Morita theory for Hopf algebroids and presheaves of groupoids
Mark Hovey
Wesleyan University
Middletown, CT
mhovey@wesleyan.edu
5/17/01
AMS classification nos: 14L05, 14L15, 16W30, 18F20, 18G15, 55N22
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 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 quasi-coherent
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 change of rings theorems in
algebraic topology.