Homotopy Theory of Comodules over a Hopf Algebroid
نویسنده
چکیده
Given a good homology theory E and a topological space X, E∗X is not just an E∗-module but also a comodule over the Hopf algebroid (E∗, E∗E). We establish a framework for studying the homological algebra of comodules over a well-behaved Hopf algebroid (A,Γ). That is, we construct the derived category Stable(Γ) of (A,Γ) as the homotopy category of a Quillen model structure on Ch(Γ), the category of unbounded chain complexes of Γ-comodules. This derived category is obtained by inverting the homotopy isomorphisms, NOT the homology isomorphisms. We establish the basic properties of Stable(Γ), showing that it is a compactly generated tensor triangulated category.
منابع مشابه
L-complete Hopf Algebroids and Their Comodules
We investigate Hopf algebroids in the category of L-complete modules over a commutative Noetherian regular complete local ring. The main examples of interest in algebraic topology are the Hopf algebroids associated to Lubin-Tate spectra in the K(n)-local stable homotopy category, and we show that these have Landweber filtrations for all finitely generated discrete modules. Along the way we inve...
متن کاملGraded Comodule Categories with Enough Projectives
It is well-known that the category of comodules over a flat Hopf algebroid is abelian but typically fails to have enough projectives, and more generally, the category of graded comodules over a graded flat Hopf algebroid is abelian but typically fails to have enough projectives. In this short paper we prove that the category of connective graded comodules over a connective, graded, flat, finite...
متن کاملTannaka-krein Duality for Hopf Algebroids
We develop the Tannaka-Krein duality for monoidal functors with target in the categories of bimodules over a ring. The Coend of such a functor turns out to be a Hopf algebroid over this ring. Using a result of [4] we characterize a small abelian, locally finite rigid monoidal category as the category of rigid comodules over a transitive Hopf algebroid.
متن کاملChromatic Phenomena in the Algebra of Bp∗bp -comodules
We describe the author’s research with Neil Strickland on the global algebra and global homological algebra of the category of BP∗BP comodules. We show, following [HS02a], that the category of E(n)∗E(n)comodules is a localization, in the abelian sense, of the category of BP∗BP comodules. This gives analogues of the usual structure theorems, such as the Landweber filtration theorem, for E(n)∗E(n...
متن کاملMorita Theory for Hopf Algebroids and Presheaves of Groupoids
Comodules over Hopf algebroids are of central importance in algebraic topology. It is well-known that a Hopf algebroid is the same thing as a presheaf of groupoids on Aff , the opposite category of commutative rings. We show in this paper that a comodule is the same thing as a quasi-coherent sheaf over this presheaf of groupoids. We prove the general theorem that internal equivalences of preshe...
متن کامل