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 left comodule [2]pg 302 is a left -module together with an -linear map
which satisfies the following two properties
- (counitary)
- (coassociative)
A right comodule is defined similarly, but instead there is a map
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 is a flat -module, then the category of comodules 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 is a Hopf-algebroid, there is an equivalence between the category of comodules and the category of quasi-coherent sheaves for the associated presheaf of groupoids
to this Hopf-algebroid.
Examples
From BP-homology
Associated to the Brown-Peterson spectrum is the Hopf-algebroid classifying p-typical formal group laws. Note
where is the localization of by the prime ideal . If we let denote the ideal
Since is a primitive in , there is an associated Hopf-algebroid
There is a structure theorem on the Adams-Novikov spectral sequence relating the Ext-groups of comodules on to Johnson-Wilson homology, giving a more tractable spectral sequence. This happens through an equivalence of categories of comodules of to the category of comodules of
giving the isomorphism
assuming and satisfy some technical hypotheses[1] pg 24.
See also
- Adams spectral sequence
- Steenrod algebra
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.