Bott periodicity in the Hit Problem

TL;DR

This paper explores the structure of modules related to the Hit Problem using Bott periodicity, revealing a 4-period cycle and identifying rare elements outside the usual hit and kernel spaces.

Contribution

It introduces a Bott periodicity pattern in the modules involved in the Hit Problem and analyzes the module of indecomposables using Bruner's resolution, advancing understanding of the $ ext{Hit}$ problem.

Findings

Modules $P_n$ exhibit a 4-period Bott periodicity with $P_{n+4} = oxed{8}$ shifts.

Identified a rare element in bidegree (5, 9) outside the $ ext{hit}$ and kernel spaces.

Determined the Hilbert series of $ ext{hit}$ elements in $ ilde{P}^{ ensor n}$.

Abstract

In this short note, we use Robert Bruner's $\mathcal{A}(1)$-resolution of $P = \mathbb{F}_2[t]$ to shed light on the Hit Problem. In particular, the reduced syzygies $P_n$ of $P$ occur as direct summands of $\widetilde{P}^{\otimes n}$, where $\widetilde{P}$ is the augmentation ideal of the map $P \to \mathbb{F}_2$. The complement of $P_n$ in $\widetilde{P}^{\otimes n}$ is free, and the modules $P_n$ exhibit a type of "Bott Periodicity" of period $4$: $P_{n+4} = \Sigma^8P_n$. These facts taken together allow one to analyze the module of indecomposables in $\widetilde{P}^{\otimes n}$, that is, to say something about the "$\mathcal{A}(1)$-hit Problem." Our study is essentially in two parts: First, we expound on the approach to the Hit Problem begun by William Singer, in which we compare images of Steenrod Squares to certain kernels of Squares. Using this approach, the author discovered a nontrivial element in bidegree $(5, 9)$ that is neither $\mathcal{A}(1)$-hit nor in $\mathrm{ker} Sq^1 + \mathrm{ker} Sq^3$. Such an element is extremely rare, but Bruner's result shows clearly why these elements exist and detects them in full generality. Second, we describe the graded $\mathbb{F}_2$-space of $\mathcal{A}(1)$-hit elements of $\widetilde{P}^{\otimes n}$ by determining its Hilbert series.