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 $i_{\ast }$ along $i:Z\to X$ is an isomorphism, and (b) when $X$ is smooth the inverse $(i_{\ast }{)}^{-1}$ can be described via Gysin pullback $i^{!}$ and cap product with $e(N{)}^{-1}$, 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 $(-)\cap e(N{)}^{-1}$. As a corollary we prove the virtual localization formula $[X{]}^{vir}=i_{\ast }([Z{]}^{vir}\cap e(N^{vir}{)}^{-1})$ 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.