Posted by Jason Polak on 15. April 2017 · Write a comment · Categories: math

I’ve made a small organizational change to this blog. I realize it might not have been easy to find series of posts on one topic, or the “post series”. They used to be on a page called “features” but I think that was too obscure. Now there’s the page “post series”, which is found above on the header. Along with the current series I have going on Non-unique Factorisation, it also lists five posts on Waldhausen categories.

Perhaps the most exciting in my mind is the six posts of “Wild Spectral Sequences”, which give little toy problems that can be solved using spectral sequences. I came up with all the proofs myself, and some of them might be published nowhere else. Though I admit some of them are quite basic, I think they are quite instructive!

What are you waiting for? Visit the Post Series page today for these topics!

Posted by Jason Polak on 15. April 2017 · Write a comment · Categories: commutative-algebra · Tags:

We are continuing the series on non-unique factorisation. For a handy table of contents, visit the Post Series directory.

In Part 1 of this series, we introduced for a commutative ring three types of relations:

  1. Associaties: $a\sim b$ means that $(a) = (b)$
  2. Strong associates: $a\approx b$ means that $a = ub$ for $u\in U(R)$
  3. Very strong associates: $a\cong B$ means that $a\sim b$ and either $a = b=0$ or $a = rb$ implies that $r\in U(R)$

Here, $U(R)$ denotes the group of units of $R$. We have already seen in a ring with nontrivial idempotents like $\Z\times \Z$, a nontrivial idempotent $e$ will be satisfy $e\sim e$ and $e\approx e$, but $e\not\cong e$ because $e = ee$ and yet $e$ is not a unit and nonzero.

Therefore, $\cong$ is not an equivalence relation for all commutative rings. But it is symmetric:

Proof. Suppose $a\cong b$. Then $a\sim b$ and so $b\sim a$. If $a$ and $b$ are not both zero, write $a = sb, b = ta$. If $b = ra$ then $a = sra = s^2rb$. Since $a\cong b$, this implies that $s^2r$ is a unit and so $r$ is a unit. Hence $b\cong a$.

Guess what? The relation $\cong$ is also transitive. Since the proof is similarly short I’ll leave the proof to the reader. So, $\cong$ is just missing being reflexive for all rings to be an equivalence relation for all rings. If $\cong$ is an equivalence relation for a ring $R$, then we say that $R$ is presimplifiable. We introduced this type of ring last time.
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Posted by Jason Polak on 11. April 2017 · Write a comment · Categories: commutative-algebra · Tags:

If $F$ is a field then the polynomial ring $F[x]$ is a unique factorisation domain: that is, every nonunit can be written uniquely as a product of irreducible elements up to a unit multiple. So in $\Q[x]$ for example, you can be sure that the polynomial $x^2 – 2 = (x-2)(x+2)$ can’t be factored any other way, and thus the only zeros of $x^2 – 2$ really are $\pm 2$.

If $F$ is not a field, then a polynomial might have a bunch of different factorisations. For example, in the ring $\Z/4[x]$ we can write $x^2 = (x)(x) = (x+2)(x+2)$. How can we make sense of factorisations in rings that are not unique factorisation domains? In order to do so, we first should make sure we understand what irreducible means in this context.

In the next several posts we’ll look at non-unique factorisation more closely, following a paper of Anderson and Valdes-Leon [1], but keeping the posts self-contained. We’ll start by looking at the concept of associates. One can in fact look at several different variations of associates. Here are three:
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Posted by Jason Polak on 21. March 2017 · Write a comment · Categories: elementary, exposition

Characteristic functions have magical properties. For example, consider a double summation:
$$\sum_{k=1}^M\sum_{r=1}^k a_{r,k}.$$
How do you switch the order of summation here? A geometric way to think of this is arrange the terms out and “see” that this sum must be equal to
$$\sum_{r=1}^M\sum_{k=r}^M a_{r,k}.$$
I find this unsatisfactory because the whole point of good notation is that you shouldn’t have to think about what it actually means. I do think it’s very important to understand what the notation means, but in doing formal manipulations it’s nice not to have to do that all the time.

A superior proof that these two sums are equal would be to write
$$\sum_{k=1}^M\sum_{r=1}^k a_{r,k} = \sum_{k=1}^M\sum_{r=1}^M a_{r,k}\delta(r\leq k)$$
where $\delta(r\leq k)$ is the function of the variables $r$ and $k$ that equals one exactly when $r\leq k$ and zero otherwise. Then we can happily switch the order of summation to get
$$\sum_{r=1}^M\sum_{k=1}^M a_{r,k}\delta(r\leq k).$$
Now, it’s trivial to get rid of the $\delta(r\leq k)$ function by writing
$$\sum_{r=1}^M\sum_{k=r}^M a_{r,k}.$$

Posted by Jason Polak on 21. March 2017 · Write a comment · Categories: commutative-algebra · Tags:

This post is a continuation of this previous one, though I repeat the main definitions for convenience.

