Category Archives: modules

## 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 $R$ […]

## Examples: Projective Modules that are Not Free

Here are nine examples of projective modules that are not free, some of which are finitely generated. Direct Products Consider the ring $R= \Z/2\times\Z/2$ and the submodule $\Z/2\times \{0\}$. It is by construction a direct summand of $R$ but certainly not free. And it's finitely generated! Another example is the submodule $\Z/2\subset \Z/6$, though this […]

## Noetherian, Artinian, but not Semisimple

There are many ways to define the propery of semisimple for a ring $R$. My favourite is the "left global dimension zero approach": a ring $R$ is left semisimple if every left $R$-module is projective, which is just the same thing as saying that every left $R$-module is injective. In particular, ideals are direct summands, […]

## Solution: Kaplansky’s Commutative Rings 4.3.2

Let $R$ be a ring and $M$ an $R$-module with a finite free resolution (an "FFR module"). That is, there exists an exact sequence $0\to F_n\to F_{n-1}\to \cdots\to F_0\to M\to 0$ with each $F_i$ a finitely generated free $R$-module. If we denote by $r_i$ the rank of $F_i$, then the Euler characteristic of $M$ is […]

## The Nonzero K-Theory of Finite Rings is Finite

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 […]

## Wild Spectral Sequences Ep. 2: Five, Isomorphism!

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 in an abelian category, if the rows […]

## Projectives and the Devious Determinant

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 Local Rings are Free

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 […]