January 15, 2013

The category U IV

Posted by Akhil Mathew under algebra, topology | Tags: , , , |
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This post is part of a series (started here) of posts on the structure of the category {\mathcal{U}} of unstable modules over the mod {2} Steenrod algebra {\mathcal{A}}, which plays an important role in the proof of the Sullivan conjecture (and its variants).

In the previous post, we introduced some additional structure on the category {\mathcal{U}}.

  • First, using the (cocommutative) Hopf algebra structure on {\mathcal{A}}, we got a symmetric monoidal structure on {\mathcal{U}}, which was an algebraic version of the Künneth theorem.
  • Second, we described a “Frobenius” functor

    \displaystyle \Phi : \mathcal{U} \rightarrow \mathcal{U},

    which was symmetric monoidal, and which came with a Frobenius map {\Phi M \rightarrow M}.

  • We constructed an exact sequence natural in {M},

    \displaystyle 0 \rightarrow \Sigma L^1 \Omega M \rightarrow \Phi M \rightarrow M \rightarrow \Sigma \Omega M \rightarrow 0, \ \ \ \ \ (4)

    where {\Sigma} was the suspension and {\Omega} the left adjoint. In particular, we showed that all the higher derived functors of {\Omega} (after {L^1}) vanish.

The first goal of this post is to use this extra structure to prove the following:

 

Theorem 39 The category {\mathcal{U}} is locally noetherian: the subobjects of the free unstable module {F(n)} satisfy the ascending chain condition (equivalently, are finitely generated as {\mathcal{A}}-modules).

 

In order to prove this theorem, we’ll use induction on {n} and the technology developed in the previous post as a way to make Nakayama-type arguments. Namely, the exact sequence (4) becomes

\displaystyle 0 \rightarrow \Phi F(n) \rightarrow F(n) \rightarrow \Sigma F(n-1) \rightarrow 0,

as we saw in the previous post. Observe that {F(0) = \mathbb{F}_2} is clearly noetherian (it’s also not hard to check this for {F(1)}). Inductively, we may assume that {F(n-1)} (and therefore {\Sigma F(n-1)}) is noetherian.

Fix a subobject {M \subset F(n)}; we’d like to show that {M} is finitely generated. (more…)

 

January 14, 2013

The category U III

Posted by Akhil Mathew under algebra, topology | Tags: , |
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This is part of a series of posts intended to understand some of the basic structure of the category {\mathcal{U}} of unstable modules over the (mod {2}) Steenrod algebra, to prepare for the proof of the Sullivan conjecture. Here’s what we’ve seen so far:

  • {\mathcal{U}} is a Grothendieck abelian category, with a set of compact, projective generators {F(n)} (the free unstable module on a generator in degree {n}). (See this post.)
  • {F(n)} has a tautological class {\iota_n} in degree {n}, and has a basis given by {\mathrm{Sq}^I \iota_n} for {I} an admissible sequence of excess {\leq n}. (This post explained the terminology and the proof.)
  • {F(1)} was the subspace {\mathbb{F}_2\left\{t, t^2, t^4, \dots\right\} \subset \widetilde{H}^*(\mathbb{RP}^\infty; \mathbb{F}_2)}.

Our goal in this post is to describe some of the additional structure on the category {\mathcal{U}}, which will eventually enable us to prove (and make sense of!) results such as {F(n) \simeq (F(1)^{\otimes n})^{ \Sigma_n}}. We’ll start with the symmetric monoidal tensor product and the suspension functor, and then connect this to the Frobenius maps (which will be defined below).

 

1. The symmetric monoidal structure

Our first order of business is to describe the symmetric monoidal structure on {\mathcal{U}}, which will be given by the {\mathbb{F}_2}-linear tensor product. In fact, recall that the Steenrod algebra is a cocommutative Hopf algebra, under the diagonal map

\displaystyle \mathrm{Sq}^n \mapsto \sum_{i+j = n} \mathrm{Sq}^i \otimes \mathrm{Sq}^j.

The Hopf algebra structure is defined according to the following rule: we have that {\theta} maps to {\sum \theta' \otimes \theta''} if and only if for every two cohomology classes {x,y } in the cohomology of a topological space, one has

\displaystyle \theta(xy) = \sum \theta'(x) \theta''(y).

