6
$\begingroup$

Let $X$ be a scheme acted upon by $\mathbf{G}_m$ and $K(X)=K_0(\text{Perf}X)$ the Thomason-Trobaugh K theory. Is there a localisation theorem in this context? By this I mean something like $$\text{id}_{K_{\mathbf{G}_m}(X)_{loc}}\ =\ i_*\frac{i^*(-)}{e(N_i)}$$ where $i:\overline{X}\to X$ is the inclusion of the fixed locus and $loc$ means localise the $K_{\mathbf{G}_m}(\text{pt})$ module $K_{\mathbf{G}_m}(X)$, and $e(N_i)$ is the Euler class of the normal bundle to $i$.

In Chriss and Ginzburg there's a localisation theorem for $K_0(\text{Coh}X)$. There are also localisation formulas for $K$, but I'm not sure if that has anything to do with abelian localisation. If there's a result I'd be quite interested in seeing a reference.

Improve this question
$\endgroup$
1
  • $\begingroup$ If $i$ has a normal bundle, the inclusion $X^{\mathbf{G}_m}\hookrightarrow X$ seems to be assumed regular. That may not be generally satisfied. Probably the question is about the situation where $X$ is not smooth? $\endgroup$
    – Matthias Wendt
    Commented Oct 11, 2020 at 8:32

0

Reset to default

You must log in to answer this question.