Virtual localization revisited

Let T be a split torus acting on an algebraic scheme X with fixed locus Z. Edidin and Graham showed that on localized T-equivariant Chow groups, (a) push-forward $i_{⁎}$ along $i:Z\to X$ is an isomorphism, and (b) when X is smooth the inverse ${\left(i_{⁎}\right)}^{-1}$ can be described via Gysin pullback $i^{!}$ and cap product with $e{\left(N\right)}^{-1}$, the inverse of the Euler class of the normal bundle N. In this paper we show that (b) still holds when X is a quasi-smooth derived scheme (or Deligne–Mumford stack), using virtual versions of the operations $i^{!}$ and $(-)\cap e{\left(N\right)}^{-1}$. As a corollary we prove the virtual localization formula ${\left[X\right]}^{\mathrm{vir}}=i_{⁎}({\left[Z\right]}^{\mathrm{vir}}\cap e{\left(N^{\mathrm{vir}}\right)}^{-1})$ of Graber–Pandharipande without global resolution hypotheses and over arbitrary base fields. We include an appendix on fixed loci of group actions on (derived) stacks which should be of independent interest.