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. Duallypg 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' pg 302 is a left -module together with an -linear mapwhich satisfies the following two properties
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A right comodule is defined similarly, but instead there is a mapsatisfying analogous axioms.
One of the main structure theorems for comodulespg 303 is if is a flat -module, then the category of comodules of the Hopf-algebroid is an abelian category.
There is a structure theorem 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 groupoidsto this Hopf-algebroid.
Examples
From BP-homology
Associated to the Brown-Peterson spectrum is the Hopf-algebroid classifying p-typical formal group laws. Notewhere is the localization of by the prime ideal. If we let denote the idealSince 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 isomorphismassuming and satisfy some technical hypotheses pg 24.