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^{[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.

Definition

Given a commutative Hopf-algebroid ${\displaystyle (A,\Gamma )}$ a left comodule ${\displaystyle M}$^{[2]pg 302} is a left ${\displaystyle A}$-module ${\displaystyle M}$ together with an ${\displaystyle A}$-linear map

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

which satisfies the following two properties

1. (counitary) ${\displaystyle (\varepsilon \otimes Id_{M})\circ \psi =Id_{M}}$
2. (coassociative) ${\displaystyle (\Delta \otimes Id_{M})\circ \psi =(Id_{\Gamma }\otimes \psi )\circ \psi }$

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

${\displaystyle \phi :M\to M\otimes _{A}\Gamma }$

satisfying analogous axioms.

Structure theorems

Flatness of Γ gives an abelian category

One of the main structure theorems for comodules^{[2]pg 303} is if ${\displaystyle \Gamma }$ is a flat ${\displaystyle A}$-module, then the category of comodules ${\displaystyle {\text{Comod}}(A,\Gamma )}$ of the Hopf-algebroid is an abelian category.

Relation to stacks

There is a structure theorem^{[1] pg 7} relating comodules of Hopf-algebroids and modules of presheaves of groupoids. If ${\displaystyle (A,\Gamma )}$ is a Hopf-algebroid, there is an equivalence between the category of comodules ${\displaystyle {\text{Comod}}(A,\Gamma )}$ and the category of quasi-coherent sheaves ${\displaystyle {\text{QCoh}}({\text{Spec}}(A),{\text{Spec}}(\Gamma ))}$ for the associated presheaf of groupoids

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

to this Hopf-algebroid.

Examples

From BP-homology

Associated to the Brown-Peterson spectrum is the Hopf-algebroid ${\displaystyle (BP_{*},BP_{*}(BP))}$ classifying p-typical formal group laws. Note

${\displaystyle {\begin{aligned}BP_{*}&=\mathbb {Z} _{(p)}[v_{1},v_{2},\ldots ]\\BP_{*}(BP)&=BP_{*}[t_{1},t_{2},\ldots ]\end{aligned}}}$

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

${\displaystyle I_{n}=(p,v_{1},\ldots ,v_{n-1})}$

Since ${\displaystyle v_{n}}$ is a primitive in ${\displaystyle BP_{*}/I_{n}}$, there is an associated Hopf-algebroid ${\displaystyle (A,\Gamma )}$

${\displaystyle (v_{n}^{-1}BP_{*}/I_{n},v_{n}^{-1}BP_{*}(BP)/I_{n})}$

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

${\displaystyle (v_{n}^{-1}E(m)_{*}/I_{n},v_{n}^{-1}E(m)_{*}(E(m)/I_{n})}$

giving the isomorphism

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

assuming ${\displaystyle M}$ and ${\displaystyle N}$ satisfy some technical hypotheses^{[1] pg 24}.

See also

• Adams spectral sequence
• Steenrod algebra