Morita theory for Hopf algebroids and presheaves of groupoids

Abstract

Abstract. Comodules over Hopf algebroids are of central importance in algebraic topology. It is well-known that a Hopf algebroid is the same thing as a presheaf of groupoids on Aff , the opposite category of commutative rings. We show in this paper that a comodule is the same thing as a quasi-coherent sheaf over this presheaf of groupoids. We prove the general theorem that internal equivalences of presheaves of groupoids with respect to a Grothendieck topology T on Aff give rise to equivalences of categories of sheaves in that topology. We then show using faithfully flat descent that an internal equivalence in the flat topology gives rise to an equivalence of categories of quasicoherent sheaves. The corresponding statement for Hopf algebroids is that weakly equivalent Hopf algebroids have equivalent categories of comodules. We apply this to formal group laws, where we get considerable generalizations of the Miller-Ravenel [MR77] and Hovey-Sadofsky [HS99] change of rings theorems in algebraic topology.

Descent for quasi-coherent sheaves on stacks

“…Part of the results here are bringing those of Hovey [13] into the homotopy theoretic framework of [11]. In [13], Hovey defines quasi-coherent sheaves on a presheaf of groupoids on ᑛff flat and prove a generalized change of rings theorem.…”

“…In [13], Hovey defines quasi-coherent sheaves on a presheaf of groupoids on ᑛff flat and prove a generalized change of rings theorem. It is a consequence of Corollary 4-6 and Lemma 5-11 that our definition of (quasi-coherent) sheaves agrees with [13, Definitions 1.1 and 1.2].…”

Morita Base Change in Hopf-Cyclic (Co)Homology

“…We denote the respective categories by U Mod and U o Mod; while U Mod is a monoidal category, U o Mod in general is not [31]. One has a forgetful functor U Mod → R e Mod using which we consider every left U -module N also as an (R, R)-bimodule with actions 17) and in both cases we usually prefer to express these actions just by juxtaposition if no ambiguity is to be expected.…”

“…The difficulty comes out when, for instance, one tries to understand how the notion of Morita equivalence between two Lie algebroids, in the sense of [10,15] and others, can be reflected to their respective associated (universal) left Hopf algebroids in such a way that invariant properties, especially homological ones, between equivalent Lie algebroids remain invariant at the level of left Hopf algebroids. In the commutative case, that is, for commutative Hopf algebroids, several notions already exist in the literature, see, e.g., [17,18].…”

The 5–local homotopy ofeo4

“…The method of faithfully flat descent allows us to reduce to an equivalent but slightly smaller Hopf algebroid [7]. The geometric underpinning is fairly simple.…”