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

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

which satisfies the following two properties

(counitary) $(\varepsilon \otimes Id_{M})\circ \psi =Id_{M}$
(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.

Structure theorems

Flatness of Γ gives an abelian category

One of the main structure theorems for comodules^[2]^{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^[1] ^{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.

Examples

From BP-homology

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

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

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^[1] ^{pg 24}.

References

^ ^a ^b ^c Hovey, Mark (2001-05-16). "Morita theory for Hopf algebroids and presheaves of groupoids". arXiv:math/0105137.
^ ^a ^b Ravenel, Douglas C. (1986). Complex cobordism and stable homotopy groups of spheres. Orlando: Academic Press. ISBN 978-0-08-087440-1. OCLC 316566772.

[:0-1] Hovey, Mark (2001-05-16). "Morita theory for Hopf algebroids and presheaves of groupoids". arXiv:math/0105137.

[Ravenel-2] Ravenel, Douglas C. (1986). Complex cobordism and stable homotopy groups of spheres. Orlando: Academic Press. ISBN 978-0-08-087440-1. OCLC 316566772.

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