[Paper Review] $L$-complete Hopf algebroids and their comodules

Andrew Baker|ArXiv.org|Jan 11, 2009

Homotopy and Cohomology in Algebraic Topology14 references3 citations

TL;DR

This paper introduces $L$-complete Hopf algebroids over commutative Noetherian regular complete local rings, focusing on their comodules in algebraic topology. It establishes that finitely generated comodules over $L$-complete Hopf algebroids—particularly those associated with Lubin-Tate spectra—admit Landweber filtrations, and introduces the notion of unipotent Hopf algebroids to generalize classical results for Hopf algebras.

ABSTRACT

We investigate Hopf algebroids in the category of $L$-complete modules over a commutative Noetherian regular complete local ring. The main examples are provided by 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 investigate the canonical Hopf algebras associated to Hopf algebroids over fields and introduce a notion of unipotent Hopf algebroid generalising that for Hopf algebras. In two appendices we continue the discussion of the connections with twisted group rings, and expand on a result of Hovey on the non-exactness of coproducts of L-complete modules.

Motivation & Objective

• To develop a theory of Hopf algebroids in the category of $L$-complete modules over a commutative Noetherian regular complete local ring.
• To understand the structure of comodules over $L$-complete Hopf algebroids, especially in the context of $K(n)$-local stable homotopy theory.
• To generalize the concept of unipotent Hopf algebras to Hopf algebroids and study their representation theory.
• To verify that the Hopf algebroid $E^ u_*E$ for a Lubin-Tate spectrum $E$ has Landweber filtrations for finitely generated comodules.
• To clarify the distinction between $L$-complete modules and other completion theories in homotopy theory, particularly in relation to Greenlees-May and Hovey-Strickland work.

Proposed method

• Adapts the notion of $L$-complete modules—defined via derived functors of $rak{m}$-adic completion—to the category of modules over a commutative Noetherian regular complete local ring $R$.
• Introduces $L$-complete Hopf algebroids as Hopf algebroids in the category of $L$-complete $R$-modules, with emphasis on the Hopf algebroid $E^ u_*E$ associated to Lubin-Tate spectra.
• Defines unipotent Hopf algebroids over a field as a generalization of unipotent Hopf algebras, using the associated Hopf algebra construction.
• Applies the Landweber filtration theorem to finitely generated discrete comodules over $L$-complete Hopf algebroids, showing the existence of composition series with simple quotients.
• Uses the Koszul resolution and $ ext{Tor}$-computations to analyze the derived functors $L_s$ and prove non-vanishing of $L_1$ for certain modules.
• Employs inverse limit techniques and diagram chasing in exact sequences involving $ ext{lim}^1$ and $ ext{lim}$ to demonstrate non-injectivity of $L_0$ on infinite sums.

Experimental results

Research questions

• RQ1 Do finitely generated comodules over $L$-complete Hopf algebroids admit composition series with simple quotients?
• RQ2 How does the notion of unipotent Hopf algebroid extend the classical theory of unipotent Hopf algebras?
• RQ3 What is the relationship between comodules over a Hopf algebroid and those over its associated Hopf algebra?
• RQ4 Is the $L_0$-functor exact on infinite sums of $L$-complete modules, and what are the implications for the category of $L$-complete modules?
• RQ5 Do the Hopf algebroids associated to Lubin-Tate spectra in the $K(n)$-local category satisfy the Landweber filtration property?

Key findings

• Finitely generated comodules over $L$-complete Hopf algebroids have composition series where each quotient is annihilated by a power of the maximal ideal.
• If the associated Hopf algebra of a Hopf algebroid over a field is unipotent, then the Hopf algebroid itself is unipotent.
• The Hopf algebroid $E^ u_*E$ for a Lubin-Tate spectrum $E$ admits Landweber filtrations for all finitely generated discrete comodules.
• The natural map $L_0(igoplus_k N) o L_0(igoplus_k M)$ is not injective, demonstrating that $L_0$ fails to be left exact on infinite sums of $L$-complete modules.
• The derived functor $L_1(igoplus_k M/N)$ is non-zero for certain $L$-complete modules $M/N$, showing that $L_1$ does not vanish even when $L_0$ is exact.
• For modules on which $p,u$ act regularly, $L_s(M) = 0$ for all $s > 0$, indicating that such modules are $L$-complete and have trivial higher derived functors.