View source for 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 [[Sheaf (mathematics)|presheaf]] of [[groupoid]]s 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 [[Module (mathematics)|modules]]. Dually<ref name=":0">{{cite arXiv|last=Hovey|first=Mark|date=2001-05-16|title=Morita theory for Hopf algebroids and presheaves of groupoids|eprint=math/0105137}}</ref><sup>pg 2</sup>, 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 <math>(A,\Gamma)</math> a '''left''' '''comodule''' <math>M</math><ref name="Ravenel">{{Cite book|last=Ravenel|first=Douglas C.|url=https://web.math.rochester.edu/people/faculty/doug/mu.html|title=Complex cobordism and stable homotopy groups of spheres|date=1986|publisher=Academic Press|isbn=978-0-08-087440-1|location=Orlando|pages=|oclc=316566772|author-link=Douglas Ravenel}}</ref><sup>pg 302</sup> is a left <math>A</math>-module <math>M</math> together with an <math>A</math>-linear map<blockquote><math>\psi: M \to \Gamma\otimes_AM</math></blockquote>which satisfies the following two properties

# ('''counitary''') <math>(\varepsilon\otimes Id_M)\circ \psi = Id_M</math>
# ('''coassociative''') <math>(\Delta\otimes Id_M) \circ \psi = (Id_\Gamma \otimes \psi) \circ \psi</math>

A right comodule is defined similarly, but instead there is a map<blockquote><math>\phi: M \to M \otimes_A \Gamma</math></blockquote>satisfying analogous axioms.

== Structure theorems ==

=== Flatness of Γ gives an abelian category ===
One of the main structure theorems for comodules<ref name="Ravenel" /><sup>pg 303</sup> is if <math>\Gamma</math> is a flat <math>A</math>-module, then the category of comodules <math>\text{Comod}(A,\Gamma)</math> of the Hopf-algebroid is an [[abelian category]].

=== Relation to stacks ===
There is a structure theorem<ref name=":0" /> <sup>pg 7</sup> relating comodules of Hopf-algebroids and modules of presheaves of groupoids. If <math>(A,\Gamma)</math> is a Hopf-algebroid, there is an equivalence between the category of comodules <math>\text{Comod}(A,\Gamma)</math> and the category of quasi-coherent sheaves <math>\text{QCoh}(\text{Spec}(A),\text{Spec}(\Gamma))</math> for the associated presheaf of groupoids<blockquote><math>\text{Spec}(\Gamma)\rightrightarrows \text{Spec}(A)</math></blockquote>to this Hopf-algebroid.

== Examples ==

=== From BP-homology ===
Associated to the [[Brown–Peterson cohomology|Brown-Peterson]] [[Spectrum (topology)|spectrum]] is the Hopf-algebroid <math>(BP_*,BP_*(BP))</math> classifying p-typical [[formal group law]]s. Note<blockquote><math>\begin{align}
BP_* &= \mathbb{Z}_{(p)}[v_1,v_2,\ldots] \\
BP_*(BP) &= BP_*[t_1,t_2,\ldots]
\end{align}</math></blockquote>where <math>\mathbb{Z}_{(p)}</math> is the [[Localization (commutative algebra)|localization]] of <math>\mathbb{Z}</math> by the prime ideal <math>(p)</math>. If we let <math>I_n</math> denote the ideal<blockquote><math>I_n = (p,v_1,\ldots, v_{n-1})</math></blockquote>Since <math>v_n</math> is a primitive in <math>BP_*/I_n</math>, there is an associated Hopf-algebroid <math>(A,\Gamma)</math><blockquote><math>(v_n^{-1}BP_*/I_n, v_n^{-1}BP_*(BP)/I_n)</math></blockquote>There is a structure theorem on the [[Adams-Novikov spectral sequence]] relating the Ext-groups of comodules on <math>(BP_*,BP_*(BP))</math> to Johnson-Wilson homology, giving a more tractable spectral sequence. This happens through an equivalence of categories of comodules of <math>(A,\Gamma)</math> to the category of comodules of <blockquote><math>(v_n^{-1}E(m)_*/I_n, v_n^{-1}E(m)_*(E(m)/I_n)</math></blockquote>giving the isomorphism<blockquote><math>\text{Ext}^{*,*}_{BP_*BP}(M,N) \cong
\text{Ext}^{*,*}_{E(m)_*E(m)}(E(m)_*\otimes_{BP_*} M,E(m)_*\otimes_{BP_*}N)</math></blockquote>assuming <math>M</math> and <math>N</math> satisfy some technical hypotheses<ref name=":0" /> <sup>pg 24</sup>.

== See also ==

* [[Adams spectral sequence]]
*[[Steenrod algebra]]

== References ==
{{Reflist}}

[[Category:Hopf algebras]]
[[Category:Homotopical algebra]]
[[Category:Algebraic topology]]
[[Category:Algebraic geometry]]