In mathematics, at the intersection of algebraic topology and algebraic geometry, there is the notion of a Hopf algebroid which encodes the information of a presheaf of groupoids whose object sheaf and arrow sheaf are represented by algebras. Because any such presheaf will have an associated site, we can consider quasi-coherent sheaves on the site, giving a topos-theoretic notion of modules. Dually<sup>pg 2</sup>, comodules over a Hopf algebroid are the purely algebraic analogue of this construction, giving a purely algebraic description of quasi-coherent sheaves on a stack: this is one of the first motivations behind the theory.
Given a commutative Hopf-algebroid a left comodule <sup>pg 302</sup> is a left -module together with an -linear map<blockquote></blockquote>which satisfies the following two properties
A right comodule is defined similarly, but instead there is a map<blockquote></blockquote>satisfying analogous axioms.
One of the main structure theorems for comodules<sup>pg 303</sup> is if is a flat -module, then the category of comodules of the Hopf-algebroid is an abelian category.
There is a structure theorem <sup>pg 7</sup> relating comodules of Hopf-algebroids and modules of presheaves of groupoids. If is a Hopf-algebroid, there is an equivalence between the category of comodules and the category of quasi-coherent sheaves for the associated presheaf of groupoids<blockquote></blockquote>to this Hopf-algebroid.
Associated to the Brown-Peterson spectrum is the Hopf-algebroid classifying p-typical formal group laws. Note<blockquote></blockquote>where is the localization of by the prime ideal . If we let denote the ideal<blockquote></blockquote>Since is a primitive in , there is an associated Hopf-algebroid <blockquote></blockquote>There is a structure theorem on the Adams-Novikov spectral sequence relating the Ext-groups of comodules on to Johnson-Wilson homology, giving a more tractable spectral sequence. This happens through an equivalence of categories of comodules of to the category of comodules of <blockquote></blockquote>giving the isomorphism<blockquote></blockquote>assuming and satisfy some technical hypotheses <sup>pg 24</sup>.