The intrinsic stable normal cone

Abstract

We construct an analog of the intrinsic normal cone of Behrend-Fantechi in the equivariant motivic stable homotopy category over a base-scheme B and construct a fundament class in E-cohomology for any cohomology theory E in SH(B). For affine B, a perfect obstruction theory gives rise to a virtual fundamental class in a twisted Borel-Moore E-homology for arbitrary E. This includes motivic cohomology (homotopy invariant) K-theory algebraic cobordism and the oriented Chow groups of Barge-Morel and Fasel. In the case of motivic cohomology, we recover the constructions of Behrend-Fantechi, with values in the Chow group.