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[1]pg 2, 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
[2]pg 302 is a left
which satisfies the following two properties
A right comodule is defined similarly, but instead there is a map
satisfying analogous axioms.
One of the main structure theorems for comodules[2]pg 303 is if
-module, then the category of comodules
of the Hopf-algebroid is an abelian category.
There is a structure theorem[1] pg 7 relating comodules of Hopf-algebroids and modules of presheaves of groupoids.
is a Hopf-algebroid, there is an equivalence between the category of comodules
and the category of quasi-coherent sheaves
Spec
{\displaystyle {\text{QCoh}}({\text{Spec}}(A),{\text{Spec}}(\Gamma ))}
for the associated presheaf of groupoids
Associated to the Brown-Peterson spectrum is the Hopf-algebroid
classifying p-typical formal group laws.
{\displaystyle {\begin{aligned}BP_{*}&=\mathbb {Z} _{(p)}[v_{1},v_{2},\ldots ]\\BP_{*}(BP)&=BP_{*}[t_{1},t_{2},\ldots ]\end{aligned}}}
by the prime ideal
denote the ideal
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
giving the isomorphism
{\displaystyle {\text{Ext}}_{BP_{*}BP}^{*,*}(M,N)\cong {\text{Ext}}_{E(m)_{*}E(m)}^{*,*}(E(m)_{*}\otimes _{BP_{*}}M,E(m)_{*}\otimes _{BP_{*}}N)}
satisfy some technical hypotheses[1] pg 24.