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…)

 

May 15, 2012

Re-learning homological algebra

Posted by Akhil Mathew under algebra, commutative algebra | Tags: , , , |
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I’ve been trying to re-understand some of the proofs in commutative and homological algebra. I never really had a good feeling for spectral sequences, but they seemed to crop up in purely theoretical proofs quite frequently. (Of course, they crop up in computations quite frequently, too.) After learning about derived categories it became possible to re-interpret many of these proofs. That’s what I’d like to do in this post.

Here is a toy example of a result, which does not use spectral sequences in its usual proof, but which can be interpreted in terms of the derived category.

Proposition 1 Let {(A, \mathfrak{m})} be a local noetherian ring with residue field {k}. Then a finitely generated {A}-module {M} such that {\mathrm{Tor}_i(M, k) = 0, i > 0} is free.

Let’s try to understand the usual proof in terms of the derived category. Throughout, this will mean the bounded-below derived category {D^-(A)} of {A}-modules: in other words, this is the category of bounded-below complexes of projectives and homotopy classes of maps. Any module {M} can be identified with an object of {D^-(A)} by choosing a projective resolution.

So, suppose {M} satisfies {\mathrm{Tor}_i(M, k) = 0, i > 0}. Another way of saying this is that the derived tensor product

\displaystyle M \stackrel{\mathbb{L}}{\otimes} k

has no homology in negative degrees (it is {M \otimes k} in degree zero). Choose a free {A}-module {P} with a map {P \rightarrow M} which induces an isomorphism {P \otimes k \simeq M \otimes k}. Then we have that

\displaystyle P \stackrel{\mathbb{L}}{\otimes} k \simeq M \stackrel{\mathbb{L}}{\otimes} k

by hypothesis. In particular, if {C} is the cofiber (in {D^-(A)}) of {P \rightarrow M}, then {C \stackrel{\mathbb{L}}{\otimes} k = 0}.

We’d like to conclude from this that {C} is actually zero, or that {P \simeq M}: this will imply the desired freeness. Here, we have:

Lemma 2 (Derived Nakayama) Let {C \in D^-(A)} have finitely generated homology. Suppose {C \stackrel{\mathbb{L}}{\otimes} k = 0}. Then {C = 0}. (more…)

 

February 17, 2010

Kahler differentials and regular local rings

Posted by Akhil Mathew under algebra, algebraic geometry, commutative algebra | Tags: , , , |
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Finally, we’re going to come to the Kahler criterion for regularity.  As far as algebraic geometry is concerned, it states that a variety over an algebraically closed field of characteristic zero is nonsingular precisely when the sheaf of differentials on it (to be defined shortly) is locally free of rank equal to the dimension.

Theorem 1 Suppose {A} is a local domain which is a localization of a finitely generated {k}-algebra for {k} a field of characteristic zero, with residue field {k}. Then {A} is a regular local ring if and only if {\Omega_{A/k}} is a free {A}-module of rank {\dim A}.

 

First, I claim {\Omega_{A/k}} is finitely generated. This follows because the corresponding claim is true for a polynomial ring, we have a conormal sequence implying it for finitely generated algebras over a field, and taking differentials commutes with localization.

Let {K} be the residue field of {A}. I claim

\displaystyle \boxed{ \dim k \otimes \Omega_{A/k} = \dim( \mathfrak{m}/\mathfrak{m}^2 ) , \quad \dim (K \otimes \Omega_{A/k}) = \dim A.}

Then, the theorem will follow from the next lemma:

Lemma 2

Let {M} be a finitely generated module over the local noetherian domain {A}, with residue field {k} and quotient field {K}. Then {M} is free iff\displaystyle \dim K \otimes M = \dim k \otimes M (more…)
 

October 20, 2009

Unramified extensions

Posted by Akhil Mathew under algebra, algebraic number theory, number theory | Tags: , , |
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As is likely the case with many math bloggers, I’ve been looking quite a bit at MO and haven’t updated on some of the previous series in a while.

Back to ANT. Today, we tackle the case {e=1}. We work in the local case where all our DVRs are complete, and all our residue fields are perfect (e.g. finite) (EDIT: I don’t think this works out in the non-local case). I’ll just state these assumptions at the outset. Then, unramified extensions can be described fairly explicitly. (more…)

 

July 22, 2009

Why simple modules are often finite-dimensional II

Posted by Akhil Mathew under algebra, representation theory | Tags: , , |
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I had a post a few days back on why simple representations of algebras over a field {k} which are finitely generated over their centers are always finite-dimensional, where I covered some of the basic ideas, without actually finishing the proof; that is the purpose of this post.

So, let’s review the notation: {k} is our ground field, which we no longer assume algebraically closed (thanks to a comment in the previous post), {A} is a {k}-algebra, {Z} its center. We assume {Z} is a finitely generated ring over {k}, so in particular Noetherian: each ideal of {Z} is finitely generated.

Theorem 1 (Dixmier, Quillen) If {A} is a finite {Z}-module, then any simple {A}-module is a finite-dimensional {k}-vector space.

(more…)