Misplaced Pages

Comodule over a Hopf algebroid

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.

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, 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 , Γ ) {\displaystyle (A,\Gamma )} a left comodule M {\displaystyle M} is a left A {\displaystyle A} -module M {\displaystyle M} together with an A {\displaystyle A} -linear map

ψ : M → Γ ⊗ A M {\displaystyle \psi :M\to \Gamma \otimes _{A}M}

which satisfies the following two properties

  1. (counitary) ( ε ⊗ I d M ) ∘ ψ = I d M {\displaystyle (\varepsilon \otimes Id_{M})\circ \psi =Id_{M}}
  2. (coassociative) ( Δ ⊗ I d M ) ∘ ψ = ( I d Γ ⊗ ψ ) ∘ ψ {\displaystyle (\Delta \otimes Id_{M})\circ \psi =(Id_{\Gamma }\otimes \psi )\circ \psi }

A right comodule is defined similarly, but instead there is a map

ϕ : M → M ⊗ A Γ {\displaystyle \phi :M\to M\otimes _{A}\Gamma }

satisfying analogous axioms.

Structure theorems

Flatness of Γ gives an abelian category

One of the main structure theorems for comodules is if Γ {\displaystyle \Gamma } is a flat A {\displaystyle A} -module, then the category of comodules Comod ( A , Γ ) {\displaystyle {\text{Comod}}(A,\Gamma )} of the Hopf-algebroid is an abelian category.

Relation to stacks

There is a structure theorem relating comodules of Hopf-algebroids and modules of presheaves of groupoids. If ( A , Γ ) {\displaystyle (A,\Gamma )} is a Hopf-algebroid, there is an equivalence between the category of comodules Comod ( A , Γ ) {\displaystyle {\text{Comod}}(A,\Gamma )} and the category of quasi-coherent sheaves QCoh ( Spec ( A ) , Spec ( Γ ) ) {\displaystyle {\text{QCoh}}({\text{Spec}}(A),{\text{Spec}}(\Gamma ))} for the associated presheaf of groupoids

Spec ( Γ ) ⇉ Spec ( A ) {\displaystyle {\text{Spec}}(\Gamma )\rightrightarrows {\text{Spec}}(A)}

to this Hopf-algebroid.

Examples

From BP-homology

Associated to the Brown-Peterson spectrum is the Hopf-algebroid ( B P ∗ , B P ∗ ( B P ) ) {\displaystyle (BP_{*},BP_{*}(BP))} classifying p-typical formal group laws. Note

B P ∗ = Z ( p ) [ v 1 , v 2 , … ] B P ∗ ( B P ) = B P ∗ [ t 1 , t 2 , … ] {\displaystyle {\begin{aligned}BP_{*}&=\mathbb {Z} _{(p)}\\BP_{*}(BP)&=BP_{*}\end{aligned}}}

where Z ( p ) {\displaystyle \mathbb {Z} _{(p)}} is the localization of Z {\displaystyle \mathbb {Z} } by the prime ideal ( p ) {\displaystyle (p)} . If we let I n {\displaystyle I_{n}} denote the ideal

I n = ( p , v 1 , … , v n − 1 ) {\displaystyle I_{n}=(p,v_{1},\ldots ,v_{n-1})}

Since v n {\displaystyle v_{n}} is a primitive in B P ∗ / I n {\displaystyle BP_{*}/I_{n}} , there is an associated Hopf-algebroid ( A , Γ ) {\displaystyle (A,\Gamma )}

( v n − 1 B P ∗ / I n , v n − 1 B P ∗ ( B P ) / I n ) {\displaystyle (v_{n}^{-1}BP_{*}/I_{n},v_{n}^{-1}BP_{*}(BP)/I_{n})}

There is a structure theorem on the Adams-Novikov spectral sequence relating the Ext-groups of comodules on ( B P ∗ , B P ∗ ( B P ) ) {\displaystyle (BP_{*},BP_{*}(BP))} to Johnson-Wilson homology, giving a more tractable spectral sequence. This happens through an equivalence of categories of comodules of ( A , Γ ) {\displaystyle (A,\Gamma )} to the category of comodules of

( v n − 1 E ( m ) ∗ / I n , v n − 1 E ( m ) ∗ ( E ( m ) / I n ) {\displaystyle (v_{n}^{-1}E(m)_{*}/I_{n},v_{n}^{-1}E(m)_{*}(E(m)/I_{n})}

giving the isomorphism

Ext B P ∗ B P ∗ , ∗ ( M , N ) ≅ Ext E ( m ) ∗ E ( m ) ∗ , ∗ ( E ( m ) ∗ ⊗ B P ∗ M , E ( m ) ∗ ⊗ B P ∗ N ) {\displaystyle {\text{Ext}}_{BP_{*}BP}^{*,*}(M,N)\cong {\text{Ext}}_{E(m)_{*}E(m)}^{*,*}(E(m)_{*}\otimes _{BP_{*}}M,E(m)_{*}\otimes _{BP_{*}}N)}

assuming M {\displaystyle M} and N {\displaystyle N} satisfy some technical hypotheses .

See also

References

  1. ^ Hovey, Mark (2001-05-16). "Morita theory for Hopf algebroids and presheaves of groupoids". arXiv:math/0105137.
  2. ^ Ravenel, Douglas C. (1986). Complex cobordism and stable homotopy groups of spheres. Orlando: Academic Press. ISBN 978-0-08-087440-1. OCLC 316566772.
Categories: