Comodule over a Hopf algebroid

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 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.

Definition
Given a commutative Hopf-algebroid $$(A,\Gamma)$$ a left comodule $$M$$ pg 302 is a left $$A$$-module $$M$$ together with an $$A$$-linear map"$\psi: M \to \Gamma\otimes_AM$"which satisfies the following two properties


 * 1) (counitary) $$(\varepsilon\otimes Id_M)\circ \psi = Id_M$$
 * 2) (coassociative) $$(\Delta\otimes Id_M) \circ \psi = (Id_\Gamma \otimes \psi) \circ \psi$$

A right comodule is defined similarly, but instead there is a map"$\phi: M \to M \otimes_A \Gamma$"satisfying analogous axioms.

Flatness of Γ gives an abelian category
One of the main structure theorems for comodules pg 303 is if $$\Gamma$$ is a flat $$A$$-module, then the category of comodules $$\text{Comod}(A,\Gamma)$$ of the Hopf-algebroid is an abelian category.

Relation to stacks
There is a structure theorem pg 7 relating comodules of Hopf-algebroids and modules of presheaves of groupoids. If $$(A,\Gamma)$$ is a Hopf-algebroid, there is an equivalence between the category of comodules $$\text{Comod}(A,\Gamma)$$ and the category of quasi-coherent sheaves $$\text{QCoh}(\text{Spec}(A),\text{Spec}(\Gamma))$$ for the associated presheaf of groupoids"$\text{Spec}(\Gamma)\rightrightarrows \text{Spec}(A)$"to this Hopf-algebroid.

From BP-homology
Associated to the Brown-Peterson spectrum is the Hopf-algebroid $$(BP_*,BP_*(BP))$$ classifying p-typical formal group laws. Note $$\begin{align} BP_* &= \mathbb{Z}_{(p)}[v_1,v_2,\ldots] \\ BP_*(BP) &= BP_*[t_1,t_2,\ldots] \end{align}$$ where $$\mathbb{Z}_{(p)}$$ is the localization of $$\mathbb{Z}$$ by the prime ideal $$(p)$$. If we let $$I_n$$ denote the ideal"$I_n = (p,v_1,\ldots, v_{n-1})$"Since $$v_n$$ is a primitive in $$BP_*/I_n$$, there is an associated Hopf-algebroid $$(A,\Gamma)$$"$(v_n^{-1}BP_*/I_n, v_n^{-1}BP_*(BP)/I_n)$"There is a structure theorem on the Adams-Novikov spectral sequence relating the Ext-groups of comodules on $$(BP_*,BP_*(BP))$$ to Johnson-Wilson homology, giving a more tractable spectral sequence. This happens through an equivalence of categories of comodules of $$(A,\Gamma)$$ to the category of comodules of "$(v_n^{-1}E(m)_*/I_n, v_n^{-1}E(m)_*(E(m)/I_n)$"giving the isomorphism $$\text{Ext}^{*,*}_{BP_*BP}(M,N) \cong \text{Ext}^{*,*}_{E(m)_*E(m)}(E(m)_*\otimes_{BP_*} M,E(m)_*\otimes_{BP_*}N)$$ assuming $$M$$ and $$N$$ satisfy some technical hypotheses pg 24.