De Rham Theory for Tame Stacks and Schemes with Linearly Reductive Singularities

Abstract

We prove that the Hodge-de Rham spectral sequence for smooth proper tame Artin stacks in characteristic p (as defined by Abramovich, Olsson, and Vistoli) which lift mod p degenerates. We push the result to the coarse spaces of such stacks, thereby obtaining a degeneracy result for schemes which are etale locally the quotient of a smooth scheme by a finite linearly reductive group scheme. Given a scheme X smooth and proper over a field k, the cohomology of the algebraic de Rham complex ΩX/k is an important invariant of X, which, when k = C, recovers the singular cohomology of X(C). When the Hodge-de Rham spectral sequence E 1 = H (ΩX/k)⇒ H (ΩX/k) degenerates, the invariants dimkH (ΩX/k) break up into sums of the finer invariants dimkH (ΩX/k). The degeneracy of this spectral sequence for smooth proper schemes in characteristic 0 was first proved via analytic methods. It was not until much later that Faltings [Fa] gave a purely algebraic proof by means of p-adic Hodge Theory. Soon afterwards, Deligne and Illusie [DI] gave a substantially simpler algebraic proof by showing that the degeneracy of the Hodge-de Rham spectral sequence in characteristic 0 is implied by its degeneracy for smooth proper schemes in characteristic p that lift mod p2. Their method therefore extends de Rham Theory to the class of smooth proper schemes in positive characteristic which lift. A version of de Rham Theory also exists for certain singular schemes. Steenbrink showed [St, Thm 1.12] that if k is a field of characteristic 0, M a proper k-scheme with quotient singularities, and j : M0 ↪→ M its smooth locus, then the hypercohomology spectral sequence E 1 = H (j∗Ω s M0/k)⇒ H (j∗Ω • M0/k) degenerates and H(j∗Ω • M0/k) agrees with H n(M(C),C) when k = C. As we explain in this paper, a version of this theorem is true in positive characteristic as well: if k has characteristic p and M is proper with quotient singularities by groups whose orders are prime to p, then the above spectral sequence degenerates for s+ t < p provided a certain liftability criterion is satisfied (see Theorem 1.15 for precise hypotheses). As a warm-up for the rest of the paper, we begin by showing how Steenbrink’s result can be reproved using the theory of stacks. The idea is as follows. Every scheme M as above is the coarse space of a smooth Deligne-Mumford stack X whose stacky structure is supported at the singular locus of M . We show that the de Rham cohomology H(ΩX/k) of the stack agrees with H(j∗Ω • M0/k). After checking that the method of Deligne-Illusie extends to Deligne-Mumford stacks, we recover Steenbrink’s result as a consequence of the degeneracy of the Hodge-de Rham spectral sequence for X. The above extends de Rham Theory to the class of schemes with quotient singularities by groups whose orders are prime to the characteristic, but in positive characteristic this class of schemes contains certain “gaps” and it is natural to ask if de Rham Theory can be 2010 Mathematics Subject Classification. 14A20, 14F40.