Toward Quillen stratification for the odd primary Steenrod algebra

Introduction

Throughout, $p$ will be an odd prime number.

Let $A$ be the mod $p$ Steenrod algebra. The cohomology of $A$, $\operatorname{Ext}^{*}_{A}(\mathbb{F}_{p},\mathbb{F}_{p})$, is the $E_{2}$-page of the classical Adams spectral sequence converging to the $p$-component of the stable homotopy groups of spheres, an object of great interest in stable homotopy theory. The goal of this paper is to try to understand the cohomology of $A$ along the lines of the so-called Quillen stratification for the mod 2 Steenrod algebra, as presented in [19] and [20 Chapter 5].

As Milnor [12] tells us, the Steenrod algebra is the Hopf algebra with graded dual

graded by $\deg\xi_{n}=2(p^{n}-1)$ and $\deg\tau_{n}=2p^{n}-1$, with coproduct

where $\xi_{0}=1$ when it appears in these formulas.

To understand a Hopf algebra like $A$, a good first step is to understand its sub-Hopf algebras. Any sub-Hopf algebra $C$ of $A$ is dual to a quotient Hopf algebra $C^{\vee}$ of $A^{\vee}$, and by a theorem of Adams and Margolis [1], those quotients must have the form

for some exponents $((c_{1},c_{2},c_{3},\ldots);(e_{0},e_{1},e_{2},\ldots))$, where $c_{i}$ is either a non-negative integer or $\infty$ and $e_{i}\in\{1,2\}$. The list of exponents is called the profile function for the Hopf algebra.

Adams and Margolis characterized precisely which such lists correspond to sub-Hopf algebras, thus providing a complete classification. We review the details in the next section and for now just mention one case: we let $D$ be the sub-Hopf algebra of $A$ with profile function

This algebra is normal in $A$, and more generally, for any sub-Hopf algebra $C$ of $A$, $C\cap D$ is normal in $C$. The inclusion map $i:C\cap D\to C$ induces a restriction map $i^{*}:\operatorname{Ext}^{*}_{C}(\mathbb{F}_{p},\mathbb{F}_{p})\to\operatorname{Ext}^{*}_{C\cap D}(\mathbb{F}_{p},\mathbb{F}_{p})$. Because $C\cap D$ is normal in $C$, the restriction map factors through the subalgebra of invariants under the action of the quotient Hopf algebra $C//(C \cap D) := C \otimes_{C \cap D} \mathbb{F}_p$:

The map $\bar{\imath}^{*}$ will make an appearance in Theorem B below.

Quillen stratification involves understanding cohomology algebras up to $F$-isomorphism, so recall:

We can strengthen the first part of the definition of $F$-isomorphism to echo the nilpotence theorem of Devinatz, Hopkins, and Smith. To do that, we work in a category which behaves like the classical stable homotopy category: given a sub-Hopf algebra $C$ of $A$, let $\operatorname{Stable}(C^\vee)$ be the category as defined in [5 Section 9.5] and examined in detail in [20]. The objects of $\operatorname{Stable}(C^\vee)$ are cochain complexes of injective graded left comodules over $C^\vee$, and the morphisms are cochain homotopy classes of maps. This category is a stable homotopy category in the sense of [5], and we may refer to its objects as “spectra.” In this category, the sphere object $S$ is an injective resolution of $\mathbb{F}_p$, and if $X$ is any object, then we may write $\pi_*(X)$ for the chain homotopy classes of maps from $S$ to $X$. If $X$ is the injective resolution of a comodule $M$, then $\pi_*(X) \cong\operatorname{Ext}^{*}_{C}(\mathbb{F}_p,M)$. (Note that if $M$ is a $C^\vee$-comodule, then $M$ can also be viewed as a $C$-comodule, and so here Ext is computed in the category of $C$-modules.) Also in this category, any sub-Hopf algebra $B$ of $C$ defines a ring spectrum called $HB$: this is an injective resolution of $C^\vee\square_B \mathbb{F}_p \cong(C \otimes_B \mathbb{F}_p)^\vee$, and its coefficient ring is $\pi_*HB = HB_* \cong\operatorname{Ext}^{*}_{B}(\mathbb{F}_p,\mathbb{F}_p)$. If $X$ is the injective resolution of a $C^\vee$-comodule $M$, then the quotient map $C^\vee\to B^\vee$ gives $M$ the structure of a $B^\vee$-comodule, and we have $HB_*(X) \cong\operatorname{Ext}^{*}_{B}(\mathbb{F}_p,M)$.

We have three main results in this paper; here are the first two.