# Weak Dimension At Most One Iff Every Ideal Is Flat

The flat dimension of an $R$-module $M$ is the infimum over lengths of flat resolutions of $M$, and the weak dimension (or $\mathrm{Tor}$-dimension) of $R$ is the supremum over all possible flat dimensions of modules. Let’s use $\mathrm{w.dim}(R)$ to denote the weak dimension of $R$. As with the global dimension, the weak dimension of $R$ can be computed as the supremum over the set of flat dimensions of the modules $R/I$ for $I$ running over the set of all left-ideals or right-ideals, either is fine!

So, if every ideal is flat, then $\mathrm{w.dim}(R) \leq 1$. What about the converse? If $\mathrm{w.dim}(R) \leq 1$, is it true that every ideal is flat? Let’s make a side remark in that if we replace weak dimension with global dimension, and flat with projective, then the answer follows from Schanuel’s lemma. However, as far as I know there is no Schanuel’s lemma when ‘projective’ is replaced by ‘flat’.

However, we can get away with using part of the proof of Schanuel’s lemma. Before continuing, the reader may wish to check out the statement and proof of Schanuel’s lemma using a double complex spectral sequence.
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# 12 Tips for Reading Math Books

Posted by Jason Polak on 07. July 2015 · Write a comment · Categories: math

Reading a math book cover to cover is a rite of passage for some, but the process can be pleasant or a nightmare; it can be rewarding or a mindless verification of theorems. This post is a guide to reading mathematics books, and is aimed for someone who is new at the process such as undergraduate or graduate students.

## 1. Find a book that suits your style of exploration

Some authors will be closer in line with your way of thinking than others. Unfortunately, it’s often quite difficult to tell which is which. One way that might help is reading through the first few pages of a book to give you a better impression of the author’s style. That, together with looking at the table of contents and introduction ought to be good enough to choose from a list of possible books on a given topic.

For example, I originally starting learning homological algebra through Mac Lane’s book ‘Homology’. After a while, it seemed as though the author was spending too much time on explicit descriptions of different homological constructions and less time on the big picture, or organising principles, though I didn’t know exactly what those were at the time. After about forty pages, I stopped and switched over to Weibel’s ‘An Introduction to Homological Algebra’, and it was much more pleasant and I ended up reading the entire book. This is, by the way, one of the rare times that I’ve actually finished an entire book.
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# Local Rings of Weak Dimension Zero are Division

Let $R$ be a ring and $M$ be an $R$-module. The flat dimension of $M$ is the infimum over all lengths of flat resolutions of $M$. Usually, the flat dimension of $M$ is denoted by $\mathrm{fd}_R(M)$. For example, $\mathrm{fd}_{\mathbb{Z}}(\mathbb{Q}) = 0$. Since $\mathbb{Q}$ has projective dimension $1$, the flat dimension and projective dimension of a module can be different. Sometimes they can be the same: $\mathbb{Z}/n$ for $n$ a positive integer has the same flat and projective dimension as $\mathbb{Z}$-modules.

The weak dimension of a ring $R$ is defined to be $\mathrm{w.dim}(R) = \sup_{M} \{ \mathrm{fd}_R(M) \}$ where $M$ runs over all left $R$-modules. Due to the symmetric nature of the tensor product, we can also take the supremum over all right $R$-modules, in contrast to the asymmetric nature of global dimension.
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# Yet Another non-Free Finitely Generated Projective

In the post Examples: Projective Modules that are Not Free, we saw nine examples of projective modules that are not free. On in particular was ‘the’ submodule $M = \oplus_{i=1}^\infty \mathbb{Z}$ of $\prod_{i=1}^\infty\mathbb{Z}$. Now, that’s a cool example to be sure, but the way we showed that $M$ was not free was to cite that $\prod_{i=1}^\infty\mathbb{Z}$ is uncountable. Actually, I like the argument a lot, but it’s possible to use the idea of that example and choose $M$ instead to be finite and different from all the examples in the aforementioned post. In fact, we’ll see a large class of examples that can be constructed from the ideas here.

The idea is to take an abelian group $A$ an consider $A$ as a module over its endomorphism ring $E = \mathrm{Hom}(A,A)$, where the endomorphisms are just homomorphisms $A\to A$ of abelian groups. Sometimes, $A$ can be projective over $E$. Actually, for a while it was believed that the projective dimension of $A$ over $E$ could only be $0$ or $1$, but eventually I.V. Bobylev showed in [1] that $A$ could have any projective dimension over $E$, including infinity!
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# Being Noetherian Is Not Local…Or Is It?

A commutative ring $R$ can be non-Noetherian and have all of its localisations at prime ideals Noetherian, such as the infamous $\prod_{i=1}^\infty \mathbb{Z}/2$. So being Noetherian is not a local property. However, there is an interesting variant of ‘local’ that does work, which I learnt from Yves Lequain’s paper [1]. It goes like this:

Theorem. Let $R$ be a ring and fix a left maximal ideal $M$ of $R$. Then $R$ is left Noetherian if and only if every left ideal contained in $M$ is finitely generated.

