##
**Injective objects in categories of unstable \(K\)-modules.**
*(English)*
Zbl 0906.55012

Heidelberg: Univ. Heidelberg, Naturwissenschaftlich-Mathematische Gesamtfakultät, iv, 125 p. (1998).

Much recent work in algebraic topology has highlighted the importance of studying the category \(\mathcal U\) of unstable modules over the Steenrod algebra. This is an abelian category, so homological algebra plays a significant rôle in its study, for which one needs to understand the projective and injective unstable modules. The projectives are easily dealt with but the injectives require more effort. Nevertheless they have been classified by J. Lannes and L. Schwartz [Topology 28, No. 2, 153-169 (1989; Zbl 0683.55016)].

In the thesis under review, the author extends this classification to the categories \(K\)-\({\mathcal U}\) of “unstable \(K\)-modules”. Here \(K\) is an unstable algebra (which the author requires to be Noetherian for the classification theorem) and an object in \(K\)-\({\mathcal U}\) is an unstable module which is also a \(K\)-module in a compatible way. The motivating example is as follows: suppose a topological group \(G\) acts on a space \(X\), then the equivariant cohomology \(H_G^*(X) = H^*(EG \times_G X)\) becomes an \(H^*BG\)-module via the map \(EG \times_G X \rightarrow BG\) collapsing \(X\) to a point. Thus \(H_G^*(X)\) gives an object in \(K\)-\({\mathcal U}\) where \(K=H^*BG\). If \(G\) is finite or a compact Lie group, then \(H^*BG\) is Noetherian so the classification applies.

When \(K\) is Noetherian, \(K\)-\({\mathcal U}\) is locally Noetherian and it suffices to consider the indecomposable injectives. The classification of these is, as one would expect, rather more complicated than that for \(\mathcal U\). Moreover, the indecomposable injectives are not given explicitly, but only as the injective envelopes of certain other (more tractable) objects. Since injective envelopes are often difficult to identify, the author gives a number of examples for \(K\) where they can be described and the indecomposables can be given in a more explicit form. Finally, she details some applications of the classification, in particular it allows the extension to \(K\)-\({\mathcal U}\) of many of the results of H.-W. Henn [J. Reine Angew. Math. 478, 189-214 (1996; Zbl 0858.55015)] previously only available for the subcategory \(K_{fg}\)-\({\mathcal U}\) of objects which are finitely generated as \(K\)-modules.

A revised version of this thesis has been published in the Bonner Mathematische Schriften, volume 316 (ISSN 0524-045X).

In the thesis under review, the author extends this classification to the categories \(K\)-\({\mathcal U}\) of “unstable \(K\)-modules”. Here \(K\) is an unstable algebra (which the author requires to be Noetherian for the classification theorem) and an object in \(K\)-\({\mathcal U}\) is an unstable module which is also a \(K\)-module in a compatible way. The motivating example is as follows: suppose a topological group \(G\) acts on a space \(X\), then the equivariant cohomology \(H_G^*(X) = H^*(EG \times_G X)\) becomes an \(H^*BG\)-module via the map \(EG \times_G X \rightarrow BG\) collapsing \(X\) to a point. Thus \(H_G^*(X)\) gives an object in \(K\)-\({\mathcal U}\) where \(K=H^*BG\). If \(G\) is finite or a compact Lie group, then \(H^*BG\) is Noetherian so the classification applies.

When \(K\) is Noetherian, \(K\)-\({\mathcal U}\) is locally Noetherian and it suffices to consider the indecomposable injectives. The classification of these is, as one would expect, rather more complicated than that for \(\mathcal U\). Moreover, the indecomposable injectives are not given explicitly, but only as the injective envelopes of certain other (more tractable) objects. Since injective envelopes are often difficult to identify, the author gives a number of examples for \(K\) where they can be described and the indecomposables can be given in a more explicit form. Finally, she details some applications of the classification, in particular it allows the extension to \(K\)-\({\mathcal U}\) of many of the results of H.-W. Henn [J. Reine Angew. Math. 478, 189-214 (1996; Zbl 0858.55015)] previously only available for the subcategory \(K_{fg}\)-\({\mathcal U}\) of objects which are finitely generated as \(K\)-modules.

A revised version of this thesis has been published in the Bonner Mathematische Schriften, volume 316 (ISSN 0524-045X).

Reviewer: Martin D.Crossley (Villetaneuse)

### MSC:

55S10 | Steenrod algebra |