Finiteness properties of injective resolutions of certain unstable modules over the Steenrod algebra and applications
type: Item
instance of: scholarly article
title: Finiteness properties of injective resolutions of certain unstable modules over the Steenrod algebra and applications
zbMATH Open document ID: 0751.55016
DOI: 10.1007/BF01445199
author: Hans-Werner Henn
published in: Mathematische Annalen
publication date: 1992-09-27T00:00:00Z
full work available at URL: https://eudml.org/doc/164858
review text: Let \(K\) be an unstable noetherian algebra over the Steenrod algebra \(A\), and let \(M\) be an unstable module over \(A\) and \(K\). The main technical result says that \(M\) has an injective resolution in the category of unstable \(A\)-modules such that each injective object in this resolution is a finite direct sum of modules of the form \(H^*V\otimes J(n)\). Here \(V\) is an elementary abelian \(p\)-group, and \(J(n)\) a dual Brown-Gitler module. Two nice applications of this result are Krull-Schmidt type results. One says that if \(K\) and \(M\) are as above with \(M\) finitely generated as a \(K\)-module, then \(M\) can be written uniquely as a finite direct sum of indecomposable unstable \(A\)-modules. The other says that if \(K\) is as above and \(H^*X\) is a finitely generated \(K\)-module, then in the stable category the \(p\)-completion of \(X\) is equivalent to a unique finite wedge of wedge-indecomposable objects.
Mathematics Subject Classification ID: 55S10
Mathematics Subject Classification ID: 55P15
zbMATH DE Number: 59955
zbMATH Keywords: unstable noetherian algebra
zbMATH Keywords: Steenrod algebra
zbMATH Keywords: unstable module
zbMATH Keywords: injective resolution
reviewed by: Donald M. Davis
MaRDI profile type: Publication
Wikidata QID: Q114233962
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description: scientific article; zbMATH DE number 59955