### algebra

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…)

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…)

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

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

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

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…)

This post is part of a series on the Sullivan conjecture in algebraic topology. The Sullivan conjecture is a topological result, which remarkably reduces — as H. Miller showed —  to a purely algebraic computation in the category of unstable modules (and eventually algebras) over the Steenrod algebra, and in particular an injectivity assertion. This is a rather formidable category, but work of Kuhn enables one to identify a quotient category of it with the category of “generic representations” of the general linear group, which can be studied using different (and often easier) means. Kuhn’s work provides an approach to proving much of the algebraic background that goes into the Sullivan conjecture. In this post, I’ll describe one of the important ingredients.

The Gabriel-Popsecu theorem is a structure theorem for Grothendieck abelian categories, a version of which will be useful in understanding the structure of the category of unstable modules over the Steenrod algebra. The purpose of this post is to discuss this result and its many-object version due to Kuhn, from the paper “Generic Representations of the Finite General Linear Groups and the Steenrod Algebra: I.” Although the proof consists mostly of a series of diagram chases, there are some subtleties that I found rather difficult to grasp, and I thought it would be worthwhile to go through it in detail here.

1. Grothendieck abelian categories

Let ${\mathcal{A}}$ be an abelian category. Then ${\mathcal{A}}$ is Grothendieck abelian if

• ${\mathcal{A}}$ has a generator: that is, there is an object ${X \in \mathcal{A} }$ such that every object in ${\mathcal{A}}$ can be built up from colimits starting with ${X}$. (More precisely, the smallest subcategory of ${\mathcal{A}}$, closed under colimits, that contains ${X}$ is ${\mathcal{A}}$ itself)
• Filtered colimits in ${\mathcal{A}}$ exist and are exact.

Many categories occurring in “nature” (e.g., categories of modules over a ring of sheaves on a site) are Grothendieck, and it is thus useful to have general results about them. The goal of this post is to describe a useful structure theorem for Grothendieck abelian categories, which will show that they are the quotients of categories of modules by Serre subcategories. (more…)

In this post, I’d like to describe a toy analog of the Sullivan conjecture. Recall that the Sullivan conjecture considers (pointed) maps from ${BG}$ into a finite complex, and states that the space of such is contractible if $G$ is finite. The stable version replaces ${BG}$ with the Eilenberg-MacLane spectrum:

Theorem 13 Let ${H \mathbb{F}_p}$ be the Eilenberg-MacLane spectrum. Then the mapping spectrum

$\displaystyle (S^0)^{H \mathbb{F}_p}$

is contractible. In particular, for any finite spectrum ${F}$, the graded group of maps ${[H \mathbb{F}_p, F] = 0}$.

In the previous post, I sketched a proof (from Ravenel’s “Localization” paper) of this result based on a little chromatic technology. The spectrum ${H \mathbb{F}_p}$ is dissonant: that is, the Morava ${K}$-theories don’t see it. However, any finite spectrum is harmonic: that is, local with respect to the wedge of Morava ${K}$-theories. It follows formally that the spectrum of maps ${H \mathbb{F}_p \rightarrow S^0}$ is contractible (and thus the same with ${S^0}$ replaced by any finite spectrum). The non-formal input was the fact that ${S^0}$ is in fact harmonic, which requires a little work.

In this post, I’d like to sketch an earlier proof of the above theorem. This proof is based on the Adams spectral sequence. In fact, the proof runs parallel to Miller’s proof of the Sullivan conjecture. There is a classical Adams spectral sequence for computing ${[H \mathbb{F}_p, S^0]}$, with ${E_2}$ page given by

$\displaystyle \mathrm{Ext}^{s,t}_{\mathcal{A}}(\mathbb{F}_p, \mathcal{A}) \implies [ H \mathbb{F}_p, S^0]_{t-s} ,$

with ${\mathcal{A}}$ the (mod ${p}$) Steenrod algebra.

It turns out, however, for purely algebraic reasons, that the ${E_2}$ term is trivial. Miller’s proof of the Sullivan conjecture relies on more complicated algebra to show that the unstable version of all this has the same vanishing property at ${E_2}$. Most of this material is from Margolis’s Spectra and the Steenrod algebra. (more…)

This is the second post intended to understand some of the ideas in Milnor’s “Note on curvature and the fundamental group.” This is the paper that introduces the idea of growth rates for groups and proves that the fundamental group in negative curvature has exponential growth (as well as a dual result on polynomial growth in nonnegative curvature). In the previous post, we described volume comparison results in negative curvature: we showed in particular that a curvature bounded above by $c < 0$ meant that the volumes of expanding balls grow exponentially in the radius. In this post, we’ll explain how this translates into a result about the fundamental group.

