Posted by Jason Polak on 22. February 2017 · Write a comment · Categories: commutative-algebra, homological-algebra · Tags:

Let $R$ be a commutative ring and $A$ and $R$-module. We say that $x_1,\dots,x_n\in R$ is a regular sequence on $A$ if $(x_1,\dots,x_n)A\not = A$ and $x_i$ is not a zero divisor on $A/(x_1,\dots,x_{i-1})A$ for all $i$. Regular sequences are a central theme in commutative algebra. Here’s a particularly interesting theorem about them that allows you to figure out a whole bunch of Ext-groups:

Theorem. Let $A$ and $B$ be $R$-modules and $x_1,\dots,x_n$ a regular sequence on $A$. If $(x_1,\dots,x_n)B = 0$ then
$$
{\rm Ext}_R^n(B,A) \cong {\rm Hom}_R(B,A/(x_1,\dots,x_n)A)$$

This theorem tells us we can calculate the Ext-group ${\rm Ext}_R^n(B,A)$ simply by finding a regular sequence of length $n$, and calculating a group of homomorphisms. We get two cool things out of this theorem: first, a corollary of this theorem is that any two maximal regular sequences on $A$ have the same length if they are both contained in some ideal $I$ such that $IA\not= A$, and second, it enapsulates a whole range of Ext-calculations in an easy package.

For example, let’s say we wanted to calculate ${\rm Ext}_\Z^1(\Z/2,\Z)$. Well, $2\in\Z$ is a regular sequence, and so the above theorem tells us that this Ext-group is just ${\rm Hom}_\Z(\Z/2,\Z/2) \cong\Z/2$.

Another example: is ${\rm Ext}_{\Z[x]}^1(\Z,\Z[x])\cong\Z$.

Of course, the above theorem is really just a special case of a Koszul complex calculation. However, it can be derived without constructing the Koszul complex in general, and so offers an instructive and minimalist way of seeing that for Noetherian rings and finitely generated modules, the notion of length of a maximal regular sequence is well-defined.

Posted by Jason Polak on 31. January 2017 · Write a comment · Categories: math

This post is a list of various books in commutative algebra (mostly called ‘Commutative Algebra’) with some comments. It is mainly geared towards students who might want to read about the subject on their own, though others might find it useful. It is not meant to be a comprehensive list, as it reflects the random process of my coming into contact with them.
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Posted by Jason Polak on 25. January 2017 · Write a comment · Categories: commutative-algebra, homological-algebra

We already saw that an abelian group with a $\Z$-direct summand is projective over its endomorphism ring. Finitely generated abelian groups are also projective over their endomorphism rings by essentially the same argument. What’s an example of an abelian group that is not projective over its endomorphism ring?

Here’s one: the multiplicative group $Z(p^\infty)$ of all $p$-power roots of unity. Another way to define this group is $\Z[p^{-1}]/\Z$. What is the endomorphism ring of this group? In fact it is the $p$-adic integers $\Z_p$. Indeed, an endomorphism $Z(p^\infty)\to\Z(p^\infty)$ has to send $1/p$ to an element $a_1$ such that $pa_1 = 0$. So we have the choice of the elements $0/p, 1/p,\cdots, (p-1)/p$, which form the cyclic subgroup $\Z/p$.

Similarly, $1/p^2$ has to be sent to an element $a_2$ such that $p^2a_2 = 0$, but also $pa_2 = a_1$. So $a_2$ has to be of the form $n/p^2$ where $n\in \Z$; in other words, $a_2$ can be in the cyclic subgroup $\Z/p^2$ generated by $1/p^2$. Hence, an endomorphism of $Z(p^\infty)$ is specified by an element of the inverse system $\cdots\to \Z/p^3\to \Z/p^2\to \Z/p$ where the transition maps are multiplication by $p$: in other words the $p$-adic integers $\Z_p$.

Now, we see that $Z(p^\infty)$ cannot be a projective $\Z_p$-module. Indeed, $\Z_p$ is a local ring and hence any projective $\Z_p$-module is in fact free (Kaplansky’s theorem) and in particular torsionfree. However, $Z(p^\infty)$ has nothing but torsion! In fact we can say more: since $\Z_p$ is a principal ideal domain, it has global dimension one, so the projective dimension of $Z(p^\infty)$ as a $\Z_p$-module is one.

Posted by Jason Polak on 25. January 2017 · Write a comment · Categories: math

I’ve spent countless hours thinking about associative rings. Yet, during my research today I read a paper by C.U. Jensen [1] and came across this elementary fact that I never thought about: if $R$ is an integral domain and $x,y$ are nonunits with $y$ divisible by every positive power of $x$ then $R$ is not Noetherian. Perhaps I’ve used this unconsciously, but I had to take a second to prove it.

