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 $(A,\mathrm{\Gamma })$ a left comodule $M$^{pg 302} is a left $A$-module $M$ together with an $A$-linear map

$\psi :M\to \mathrm{\Gamma }{\otimes }_{A}M$

which satisfies the following two properties

1. (counitary) $(\epsilon \otimes I{d}_M)\circ \psi =I{d}_M$
2. (coassociative) $(\mathrm{\Delta }\otimes I{d}_M)\circ \psi =(I{d}_{\mathrm{\Gamma }}\otimes \psi )\circ \psi$

A right comodule is defined similarly, but instead there is a map

$\varphi :M\to M{\otimes }_A\mathrm{\Gamma }$

satisfying analogous axioms.

Structure theorems

Flatness of Γ gives an abelian category

One of the main structure theorems for comodules^{pg 303} is if $\mathrm{\Gamma }$ is a flat $A$-module, then the category of comodules $\text{Comod}(A,\mathrm{\Gamma })$ 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 $(A,\mathrm{\Gamma })$ is a Hopf-algebroid, there is an equivalence between the category of comodules $\text{Comod}(A,\mathrm{\Gamma })$ and the category of quasi-coherent sheaves $\text{QCoh}(\text{Spec}(A),\text{Spec}(\mathrm{\Gamma }))$ for the associated presheaf of groupoids

$\text{Spec}(\mathrm{\Gamma })\rightrightarrows \text{Spec}(A)$

to this Hopf-algebroid.

Examples

From BP-homology

Associated to the Brown-Peterson spectrum is the Hopf-algebroid $(B{P}_{\ast },B{P}_{\ast }(BP))$ classifying p-typical formal group laws. Note

$\begin{array}{cc}B{P}_{\ast }& ={\mathbb{Z}}_{(p)}[v_1,v_2,\dots ]\\ B{P}_{\ast }(BP)& =B{P}_{\ast }[t_1,t_2,\dots ]\end{array}$

where ${\mathbb{Z}}_{(p)}$ is the localization of $\mathbb{Z}$ by the prime ideal $(p)$. If we let $I_n$ denote the ideal

$I_n=(p,v_1,\dots ,v_{n-1})$

Since $v_n$ is a primitive in $B{P}_{\ast }/I_n$, there is an associated Hopf-algebroid $(A,\mathrm{\Gamma })$

$(v_n^{-1}B{P}_{\ast }/I_n,v_n^{-1}B{P}_{\ast }(BP)/I_n)$

There is a structure theorem on the Adams-Novikov spectral sequence relating the Ext-groups of comodules on $(B{P}_{\ast },B{P}_{\ast }(BP))$ to Johnson-Wilson homology, giving a more tractable spectral sequence. This happens through an equivalence of categories of comodules of $(A,\mathrm{\Gamma })$ to the category of comodules of

$(v_n^{-1}E(m{)}_{\ast }/I_n,v_n^{-1}E(m{)}_{\ast }(E(m)/I_n)$

giving the isomorphism

${\text{Ext}}_{B{P}_{\ast }BP}^{\ast ,\ast }(M,N)\cong {\text{Ext}}_{E(m{)}_{\ast }E(m)}^{\ast ,\ast }(E(m{)}_{\ast }{\otimes }_{B{P}_{\ast }}M,E(m{)}_{\ast }{\otimes }_{B{P}_{\ast }}N)$

assuming $M$ and $N$ satisfy some technical hypotheses ^{pg 24}.

See also

• Adams spectral sequence
• Steenrod algebra