lambda, where L_\lambda in \sigma[N_\lambda]. We call \sigma[M] \sigma-indecomposable if no such non-trivial decomposition exists.

As a special case of the situation described we mention

\sigma[Q/Z] = Å p   prime  \sigma[Zp\infty],

the decomposition of the category of torsion abelian groups as a direct sum of the categories of p-groups,

It is obvious that any \sigma-decomposition of M is also a fully invariant decomposition. The reverse implication holds in case M is a projective generator or an injective cogenerator in \sigmaM. Hence the following observation is of use in this context:

Lemma. Let M have a decomposition which complements direct summands. Then M has a fully invariant decomposition, where the fully invariant submodules do not decompose non-trivially into fully invariant submodules.

It is well-known that locally noetherian self-injective modules, and modules M which are perfect and projective in M, have decompositions which complement direct summands.

\sigma-decomposition for locally noetherian modules

Let M be a locally noetherian A-module. Then M has a \sigma-decomposition M=\oplus_\Lambda M_\lambda, where each M_\lambda is \sigma-indecomposable.

A similar result holds for categories with a (semi-)perfect projective generator. In particular, every semiperfect ring A has a \sigma-decomposition A=Ae₁\oplus ... \oplusAe_k, where the e_i are central idempotents of A which are not a non-trivial sum of central idempotents.

Now let C be an R-coalgebra which is projective as R-module. Then the category of right C-comodules can be identified with CC and the above theorem yields:

\sigma-decomposition of coalgebras

Let C be a coalgebra over a noetherian ring R with C_R projective.

There exist a \sigma-decomposition C=\oplus_\Lambda C_\lambda, i.e., CC = \oplus_\Lambda \sigma[_C^* C_\lambda]. Each C_\lambda is a sub-coalgebra of C, and \sigma[_C^*C_\lambda]=\sigma[_C\lambda^*C_\lambda].

Corollary. Let R be a QF ring and C an R-coalgebra with C_R projective.

C has fully invariant decompositions with \sigma-indecomposable summands.

Each fully invariant decomposition is a \sigma-decomposition.

C is \sigma-indecomposable if and only if C has no non-trivial fully invariant decomposition. If C is cocommutative then C = \oplus_\Lambda \wE_\lambda is a fully invariant decomposition, where { E_\lambda }_\Lambda is a minimal representing set of simple C-comodules.

Date received: January 22, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cabw-53.