Here’s the proof: if $y = z_1x = z_2x^2 = \cdots$ then the ideal $(z_1,z_2,\cdots)$ cannot be finitely generated, because $z_n = r_1z_1 + \cdots + r_{n-1}z_{n-1}$ and $y = z_ix^i$ for all $i$ implies that $x$ is actually invertible.

This fact is curious, because it plays into constructing some nonstandard models of the integers as I learned from Jensen’s paper: If $\Ucl$ is a nonprincipal ultrafilter on the natural numbers $\N$, then the product $\Z^\N/\Ucl$ is one such nonstandard model: it satisfies precisely the same first-order sentences that $\Z$ does. Yet, it’s not Noetherian, because the element represented by $(2,4,8,16,32,64,…)$ is not a unit and divisible by every power of $2$.

In fact, not only is the ultraproduct $\Z^\N/\Ucl$ not Noetherian, it actually has global dimension two. So, neither Noetherian nor having global dimension one is expressible in first-order logic.

It’s amazing what basic facts you can still come across in this era. Perhaps someone should write the book 100 Quick Facts You Didn’t Know About Rings?!

[1] Jensen. Peano rings of arbitrary global dimension. Journal of the London Mathematical Society, 1980, 2, 39-44

Posted by Jason Polak on 16. January 2017 · Write a comment · Categories: commutative-algebra · Tags:

An abelian group $A$ is a left $E = {\rm End}(A)$-module via $f*a = f(a)$. If $B$ is a direct summand of $A$ as an abelian group, then ${\rm Hom}(B,A)$ is also a left $E$-module and is in fact a direct summand of $E$ as an $E$-module, so it is $E$-projective. In particular, if $B = \Z$, then ${\rm Hom}(B,A)\cong A$ as $E$-modules. Thus $A$ is a projective $E$-module whenever $A$ has $\Z$ as a direct-summand.

These observations allow us to construct projective modules that often aren’t free over interesting rings. Take the abelian group $A = \Z\oplus \Z$ for instance. Its endomorphism ring $E$ is the ring $M_2(\Z)$ of $2\times 2$ matrices with coefficients in $\Z$. As we have remarked, $\Z\oplus \Z$ must be projective as an $M_2(\Z)$-module.

Is $\Z\oplus\Z$ free as an $M_2(\Z)$-module? On the surface, it seems not to be, but of course we need proof. And here it is: for each element of $\Z\oplus \Z$, there exists an element of $M_2(\Z)$ annihilating it. Such a thing can’t happen for free modules.

One might wonder, is every $M_2(\Z)$-module projective? Or in other words, is $M_2(\Z)$ semisimple? Let’s hope not! But $M_2(\Z)$ is thankfully not semisimple: $\Z/2\oplus\Z/2$ is a $M_2(\Z)$-module that is not projective: any nonzero element of $M_2(\Z)$ spans a submodule of infinite order, and therefore so must any nonzero element of a nonzero projective.

Posted by Jason Polak on 16. January 2017 · Write a comment · Categories: exposition · Tags:

A neutrino is a ultra low mass chargless subatomic particle that is produced in a variety of nuclear reactions such as beta decay. Neutrinos are incredibly abundant, but because of their size and lack of charge, they pass through almost anything and are extremely difficult to detect. Ray Jayawardhana’s book Neutrino Hunters is a fascinating glimpse into the great strides made by physicists to actually detect and understand these mysterious particles.

Neutrino Hunters progresses historically from Wolfgang Pauli’s hypothesising the existence of neutrino to its experimental confirmation and analysis as a central player in the workings of the universe. Several colourful and intriguing portraits of physicists appear along with the incredible experiments that were devised to understand the neutrino. Particularly fascinating were the early neutrino detectors, filled with hundreds of liters of dry-cleaning fluid that had to be placed deep underground to avoid interference. As time goes on, the detectors become more complex and even more ingenious, though I’ll leave out specifics so as not to spoil the book. Suffice it to say, some of the feats accomplished with complicated apparatus are truly astounding.

After the story of the then-state of the art is told, the author explains some future experiments and unresolved speculations, such as the possibility of a fourth neutrino flavour and neutrino communication.

The author manages to keep an excellent balance between precise scientific explanation for the nonspecialist and lively historical recounting. As someone who usually is terribly bored with history, I was never bored reading this book. This is not surprising, as the author is a physicist himself and does not fall into the habit (usually possessed by journalists without scientific training) of including vast amounts of irrelevant detail.

Overall, Jayawardhana is precise without ever being overly technical, and Neutrino Hunters should be easily readable by anyone with a basic knowledge of the atom and a yearning to glimpse into the subatomic world and the origins of the universe. Highly recommended.

Posted by Jason Polak on 11. January 2017 · Write a comment · Categories: commutative-algebra

Wolmer Vasconcelos [1] gave the following classification theorem about commutative local rings of global dimension two:

Theorem. Let $A$ be a commutative local ring of global dimension two with maximal ideal $M$. If $M$ is principal or not finitely generated, then $A$ is a valuation domain. Otherwise $M$ is generated by a regular sequence of two elements, and $A$ will be Noetherian if and only if it is completely integrally closed.

