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
(counitary) (coassociative)
A right comodule is defined similarly, but instead there is a mapsatisfying analogous axioms.
Structure theorems
Flatness of Γ gives an abelian category 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.
Relation to stacks 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.
See also
Adams spectral sequence Steenrod algebra
References