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→X is an isomorphism, and (b) when X is smooth the inverse (i_⁎)⁻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 (−)∩e(N)⁻¹. As a corollary we prove the virtual localization formula [X]^vir=i_⁎([Z]^vir∩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.