In this post we shall prove a small part of this theorem: that if $A$ is a commutative local ring of global dimension two and $M$ is a principal ideal, then $A$ is a valuation domain (i.e. for all $a,b\in A$ either $a | b$ or $b | a). As always, we use the term local ring to mean a commutative ring with a unique maximal ideal.
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Posted by Jason Polak on 07. January 2017 · Write a comment · Categories: exposition · Tags:

Math blogging is a fun part of being a mathematician. For me, it’s an aid to reading literature and an outlet for my writing prediliction. Blogging is cool because you write whatever you want without having to worry about the sometimes arbitrary and muddled standards of publication. But how do you do it? In this post I’ll give six tips on math blogging that should help.

1. Don’t worry about failure

Blogging is such a carefree medium that there’s no reason to worry about failed or unfinished posts. I have a folder of such posts just in case I want to resurrect any of them, and it contains around 80 posts–or just under half of the number of all the posts published. These range from very preliminary to finished and polished, totalling about 48000 words, or about twice the number of words in Hampton Fancher’s script for the movie Blade Runner. And you know what? It was fun to write those too, but in the end I decided they were not the right material for Aleph Zero Categorical.

2. Don’t worry about sophistication

Surprisingly, it’s actually fine to write about finitely generated abelian groups even though you’re working on interuniversal Teichmuller theory. It’s also fine to have a sophisticated blog. Mathematics actually needs much more exposition, so there’s really no need to restrict yourself to certain topics because you think other mathematicians will look down on it.

3. Keep your blog focused

Pick a topic and stick with it. For my blog, it’s math and related fields, like computer science and applications, though mostly I just write about algebra. There have been occasions where I’ve been tempted to post reviews of books I’ve read in other fields like biology but I’ve resisted because that was outside the scope I set out, and I doubt it would make sense to my audience. Writing outside the scope is a slipperly slope: first it’ll be one or two posts on chemistry, and then pretty soon you’ll be writing on bizarre topics like politics and South American mushrooms.

4. Update regularly

I require myself to produce one post per month. Only once or twice did that fail in the past five or so years, and on average I’m way above that. Not only will a regular update requirement keep you blogging, but it will keep your readers around. Most math blogs miss months, so I figure I’m safe.

5. Heed the format

A blog post is not supposed to be long, and people don’t visit blogs to read proofs of the four colour theorem. I’ve definitely written posts that were too long, and in the end those were not so popular. Keep posts to a main idea, keep it under a thousand words, and your blog will be far more readable for it.

6. In the end, it doesn’t have to be math

Weird advice for a post on math blogging right? But in the end if you don’t enjoy math blogging, you might still enjoy food blogging or posting pictures of rocks.

A ring of left global dimension zero is a ring $R$ for which every left $R$-module is projective. These are also known as semisimple rings of the Wedderburn-Artin theory fame, which says that these rings are precisely the finite direct products of full matrix rings over division rings. Note the subtle detail that “semisimple” is used here instead of “left semisimple” because left semisimple is the same thing as right semsimple.

In the commutative world, the story for Krull dimension zero is not so simple. For example, every finite commutative ring has Krull dimension zero. Indeed, if $R$ is a ring with Krull dimension greater than zero, then there would exist two distinct primes $P\subset Q$ so that $R/P$ is an integral domain that is not a field. Thus, $R$ is infinite, as every finite integral domain is a field.

The story becomes simpler if we require $R$ to have no nilpotent elements: i.e., that $R$ is reduced. In this case, a commutative ring is reduced and of Krull dimension zero if and only if every principal ideal is idempotent. Every principal ideal being idempotent means that for every $x\in R$ there is an $a\in R$ such that $xax = x$. Rings, commutative or not, satisfying this latter condition are called von Neumann regular. So:

Theorem. A commutative ring has Krull dimension zero and is reduced if and only if it is von Neumann regular.

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Posted by Jason Polak on 29. December 2016 · Write a comment · Categories: commutative-algebra · Tags:

Let $R$ be a commutative ring. We say that an $R$-algebra $A$ is separable if it is projective as an $A\otimes_R A^{\rm op}$-module. Examples include full matrix rings over $R$, finite separable field extensions, and $\Z[\tfrac 12,i]$ as a $\Z[\tfrac 12]$-algebra.

The 1970 classic Separable Algebras by deMeyer and Ingraham acquaints the reader with this important class of algebras from two viewpoints: the noncommutative one through structure theory and the Brauer group, and the commutative one through Galois theory.

This book accomplished the rare feat of keeping me interested; throughout its pages I found I could apply its results to familiar situations: Why are the only automorphisms of full matrix rings over fields inner? Why are such rings simple? What makes Galois theory tick? Separable Algebras explains with clarity how familiar algebra works through the lens of separable algebras.
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