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

$(B{P}_{*},B{P}_{*}(BP))$

classifying p-typical formal group laws. Note

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

where

${\mathbb{\mathbb{Z}}}_{(p)}$

is the localization of

$\mathbb{\mathbb{Z}}$

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

$B{P}_{*}/I_n$

, there is an associated Hopf-algebroid

$(A,\mathrm{\Gamma })$

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

There is a structure theorem on the Adams-Novikov spectral sequence relating the Ext-groups of comodules on

$(B{P}_{*},B{P}_{*}(BP))$

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

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

giving the isomorphism

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

assuming

$M$

and

$N$

satisfy some technical hypotheses^{pg 24}.

See also

• Adams spectral sequence
• Steenrod algebra