The nice thing about this statement is that it avoids localisation so it’s easy to state for noncommutative rings.
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# Krull vs Global Dimension in Commutative Noetherian Rings

Let $R$ be a ring. The projective dimension $\mathrm{pd}_R(M)$ of an $R$-module $M$ is the infimum over the lengths of projective resolutions of $M$. The left global dimension of $R$ is the supremum over the projective dimensions of all left $R$-modules. There is a notion of right global dimension where left modules are replaced with right modules. Since we’ll be talking about commutative rings only, we’ll just use global dimension to refer to both kinds, and write $\mathrm{g\ell.dim}(R)$ for the global dimension of $R$.

As an example, if $R$ is a field then $\mathrm{g\ell.dim}(R) = 0$ because every module is free. If $R$ is a principal ideal domain (PID), then $\mathrm{g\ell.dim}(R) = 1$. This is because any module $M$ admits a surjection $F\to M$ where $F$ is a free module. But the kernel of this map is also free since over a PID, a submodule of a free module is free. One of the first results in the theory of global dimension is that $\mathrm{g\ell.dim}(R[x]) = 1 + \mathrm{g\ell.dim}(R)$. So far then we have examples of rings with any finite global dimension.
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# Pretty Plots with Pygame and Python

Posted by Jason Polak on 01. April 2015 · Write a comment · Categories: math

The other day I learned a little about the pygame python module that provides access to sdl functions on Linux along with some higher level functions. Basically, this means that you can use pygame to access some pretty low level graphics to draw things. I thought it might be fun to use it to make pretty pictures. Here’s an example where points are plotted, with the colour of each point a function of the coordinates. These simple examples should indicate how this type of low-level drawing can be used, perhaps to visualise complicated data or make interesting animations.

In sdl as with many computer graphics systems, coordinates start from the top left with $(0,0)$, and proceed positively right and down. The equations presented below give the red-green-blue (RGB) values of the color of the point at $(x,y)$. Each value is taken modulo $255$ (actually, I should have taken the value modulo $256$, since each color goes from $0$ to $255$, but I didn’t think carefully about this when I wrote the program!).

 \begin{align*} r &= x\\ g &= x-y\\ b &= x + y \end{align*}

Posted by Jason Polak on 25. March 2015 · Write a comment · Categories: algebraic-topology, books

Doug Ravenel has made his book Nilpotence and periodicity in stable homotopy theory available for free download along with a list of errata, also available at the same page as the book.

Here is the official description from Princeton University Press:

Nilpotence and Periodicity in Stable Homotopy Theory describes some major advances made in algebraic topology in recent years, centering on the nilpotence and periodicity theorems, which were conjectured by the author in 1977 and proved by Devinatz, Hopkins, and Smith in 1985. During the last ten years a number of significant advances have been made in homotopy theory, and this book fills a real need for an up-to-date text on that topic.

Ravenel’s first few chapters are written with a general mathematical audience in mind. They survey both the ideas that lead up to the theorems and their applications to homotopy theory. The book begins with some elementary concepts of homotopy theory that are needed to state the problem. This includes such notions as homotopy, homotopy equivalence, CW-complex, and suspension. Next the machinery of complex cobordism, Morava K-theory, and formal group laws in characteristic p are introduced. The latter portion of the book provides specialists with a coherent and rigorous account of the proofs. It includes hitherto unpublished material on the smash product and chromatic convergence theorems and on modular representations of the symmetric group.

At the official page of the book, you can also buy a paperback copy.

# Dokchitser’s Notes on l-adic Representations

Posted by Jason Polak on 18. March 2015 · Write a comment · Categories: math

It’s always a treat when a set of lecture notes for a workshop find their way to my arXiv RSS feeds. Today I found notes written by Samuele Anni on Vladimir Dokchitser’s lectures $l$-adic representations and their associated invariants (20pp). Here is the abstract:

These are notes from a 3-lecture course given by V. Dokchitser at the ICTP in Trieste, Italy, 1st–5th of September 2014, as part of a graduate summer school on “L-functions and modular forms”. The course is meant to serve as an introduction to l-adic Galois representations over local fields with “l not equal to p”, and has a slightly computational bent. It is worth mentioning that the course is not about varieties and their etale cohomology, but merely about the representation theory.

Here is a quotation from the introduction containing the prerequisites:

The prerequisites for the course are fairly modest: the theory of local fields and representation theory of finite groups, such as might be covered in a Masters level course, and, for the sake of examples and motivation, the theory of elliptic curves over local fields (Silverman’s book is more than sufficient). Some previous exposure to L-functions is desirable, as, after all, this was the main topic of the school.

# A Serre Fibration that is not a Hurewicz Fibration

Posted by Jason Polak on 18. March 2015 · Write a comment · Categories: math

Given two similar definitions, it is very valuable to have a counterexample distinguishing them. Here is one case where this arises: A Hurewicz fibration $p:E\to B$ of unbased spaces is a continuous map such that for every space $X$ and every commutative diagram
$$\begin{matrix} X & \longrightarrow & E\\ \downarrow & ~ & \downarrow \\ X\times I & \longrightarrow & B\end{matrix}$$
there exists a map $X\times I\to E$ making the resulting diagram commute. On the other hand, a Serre fibration $p:E\to B$ is a continuous map with the same property for every CW-complex $X$. From the definition it seems that there might be Serre fibrations that are not Hurewicz fibrations. In fact, there are! An example first appeared in R. Brown’s paper [1] in 1966. We define $E$ to be the subset of the Euclidean plane $\R^2$ defined by More »