# Localizations over the Steenrod algebra The lost chapter

DOI: 10.1007/s002090000155

- Cite this article as:
- Neusel, M. Math Z (2000) 235: 353. doi:10.1007/s002090000155

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## Abstract.

Let H\(^*\) be an unstable algebra over the Steenrod algebra, and let \(S\subset{\rm H^*}\) be a multiplicatively closed subset. inherits an action of the Steenrod algebra from H\(^*\), which is, however, in general no longer unstable. In this note we consider the following three statements.

(1)] H\(^*\) is Noetherian,

(2) the integral closure, \(\overline{{\rm H^*}_{S^{-1}{\rm H^*}}}\), of H\(^*\) in the localization with respect to *S* is Noetherian,

(3) \(\overline{{\rm H^*}_{S^{-1}{\rm H^*}}}={\cal Un}(S^{-1}{\rm H^*})\), where \({\cal Un}(-)\) denotes the unstable part.

If the set *S* contains only (nonzero) nonzero divisors and the algebras are reduced then

\(\)

If *S* contains zerodivisors, then only \((1) \Rightarrow (2)\) remains true, to show the converse is false we construct a counterexample. The implication \((1) \Rightarrow (3)\) is always true, while \((3) \Rightarrow (2)\) needs a bunch of technical assumptions to remain true. However, none of them can be removed: we illustrate this also with examples. Finally, as a technical tool, we characterize \(\Delta\)-finite algebras.