The equivariant Atiyah class

Let X be a complex scheme acted on by an affine algebraic group G. We prove that the Atiyah class of a G-equivariant perfect complex on X, as constructed by Huybrechts and Thomas, is G-equivariant in a precise sense. As an application, we show that, if G is reductive, the obstruction theory on the fine relative moduli space M → B of simple perfect complexes on a G-invariant smooth projective family Y → B is Gequivariant. The results contained here are meant to suggest how to check the equivariance of the natural obstruction theories on a wide variety of moduli spaces equipped with a torus action, arising for instance in Donaldson-Thomas theory and Vafa-Witten theory.