Posted by Jason Polak on 03. May 2014 · Write a comment · Categories: modules · Tags:

Let \(R\) be a finite ring. The example we’ll have in mind at the end is the ring of \(2\times 2\) matrices over a finite field, and subrings. A. Kuku proved that \(K_i(R)\) for \(i\geq 1\) are finite abelian groups. Here, \(K_i(R)\) denotes Quillen’s \(i\)th \(K\)-group of the ring \(R\). In this post we will look at an example, slightly less simple than \(K_1\) of finite fields, showing that these groups can be arbitrarily large. Before we do this, let us briefly go over why this is true

But even before this, can you think of an example showing why this is false for \(i=0\)?
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A commutative Noetherian local ring $ R$ with maximal ideal $ M$ is called a regular local ring if the Krull dimension of $ R$ is the same as the dimension of $ M/M^2$ as a $ R/M$-vector space.

In studying regular local rings one often uses the following lemma in inductive arguments: if $ R$ is an arbitrary commutative Noetherian local ring with maximal ideal $ M$, and $ M$ consists entirely of zero divisors, then the projective dimension $ \mathrm{pd}_R(A)$ is either zero or infinity for any finitely-generated $ R$-module $ A$. In other words, the only $ R$-modules of finite projective dimension are the projective (hence free) modules.

This is a neat little result that has a fun More »

It’s time for another installment of Wild Spectral Sequences! We shall start our investigations with a classic theorem useful in many applications of homological algebra called Schanuel’s lemma, named after Stephen Hoel Schanuel who first proved it.

Consider for a ring $ R$ the category of left $ R$-modules, and let $ A$ be any $ R$-module. Schanuel’s lemma states: if $ 0\to K_1\to P_1\to A\to 0$ and $ 0\to K_2\to P_2\to A\to 0$ are exact sequences of $ R$-modules with $ P$ projective, then $ K_1\oplus P_2\cong K_2\oplus P_1$.

We shall prove this using spectral sequences. I came up with this proof while trying to remember the “usual” proof of Schanuel’s lemma and I thought that this would be a good illustration of how spectral sequences can be used to eliminate the dearth of clarity in the dangerous world of diagram chasing.

Before I start, I’d like to review a pretty cool fact I which I think of as expanding the kernel, which is pretty useful More »

Last time on Wild Spectral Sequences, we conquered the snake lemma using a spectral sequence argument. This time, we meet a new beast: the five lemma. The objective is the usual: prove the five lemma using spectral sequences.

Recall that the five lemma states that given a diagram

fivediag

in an abelian category, if the rows are exact and $ a,b,d,e$ are isomorphisms, then so is $ c$. Actually, the hypotheses are too strong. It suffices to have $ b,d$ isomorphisms, $ a$ an epimorphism and $ e$ a monomorphism. One can deduce this via J. Leicht’s “strong four lemma” (which we might try and prove via a spectral sequence too) or just by using the regular diagram-chasing proof of the five lemma.
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The determinant is certainly a fascinating beast. But what is the determinant? Is it a really just a number or a function on matrices? In this post I hope to convince you that the answer is ‘no’. In fact, we will see that the determinant, suitably modified, can be used to classify certain types of projective modules over nice rings.

Determinants of Matrices

Let $ R$ be a commutative ring and $ n$ be a natural number. Just as in the case of vector spaces, an $ R$-module map $ f:R^n\to R^n$ can be given by an $ n\times n$ matrix with coefficients in $ R$. Moreover, we can compute the determinant of this matrix just as in linear algebra. In fact, various notions of “determinants” also exist when $ R$ is not commutative, but we will stick with the commutative case.
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Posted by Jason Polak on 06. May 2012 · 2 comments · Categories: commutative-algebra, modules · Tags: , , , ,

Conventions and definitions: Rings are unital and not necessarily commutative. Modules over rings are left modules. A local ring is a ring in which the set of nonunits form an ideal. A module is called projective if it is a direct summand of a free module.

Today I shall share with you the wonderful result that any projective module over a local ring is free. We shall follow Kaplansky (reference given below), who first proved this result.

Now modules are in fact my favourite mathematical objects. They are like vector spaces, except that they are interesting. Of course, this “interesting” can be irksome if one has to solve a problem and these interesting properties throw a wrench in the works. However, by themselves modules are certainly curious creatures worthy of intense and gruelling analysis!

Of course, when the idea of a module was first conceived, mathematicians attempted to port all kinds of ideas from vector spaces into the world of modules. Some, like the direct sum construction, worked flawlessly. Other concepts such as rank, fortunately or unfortunately depending on your perspective, did not turn out so well (think about it: if everything worked well with modules then there’d be much less interesting math).
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Posted by Jason Polak on 01. February 2012 · Write a comment · Categories: commutative-algebra, modules · Tags: ,

Introduction

Assuming the axiom of choice, any vector space possesses the pleasant but prosaic property* that it is determined up to isomorphism by the cardinality of its basis.

For instance, consider $ \prod_\omega \mathbb{Z}/2$ and $ \oplus_{2^\omega} \mathbb{Z}/2$. Both are vector spaces over the finite field $ \mathbb{Z}/2$ so to show that they are isomorphic, we need to show that their respective bases have the same cardinality. The vector space on the right is written as a direct summand and so we can see that its basis must have size $ \mathfrak{c}$. On the other hand, the vector space $ \prod_\omega \mathbb{Z}/2$ has cardinality $ \mathfrak{c}$ over a finite field, so its basis must have the same cardinality as the space itself. Aren’t vector spaces a walk in the park; a piece of cake; easy as pie (ok, enough metaphors?!)?

From $ \mathbb{Z}/2$ to $ \mathbb{Z}$ Modules

But what if we sent the above proof to a publisher who didn’t yet have the “2” character or the “/” installed on her printing press? Then all hell would break loose because $ \prod_\omega \mathbb{Z}$ and $ \oplus_{2^\omega} \mathbb{Z}$ aren’t vector spaces any more, and the previous paragraph would be rife with errors. But they certainly are abelian groups, and they have a bit more spice than those vector spaces. So are they isomorphic? They do have the same cardinality. Fortunately for us, Baer (“Abelian Groups without Elements of Finite Order”, Duke Math J. 3 (1937), pp. 88-122) answered this question in the negative (fortunately, because otherwise abelian groups would be less exciting). In fact, this question is particularly interesting to me because I had wondered about it a few months ago, and now I have the answer, thanks to Faith’s book for the references.
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Posted by Jason Polak on 28. January 2012 · Write a comment · Categories: books, modules · Tags: , ,

As it happens every so often, I browse the mathematical library pseudorandomly, and look out for interesting titles; usually a prerequisite for interesting is that they have something to do with the realm of algebra. This is exactly how I found Faith’s book, with its captivating title urging me to borrow it.

Now, inevitably in mathematical research, one has to efficiently skim through papers and books to find specific ideas and facts. The unfortunate thing is that sometimes it is easy to neglect the stimulation of the idle curiosity that probably brought most mathematicians into their fields in the first place, and so I try to combat this neglect by my idle browsing and blogging.

I try not to spend too much time on this so that I progress with my degree, but I try to nurture my curiosity through reading anything that looks interesting. Returning to books, I do believe there are few worse literary follies than a graduate algebra textbook that lacks imagination in its examples and theorems and passion in its explication. I only fear that such books will tend to promote in the learning of higher algebra what most institutions have done with calculus, and that is to make it a tiresome mechanical effort, washing away the once vibrant and fanciful colours from the gentle tendrils of the mind.

But fear not! Should the mental dessication start to occur in a young algebraist’s mind; should the flames of passion dim for the wonders of the injective module, she can always turn to the entire object of this post, videlicet Faith’s “Rings and Things and a Fine Array of Twentieth Century Associative Algebra”
. I refer to the second edition, incidentally, which corrects many errors from the 1st edition.

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