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Talk

April 21, 2026
Georg Lehner (Universität Münster): Algebraic K-theory of stably locally compact spaces

Abstract

Recent advances by Efimov, Lurie, Clausen, Nikolaus, and others have made it possible to extend algebraic K-theory, originally defined for small stable categories, to the broader class of so-called dualizable categories. These include categories of sheaves on locally compact Hausdorff spaces, which play a role analogous to that of algebras of continuous functions in the setting of C*-algebras. This perspective enables a transfer of techniques between operator-theoretic K-theory and algebraic K-theory and provides a promising new framework for understanding assembly conjectures such as the Farrell–Jones and Baum–Connes conjectures.

There is a larger class of spaces than that of locally compact Hausdorff spaces for which the associated categories of sheaves remain dualizable: The class of stably locally compact spaces, which includes locally compact Hausdorff spaces, as well as spectra of rings. The category of stably locally compact spaces is particularly well suited to analysis: analogues of Tychonoff’s theorem and the Urysohn lemma hold in this setting. We give a formula for the algebraic K-theory of stably locally compact spaces that not only generalizes the corresponding formula for locally compact Hausdorff spaces, but also recovers the additivity theorem and the vanishing of K-theory for sheaves with singular support as special cases.

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