1. Growth rates of groups

Let ${G}$ be a finitely generated group, and let ${S}$ be a finite set of generators such that ${S^{-1} = S}$. We define the norm

$\displaystyle \left \lVert\cdot \right \rVert_S: G \rightarrow \mathbb{Z}_{\geq 0}$

such that ${\left \lVert g \right \rVert_S}$ is the length of the smallest word in ${S}$ that evaluates to ${g}$. We note that

$\displaystyle d(g, h) \stackrel{\mathrm{def}}{=} \left \lVert gh^{-1} \right \rVert_S$

defines a metric on ${G}$. The metric depends on the choice of ${S}$, but only up to scaling by a positive constant. That is, given ${S, S'}$, there exists a positive constant ${p}$ such that ${\left \lVert \cdot \right \rVert_S \leq p \left \lVert\cdot \right \rVert_{S'}}$. The metric space structure on ${G}$ is thus defined up to quasi-isometry. (more…)

I’ve been reading Wall’s “Finiteness conditions for CW complexes.” This paper gives necessary and sufficient conditions for a space to be homotopy equivalent to a finite cell complex. Alternatively, it gives an obstruction in ${K}$-theory for when a retract (in the homotopy category) of a finite cell complex has the homotopy type of a finite cell complex. I’d like to describe this result, and try to motivate why the existence of such an obstruction is a natural thing to expect by a simpler analogy with algebra.

There is a fruitful analogy between spaces and chain complexes. Let ${R}$ be a ring, and consider the derived category ${D(R)}$ of chain complexes of ${R}$-modules. There are various interesting subcategories of ${D(R)}$:

1. The finitely presented derived category ${D_{fp}(R)}$; this is the smallest triangulated (or stable) full subcategory of ${D(R)}$ containing ${R}$ and closed under cofiber sequences. In other words, ${D_{fp}(R)}$ consists of complexes which are quasi-isomorphic to finite complexes of finitely generated free modules.
2. The perfect derived category ${D_{pf}(R)}$: this is the category of objects ${X \in D(R)}$ such that ${\hom(X, \cdot)}$ commutes with direct sums (i.e., the compact objects). It turns out that so-called perfect complexes are those that can be represented as finite complexes of finitely generated projectives.

One should think of the finitely presented objects as analogous to the finite cell complexes in topology, and the perfect objects as analogous to the retracts of finite cell complexes. (To push the analogy: the finite cell complexes are the smallest subcategory of the ${\infty}$-category of spaces containing ${\ast}$ and closed under finite colimits. The retracts of finite cell complexes are the compact objects in this ${\infty}$-category.) (more…)

This post continues the series on local cohomology.

Let ${A}$ be a noetherian ring, ${\mathfrak{a} \subset A}$ an ideal. We are interested in the category ${\mathrm{QCoh}(\mathrm{Spec} A \setminus V(\mathfrak{a}))}$ of quasi-coherent sheaves on the complement of the closed subscheme cut out by ${\mathfrak{a}}$. When ${\mathfrak{a} = (f)}$ for ${f \in A}$, then

$\displaystyle \mathrm{Spec} A \setminus V(\mathfrak{a}) = \mathrm{Spec} A_f,$

and so ${\mathrm{QCoh}(\mathrm{Spec} A \setminus V(\mathfrak{a}))}$ is the category of modules over ${A_f}$. When ${\mathfrak{a}}$ is not principal, the open subschemes ${\mathrm{Spec} A \setminus V(\mathfrak{a})}$ are generally no longer affine, but understanding quasi-coherent sheaves on them is still of interest. For instance, we might be interested in studying vector bundles on projective space, which pull back to vector bundles on the complement ${\mathbb{A}^{n+1} \setminus \left\{(0, 0, \dots, 0)\right\}}$. This is not affine once ${n > 0}$.

In order to do this, let’s adopt the notation

$\displaystyle X' = \mathrm{Spec} A \setminus V(\mathfrak{a}) , \quad X = \mathrm{Spec} A,$

and let ${i: X' \rightarrow X}$ be the open imbedding. This induces a functor

$\displaystyle i_* : \mathrm{QCoh}(X') \rightarrow \mathrm{QCoh}(X)$

which is right adjoint to the restriction functor ${i^* : \mathrm{QCoh}(X) \rightarrow \mathrm{QCoh}(X')}$. Since the composite ${i^* i_* }$ is the identity on ${\mathrm{QCoh}(X')}$, we find by a formal argument that ${i_*}$ is fully faithful.

Fully faithful right adjoint functors have a name in category theory: they are localization functors. In other words, when one sees a fully faithful right adjoint ${\mathcal{C} \rightarrow \mathcal{D}}$, one should imagine that ${\mathcal{C}}$ is obtained from ${\mathcal{D}}$ by inverting various morphisms, say a collection ${S}$. The category ${\mathcal{C}}$ sits inside ${\mathcal{D}}$ as the subcategory of ${S}$-local objects: in other words, those objects ${x}$ such that ${\hom(\cdot, x)}$ turns morphisms in ${S}$ into isomorphisms. (more…)

Let ${X \subset \mathbb{P}^r_{\mathbb{C}}}$ be a smooth projective variety, and let ${H}$ be a generic hyperplane. For generic enough ${H}$, the intersection ${X \cap H}$ is itself a smooth projective variety of dimension one less. The Lefschetz hyperplane theorem asserts that the map

$\displaystyle H \cap X \rightarrow X$

induces an isomorphism on ${\pi_1}$, if ${\dim X \geq 3}$.

