Category Archives: homological-algebra

## Regular Sequences and Ext Calculations

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 […]

## An Abelian Group of Endoprojective Dimension One

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 […]

## Commutative von Neumann Regular Rings

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 […]

## Self Injective Integral Domains are Fields: Two Proofs

For finite commutative rings, integral domains are the same as fields. This isn't too surprising, because an integral domain $R$ is a ring such that for every nonzero $a\in R$ the $R$-module homomorphism $R\to R$ given by $r\mapsto ra$ is injective. Fields are those rings for which all these maps are surjective. But injective and […]

## Flat Modules and Finitely Generated Submodules

Let $R$ be a ring and $M$ and $R$-module. If every finitely generated submodule of $M$ is flat, then so is $M$, because direct limits commute with the $\mathrm{Tor}$-functor. What about the converse? If $M$ is flat, are all its finitely generated submodules flat too? Not necessarily! In fact, here's a roundabout argument without an […]

## Projective Principal Ideals, Idempotent Annihilators

Given an idempotent $e$ in a ring $R$, the right ideal $eR$ is projective as a right $R$-module. In fact, $eR + (1-e)R$ is actually a direct sum decomposition of $R$ as a right $R$-module. An easy nontrivial example is $\Z\oplus\Z$ with $e = (1,0)$. Fix an $a\in R$. If $aR$ is a projective right […]

## Projectivity and the Double Dual

Projective modules are the algebraic analogues of vector bundles, and they satisfy some strong properties. To state one we will first introduce the notation $P^* := {\rm Hom}_R(P,R)$ for any right $R$-module $P$. (Working with right $R$-modules is just a convention) Here's one property that projective modules satisfy: if $P$ is a right projective module […]

## The "Fractional" Isomorphism Theorem

For modules one has the isomorphism theorem $(A/C)/(B/C) \cong A/B$ for $C\leq B\leq A$. One way to remember it is through analogy with canceling of fractions. Another way to remember and prove it is to put all the modules in a 3×3 commutative diagram  \begin{matrix} C & \to & B & \to & B/C\\ […]

## The Torsion Subgroup of an Abelian Group

Let $A$ be an abelian group. We call an element $a\in A$ torsion if there exists a natural number $n$ such that $na = 0$. The set of all torsion elements $T(A)$ of $A$ form a subgroup of $A$, and we can think of $T$ as an endofunctor on the category of abelian groups. Here […]

## Gaussian Rings and Flat Dimension

Let $R$ be any commutative ring. The content of a polynomial $f\in R[x]$ is by definition the two-sided ideal in $R$ generated by the coefficients of $f$. If $f,g\in R[x]$, then $c(fg)\subseteq c(f)c(g)$, because each coefficient of $fg$ is a linear combination of elements of $c(f)c(g)$. Sometimes, however, this inclusion is strict. For example, if […]