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 [math]\displaystyle{ (A,\Gamma) }[/math] a left comodule [math]\displaystyle{ M }[/math][2]pg 302 is a left [math]\displaystyle{ A }[/math]-module [math]\displaystyle{ M }[/math] together with an [math]\displaystyle{ A }[/math]-linear map
[math]\displaystyle{ \psi: M \to \Gamma\otimes_AM }[/math]
which satisfies the following two properties
- (counitary) [math]\displaystyle{ (\varepsilon\otimes Id_M)\circ \psi = Id_M }[/math]
- (coassociative) [math]\displaystyle{ (\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
[math]\displaystyle{ \phi: M \to M \otimes_A \Gamma }[/math]
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 [math]\displaystyle{ \Gamma }[/math] is a flat [math]\displaystyle{ A }[/math]-module, then the category of comodules [math]\displaystyle{ \text{Comod}(A,\Gamma) }[/math] 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 [math]\displaystyle{ (A,\Gamma) }[/math] is a Hopf-algebroid, there is an equivalence between the category of comodules [math]\displaystyle{ \text{Comod}(A,\Gamma) }[/math] and the category of quasi-coherent sheaves [math]\displaystyle{ \text{QCoh}(\text{Spec}(A),\text{Spec}(\Gamma)) }[/math] for the associated presheaf of groupoids
[math]\displaystyle{ \text{Spec}(\Gamma)\rightrightarrows \text{Spec}(A) }[/math]
to this Hopf-algebroid.
Examples
From BP-homology
Associated to the Brown-Peterson spectrum is the Hopf-algebroid [math]\displaystyle{ (BP_*,BP_*(BP)) }[/math] classifying p-typical formal group laws. Note
[math]\displaystyle{ \begin{align} BP_* &= \mathbb{Z}_{(p)}[v_1,v_2,\ldots] \\ BP_*(BP) &= BP_*[t_1,t_2,\ldots] \end{align} }[/math]
where [math]\displaystyle{ \mathbb{Z}_{(p)} }[/math] is the localization of [math]\displaystyle{ \mathbb{Z} }[/math] by the prime ideal [math]\displaystyle{ (p) }[/math]. If we let [math]\displaystyle{ I_n }[/math] denote the ideal
[math]\displaystyle{ I_n = (p,v_1,\ldots, v_{n-1}) }[/math]
Since [math]\displaystyle{ v_n }[/math] is a primitive in [math]\displaystyle{ BP_*/I_n }[/math], there is an associated Hopf-algebroid [math]\displaystyle{ (A,\Gamma) }[/math]
[math]\displaystyle{ (v_n^{-1}BP_*/I_n, v_n^{-1}BP_*(BP)/I_n) }[/math]
There is a structure theorem on the Adams-Novikov spectral sequence relating the Ext-groups of comodules on [math]\displaystyle{ (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]\displaystyle{ (A,\Gamma) }[/math] to the category of comodules of
[math]\displaystyle{ (v_n^{-1}E(m)_*/I_n, v_n^{-1}E(m)_*(E(m)/I_n) }[/math]
giving the isomorphism
[math]\displaystyle{ \text{Ext}^{*,*}_{BP_*BP}(M,N) \cong \text{Ext}^{*,*}_{E(m)_*E(m)}(E(m)_*\otimes_{BP_*} M,E(m)_*\otimes_{BP_*}N) }[/math]
assuming [math]\displaystyle{ M }[/math] and [math]\displaystyle{ N }[/math] satisfy some technical hypotheses[1]pg 24.
See also
References
- ↑ 1.0 1.1 1.2 Hovey, Mark (2001-05-16). "Morita theory for Hopf algebroids and presheaves of groupoids". arXiv:math/0105137.
- ↑ 2.0 2.1 Ravenel, Douglas C. (1986). Complex cobordism and stable homotopy groups of spheres. Orlando: Academic Press. ISBN 978-0-08-087440-1. OCLC 316566772. https://web.math.rochester.edu/people/faculty/doug/mu.html.
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