Mathematische Zeitschrift

, Volume 235, Issue 2, pp 353–378

Localizations over the Steenrod algebra The lost chapter

The lost chapter
  • Mara D. Neusel
Original article

DOI: 10.1007/s002090000155

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


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.

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Mara D. Neusel
    • 1
  1. 1.Yale University, Department of Mathematics, 10 Hillhouse Avenue, P.O. Box 208283, New Haven, CT 06520–8283, USA (e-mail:

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