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 $i$. Last time, we looked at the following theorem:

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
$$
{\rm Ext}_R^n(B,A) \cong {\rm Hom}_R(B,A/(x_1,\dots,x_n)A)$$

When $R$ is a Noetherian ring, $I$ a proper ideal of $R$, and $A$ a finitely-generated $R$-module, this theorem for $B = R/I$ says that the least integer $n$ such that ${\rm Ext}_R^n(R/I,A)\not= 0$ is exactly the length of a maximal regular sequence in $I$ on $A$.

The Noetherian and finitely generated hypotheses are crucial. Why is this? It’s because you need to have control over zero divisors. In fact you can see this by looking at the case $n = 0$:

Theorem. Let $R$ be a Noetherian ring, $I$ a proper ideal of $R$, and $A$ a finitely-generated $R$-module. Then every element of $I$ is a zero divisor on $A$ if and only if ${\rm Hom}_R(R/I,A)\not= 0$.
Proof. Since $A$ is a finitely generated $R$-module, that every element of $I$ is a zero divisor on $A$ is equivalent to $I$ being contained in the annihilator of a single nonzero element $a\in A$, which is in turn equivalent to every element of $I$ being sent to zero under the homomorphism
$$R\to A\\ 1\mapsto a.$$
Such homomorphisms are the same as nonzero homomorphisms $R/I\to A$. QED.

Here we are using this crucial fact:

Cool Theorem. For a finitely generated module $A$ over a Noetherian ring $R$, the zero divisors $Z(A)$ of $A$ in $R$ are a union of prime ideals of $R$, each of which are ideals maximal with respect to the property of being in $Z(A)$. Furthermore, each such prime is the annihilator of a single nonzero element of $A$.

In general, primes that are equal to the annihilator of a single element of a module $M$ are called the associated primes of $M$, and of course the theory of associated primes and primary decomposition is much more vast than this simple ‘Cool Theorem’, as is evident from Eisenbud’s 30-page treatment of them in his book Commutative Algebra. In practice however, I only ever seem to need this simple version of the ‘Cool Theorem’.

Posted by Jason Polak on 08. March 2017 · Write a comment · Categories: math

If $n \geq 0$ is an integer, define the number $n!$, pronounced “$n$ factorial”, as the number of bijections from an $n$-element set to itself. Therefore, $0! = 1$ and if $n > 0$ then
$$n! = 1\times \cdots\times n$$.
What’s the best way to calculate this quantity? The following “First Method” could possibly be the most straightforward method in the C language:

On other other hand, one might be tempted to reduce the number More »

Posted by Jason Polak on 05. March 2017 · Write a comment · Categories: commutative-algebra · Tags:

I’d like to invite readers of this blog to download my latest paper, to appear in the Canadian Mathematical Bulletin:

What is this paper about? It uses the theory of separable algebras to study separable polynomials in $\Z/n[x]$, which extends the usual definition of separability for polynomials over a field.

Let $d\geq 2$. A classical theorem of Leonard Carlitz says that for a prime $p$ with $q=p^k$, the number of monic separable polynomials of degree $d$ in $\F_q[x]$ is $q^d – q^{d-1}$. One can also define separable for polynomials in $\Z/n[x]$. In this case, since a polynomial cannot always be converted to a monic one by multiplying by a unit, it makes more sense count all separable polynomials. Deriving a formula for this number is exactly what my paper does.

Read the paper to see how it’s done! Although it talks about separable algebras, you can actually read it without knowing anything about this more advanced stuff as the interface between separable algebra theory and the concrete combinatorics is pretty clean. Or, you can just look at the final answer: the number of separable polynomials in $\Z/n[x]$ of degree at most $d$ for $d\geq 1$ is given by
$$\phi(n)n^d\prod_{i=1}^m(1 + p_i^d)$$
where $n = p_1^{k_1}\cdots p_m^{k_m}$ is the prime factorization of $n$ and $\phi(n) = |(\Z/n)^\times|$ is Euler’s phi function. The formulas in the paper have been checked mutliple times with Sage.

Posted by Jason Polak on 01. March 2017 · Write a comment · Categories: combinatorics

In this post we’ll give formulas for the number of bijective, injective, and surjective functions from one finite set to another. Although it’s not difficult, a formula for the number of surjective functions was one of the first problems I solved as an undergrad that got me interested in recurrence relations and combinatorics.

Let’s use the notation $[n] = \{ 1,2,\dots,n\}$ for an $n$-element set.
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Posted by Jason Polak on 01. March 2017 · Write a comment · Categories: math

Number theory (NT) on the arXiv is a big section. In fact over the last two years, a little over 5700 papers have been posted there. Just for fun I wrote a Python program to help me find the most popular words in the titles and abstracts of the papers. Excluding common english words and common mathematical words such prove, function, and theorem, here are the top twenty keywords I found:

  1. prime
  2. group
  3. integer
  4. modular
  5. rational
  6. elliptic
  7. quadratic
  8. series
  9. polynomial
  10. zeta
  11. class
  12. bound
  13. galois
  14. algebraic
  15. abelian
  16. representation
  17. adic
  18. sequence
  19. diophantine
  20. linear

And how does this compare to algebraic geometry (AG), with almost the same number of papers posted?
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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.