Finiteness properties of injective resolutions of certain unstable modules over the Steenrod algebra and applications
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.
scientific article; zbMATH DE number 59955