Descent for quasi-coherent sheaves on stacks

Abstract

We give a homotopy theoretic characterization of sheaves on a stack and, more generally, a presheaf of groupoids on an arbitary small site ᑝ. We use this to prove homotopy invariance and generalized descent statements for categories of sheaves and quasi-coherent sheaves. As a corollary we obtain an alternate proof of a generalized change of rings theorem of Hovey.

Geometric criteria for Landweber exactness

“…A consequence of Theorem 1.2 is that the category of comodules over (B, Γ B ) is a function of the orbit of Spec B in Spec A. Theorem 1.2 also implies that the category of comodules over (B, Γ B ) is a localization of the category of (A, Γ)-comodules [13,Theorem A]. This is analogous to the fact that when U ⊂ X is an intersection of open subschemes, the category of quasi-coherent sheaves on U is a localization of the category of quasi-coherent sheaves on X, and follows from the description of quasi-coherent sheaves on a stack via descent in [8]. This result is described in Subsection 3.3.…”

“…It is an observation of Laures (see Remark 3.8 The importance of Hopf algebroids in Algebraic Topology stems from the fact that the main computational tool for stable homotopy, namely the Adams Spectral Sequence, takes as input Ext over the categories of comodules. The category of comodules over (A, Γ) is equivalent to the category of quasi-coherent sheaves on M A,Γ and depends only on the weak homotopy type of the presheaf (Spec A, Spec Γ) (see [8,Proposition 5.15] or [12]). The following result classifies the homotopy types of stacks arising from a Landweber exact map of rings.…”

Diagrams indexed by Grothendieck constructions

“…1 Covers in C/M are the collections of morphisms which forget to covers in C. The site C/M makes sense for any presheaf of groupoids M (or even a presheaf of categories) on a site C (see [H2,Section 2.1]). If M is represented by an object X ∈ C, then the site described above is just the usual topology on the over category C/X.…”

On affine morphisms of Hopf algebroids

“…Foundational work has been carried out on a category of algebraic stacks suitable for the application to stable homotopy theory by Hopkins and Miller [8], Goerss [3], Pribble in his thesis [12], Naumann [11] and by others. There is related foundational work on stacks from the homotopy theoretic or derived viewpoint by Hollander in [6,5,4]. The algebraic stacks which are considered differ from the usual notion from algebraic geometry, in that there are no finiteness assumptions.…”

“…The arguments considering affine morphisms are explicit (constructive), without appeal to a general faithfully flat descent argument. The treatment of descent data given here is related to the homotopy theoretic approach to stacks, as in the work of Hollander [6,5,4].…”