[Paper Review] Idempotents, Localizations and Picard groups of A(1)-modules

Robert Bruner|arXiv (Cornell University)|Nov 1, 2012

Homotopy and Cohomology in Algebraic Topology10 references3 citations

TL;DR

This paper analyzes the stable isomorphism type of polynomial rings over $Δ(1)$-modules using Margolis localizations and periodicity. It shows that all bounded-below $Δ(1)$-modules decompose uniquely into $Q_0$- and $Q_1$-local components, and fully determines the Picard groups of the local subcategories, proving that the only stably idempotent $Δ(1)$-modules are those arising from these localizations.

ABSTRACT

We analyze the stable isomorphism type of polynomial rings on degree 1 generators as modules over the sub-algebra A(1) = of the mod 2 Steenrod algebra. Since their augmentation ideals are Q_1-local, we do this by studying the Q_i-local subcategories and the associated Margolis localizations. The periodicity exhibited by such modules reduces the calculation to one that is finite. We show that these are the only localizations which preserve tensor products, by first computing the Picard groups of these subcategories and using them to determine all idempotents in the stable category of bounded-below A(1)-modules. We show that the Picard groups of the whole category are detected in the local Picard groups, and show that every bounded-below A(1) -module is uniquely expressible as an extension of a Q_0-local module by a Q_1-local module, up to stable equivalence. Applications include correct, complete proofs of Ossa's theorem, applications to Powell's work describing connective K-theory of classifying spaces of elementary abelian groups in functorial terms, and Ault's work on the hit problem.

Motivation & Objective

• To systematically analyze the stable isomorphism type of polynomial rings on degree 1 generators as $Δ(1)$-modules.
• To understand the role of $Q_i$-local subcategories and their associated Margolis localizations in decomposing $Δ(1)$-modules.
• To classify all stably idempotent $Δ(1)$-modules and determine which localizations preserve tensor products.
• To compute the Picard groups of $Q_i$-local subcategories and show that the global Picard group is detected in the local ones.
• To establish a unique decomposition of bounded-below $Δ(1)$-modules into $Q_0$-local and $Q_1$-local summands up to stable equivalence.

Proposed method

• Uses periodicity in $Q_0$- and $Q_1$-local modules: $ΩM \simeq \Sigma M$ for $Q_0$-local $M$, and $\u03a9^4M \simeq \Sigma^{12}M$ for $Q_1$-local $M$.
• Introduces modules $R$ and $P_0$ as $Q_0$- and $Q_1$-local representatives of $H^*BC_2$ and its suspension.
• Establishes a non-split triangle $\Sigma R \xrightarrow{\epsilon} \mathbf{F}_2 \xrightarrow{\eta} P_0$, showing $\mathbf{L}_0\mathbf{F}_2 \simeq \Sigma R$ and $\mathbf{L}_1\mathbf{F}_2 \simeq P_0$.
• Reduces the study of tensor powers $P^{igotimes n}$ to finitely many cases via periodicity, using explicit stable equivalences.
• Computes Picard groups of $Q_i$-local subcategories: $\operatorname{Pic}^{(0)}(\mathcal{A}(1)) = \mathbb{Z}$, $\operatorname{Pic}^{(1)}(\mathcal{A}(1)) = \mathbb{Z} \oplus \mathbb{Z}/(4)$.
• Uses the localization map $\operatorname{Pic} \to \operatorname{Pic}^{(0)} \oplus \operatorname{Pic}^{(1)}$ to show it is a monomorphism, detecting the global Picard group.

Experimental results

Research questions

• RQ1 Which $Δ(1)$-modules are stably idempotent, i.e., satisfy $I \otimes I \simeq I$?
• RQ2 How do $Q_i$-local subcategories of $Δ(1)$-modules behave under tensor products and periodicity?
• RQ3 What is the structure of the Picard group of the category of bounded-below $Δ(1)$-modules?
• RQ4 Can every bounded-below $Δ(1)$-module be uniquely decomposed into $Q_0$-local and $Q_1$-local components up to stable equivalence?
• RQ5 Are the Margolis localizations $\mathbf{L}_i\mathbf{F}_2$ the only localizations that preserve tensor products in the stable category?

Key findings

• The only stably idempotent bounded-below $Δ(1)$-modules are those isomorphic to suspensions and loops of $\mathbf{L}_0\mathbf{F}_2 \simeq \Sigma R$ and $\mathbf{L}_1\mathbf{F}_2 \simeq P_0$, as shown in Theorem 10.1.
• The Picard group of the $Q_0$-local subcategory of bounded-below $Δ(1)$-modules is $\mathbb{Z}$, while that of the $Q_1$-local subcategory is $\mathbb{Z} \oplus \mathbb{Z}/(4)$, with the torsion part arising from four-fold periodicity.
• Every bounded-below $Δ(1)$-module is uniquely expressible as an extension of a $Q_0$-local module by a $Q_1$-local module up to stable equivalence.
• The localization map $\operatorname{Pic}(\mathcal{A}(1)) \to \operatorname{Pic}^{(0)} \oplus \operatorname{Pic}^{(1)}$ is a monomorphism, showing that the global Picard group is fully detected in the local Picard groups.
• The tensor powers of $H^*BC_2$ are stably equivalent to their algebraic loops, exhibiting four-fold periodicity: $(\Omega I)^{\otimes n} \simeq \Omega^n I$ for $I = H^*BC_2$.
• The free summand in $P^{\otimes n}$ is fully described via Hilbert series: $H(F_n) = H(\mathcal{A}(1)) \cdot \frac{t^n(1 - t^n(1-t)^{n-1}Q_n(t))}{(1-t)^{n-1}(1-t^4)(1+t^3)}$, with explicit formulas for $Q_n(t)$ modulo 4.