The cocommutative Hopf algebra structure on {\mathcal{A}} gives a tensor product on the category of (graded) {\mathcal{A}}-modules, which is symmetric monoidal. It’s easy to check that if {M, N} are {\mathcal{A}}-modules satisfying the unstability condition, then so does {M \otimes N}. This is precisely the symmetric monoidal structure on {\mathcal{U}}. (more…)

 

January 14, 2013

The category U II

Posted by Akhil Mathew under algebra, topology | Tags: , , |
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The purpose of this post (like the previous one) is to go through some of the basic properties of the category {\mathcal{U}} of unstable modules over the (mod {2}) Steenrod algebra. An analysis of {\mathcal{U}} will ultimately lead to the proof of the Sullivan conjecture. Most of this material, again, is from Schwartz’s Unstable modules over the Steenrod algebra and Sullivan’s fixed point set conjecture; another useful source is Lurie’s notes. 

1. The modules {F(n)}

In the previous post, we showed that the category {\mathcal{U}} had enough projectives. More specifically, we constructed — using the adjoint functor theorem — an object {F(n)}, for each {n}, which we called the free unstable module on a class of degree {n}.The object {F(n)} had the universal property

\displaystyle \hom_{\mathcal{U}}(F(n), M) \simeq M_n,\quad M \in \mathcal{U}.

To start with, we’d like to have a more explicit description of the module {F(n)}.

To do this, we need a little terminology. A sequence of positive integers

\displaystyle i_k, i_{k-1}, \dots, i_1

is called admissible if

\displaystyle i_j \geq 2 i_{j-1}

for each {j}. It is a basic fact, which can be proved by manipulating the Adem relations, that the squares

\displaystyle \mathrm{Sq}^I \stackrel{\mathrm{def}}{=} \mathrm{Sq}^{i_k} \mathrm{Sq}^{i_{k-1}} \dots \mathrm{Sq}^{i_1}, \quad I = (i_k, \dots, i_1) \ \text{admissible}

form a spanning set for {\mathcal{A}} as {I} ranges over the admissible sequences. In fact, by looking at the representation on various cohomology rings, one can prove:

Proposition 29 The {\mathrm{Sq}^I} for {I } admissible form a basis for the Steenrod algebra {\mathcal{A}}. (more…)

 

January 13, 2013

The category U

Posted by Akhil Mathew under topology | Tags: , , |
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The purpose of this post is to introduce the basic category that enters into the unstable Adams spectral sequence and the proof of the Sullivan conjecture: the category of unstable modules over the Steenrod algebra. Throughout, I’ll focus on the (simpler) case of {p=2}.

1. Recap of the Steenrod algebra

Let {X} be a space. Then the cohomology {H^*(X; \mathbb{F}_2)} has a great deal of algebraic structure:

  • It is a graded {\mathbb{F}_2}-vector space concentrated in nonnegative degrees.
  • It has an algebra structure (respecting the grading).
  • It comes with an action of Steenrod operations

    \displaystyle \mathrm{Sq}^i: H^*(X; \mathbb{F}_2 ) \rightarrow H^{*+i}(X; \mathbb{F}_2), \quad i \geq 0.

The Steenrod squares, which are constructed from the failure of strict commutativity in the cochain algebra {C^*(X; \mathbb{F}_2)}, are themselves subject to a number of axioms:

  • {\mathrm{Sq}^0} acts as the identity.
  • {\mathrm{Sq}^i} is compatible with the suspension isomorphism between {H_*(X; \mathbb{F}_2), \widetilde{H}_*(\Sigma X; \mathbb{F}_2)}.
  • One has the Adem relations: for {i < 2j},

    \displaystyle \mathrm{Sq}^i \mathrm{Sq}^j = \sum_{0 \leq 2k \leq i} \binom{j-k-1}{i-2k} \mathrm{Sq}^{i+j-k}\mathrm{Sq}^k. \ \ \ \ \ (3)

In other words, there is a (noncommutative) algebra of operations, which is the Steenrod algebra {\mathcal{A}}, such that the cohomology of any space {X} is a module over {\mathcal{A}}. The Steenrod algebra can be defined as

\displaystyle \mathcal{A} \stackrel{\mathrm{def}}{=} T( \mathrm{Sq}^0, \mathrm{Sq}^1, \dots )/ \left( \mathrm{Sq}^0 = 1, \ \text{Adem relations}\right) . (more…)