On the stack of 0-dimensional coherent sheaves: motivic aspects

Let X be a variety. We study (decompositions of) the motivic class, in the Grothendieck ring of stacks, of the stack Cohⁿ(X) of $0$-dimensional coherent sheaves of length n on X. To do so, we review the construction of the support map Cohⁿ(X) → Symⁿ(X) to the symmetric product and we prove that, for any closed point p ∈ X, the punctual stack Cohⁿ(X)ₚ parametrising sheaves supported at p only depends on a formal neighbourhood of p. We perform the same analysis for the Quot-to-Chow morphism QuotX( E,n) → Symⁿ(X), for a fixed sheaf E ∈ Coh(X).