We might be interested in analog over any field, possibly of characteristic ${p}$. Here ${\pi_1}$ has to be replaced with its étale analog, but otherwise it is a theorem of Grothendieck that ${H \cap X \rightarrow X}$ still induces an isomorphism on ${\pi_1}$, under the same hypotheses. This is one of the main results of SGA 2, and it uses the local cohomology machinery developed there. One of my goals in the next few posts is to understand some of the ideas that go into Grothendieck’s argument.

More generally, suppose ${Y \subset X}$ is a subvariety. To say that ${\pi_1(Y) \simeq \pi_1(X)}$ (always in the étale sense) is to say that there is an equivalence of categories

$\displaystyle \mathrm{Et}(X) \simeq \mathrm{Et}(Y)$

between étale covers of ${X}$ and étale covers of ${Y}$. How might one prove such a result? Grothendieck’s strategy is to attack this problem in three stages:

1. Compare ${\mathrm{Et}(X)}$ to ${\mathrm{Et}(U)}$, where ${U \supset Y}$ is a neighborhood of ${Y}$ in ${X}$.
2. Compare ${\mathrm{Et}(U)}$ to ${\mathrm{Et}(\hat{X}_Y)}$, where ${\hat{X}_Y}$ is the formal completion of ${X}$ along ${Y}$ (i.e., the inductive limit of the infinitesimal thickenings of ${Y}$).
3. Compare ${\mathrm{Et}( \hat{X})_Y)}$ and ${\mathrm{Et}(Y)}$.

In other words, to go from ${Y}$ to ${X}$, one first passes to the formal completion along ${Y}$, then to an open neighborhood, and then to all of ${Y}$. The third step is the easiest: it is the topological invariance of the étale site. The second step is technical. In this post, we’ll only say something about the first step.

The idea behind the first step is that, if ${Y}$ is not too small, the passage from ${U}$ to ${X}$ will involve adding only subvarieties of codimension ${\geq 2}$, and these will unaffect the category of étale covers. There are various “purity” theorems to this effect.

The goal of  this post is to sketch Grothendieck’s proof of the following result of Zariski and Nagata.

Theorem 10 (Purity in dimension two) Let ${(A, \mathfrak{m})}$ be a regular local ring of dimension ${2}$, and let ${X = \mathrm{Spec} A}$. Then the map

$\displaystyle \mathrm{Et}(X) \rightarrow \mathrm{Et}(X \setminus \left\{\mathfrak{m}\right\})$

is an equivalence of categories.

In other words, “puncturing” the spectrum of a regular local ring does not affect the fundamental group. (more…)

Let ${(A, \mathfrak{m})}$ be a regular local (noetherian) ring of dimension ${d}$. In the previous post, we described loosely the local cohomology functors

$\displaystyle H^i_{\mathfrak{m}}: \mathrm{Mod}(A) \rightarrow \mathrm{Mod}(A)$

(in fact, described them in three different ways), and proved a fundamental duality theorem

$\displaystyle H^i_{\mathfrak{m}}(M) \simeq \mathrm{Ext}^{d-i}(M, A)^{\vee}.$

Here ${\vee}$ is the Matlis duality functor ${\hom(\cdot, Q)}$, for ${Q}$ an injective envelope of the residue field ${A/\mathfrak{m}}$. This was stated initially as a result in the derived category, but we are going to use the above form.

The duality can be rewritten in a manner analogous to Serre duality. We have that ${H^d_{\mathfrak{m}}(A) \simeq Q}$ (in fact, this could be taken as a definition of ${Q}$). For any ${M}$, there is a Yoneda pairing

$\displaystyle H^i_{\mathfrak{m}}(M) \times \mathrm{Ext}^{d-i}(M, A) \rightarrow H^d_{\mathfrak{m}}(A) \simeq Q,$

and the local duality theorem states that it is a perfect pairing.

Example 1 Let ${k}$ be an algebraically closed field, and suppose that ${(A, \mathfrak{m})}$ is the local ring of a closed point ${p}$ on a smooth ${k}$-variety ${X}$. Then we can take for ${Q}$ the module

$\displaystyle \hom^{\mathrm{top}}_k(A, k) = \varinjlim \hom_k(A/\mathfrak{m}^i, k):$

in other words, the module of ${k}$-linear distributions (supported at that point). To see this, note that ${\hom_k(\cdot, k)}$ defines a duality functor on the category ${\mathrm{Mod}_{\mathrm{sm}}(A)}$ of finite length ${A}$-modules, and any such duality functor is unique. The associated representing object for this duality functor is precisely ${\hom^{\mathrm{top}}_k(A, k)}$.

In this case, we can think intuitively of ${H^i_{\mathfrak{m}}(A)}$ as the cohomology

$\displaystyle H^i(X, X \setminus \left\{p\right\}).$

These can be represented by meromorphic ${d}$-forms defined near ${p}$; any such ${\omega}$ defines a distribution by sending a function ${f}$ defined near ${p}$ to ${\mathrm{Res}_p(f \omega)}$. I’m not sure to what extent one can write an actual comparison theorem with the complex case. (more…)

Next Page »