Analogues of Gersten's conjecture for singular schemes

TL;DR

This paper proposes and proves analogues of Gersten's conjecture for certain singular schemes, indicating that their algebraic K-theory can be understood via regular parts and infinitesimal thickenings of singularities.

Contribution

It formulates new conjectures for non-regular Noetherian local $Q$-algebras and proves them in various cases, extending the understanding of algebraic K-theory for singular schemes.

Findings

Analogues of Gersten's conjecture are established for specific singular schemes.

K-theory of such rings can be reconstructed from regular locus and infinitesimal neighborhoods.

Results support a combined approach to understanding K-theory in singular settings.

Abstract

We formulate analogues, for Noetherian local $\mathbb Q$-algebras which are not necessarily regular, of the injectivity part of Gersten's conjecture in algebraic $K$-theory, and prove them in various cases. Our results suggest that the algebraic $K$-theory of such a ring should be detected by combining the algebraic $K$-theory of both its regular locus and the infinitesimal thickenings of its singular locus.