A Very Short Introduction to Regular Sequences

Take yourself away from this cold day in December and transport yourself to the world of commutative rings with identity. In this land there is a wonderful tool called the theory of regular sequences, which we will examine in this post. Our aim will be to get a quick idea of what regular sequences are, without going into too much tedious detail, with the hope that everyone reading this will think regular sequences are cool.

Now before I even define regular sequences, let us look at some examples of regular sequences:

1. In the ring $k[x,y,z]$, the sequence $x,y,z$.
2. In the ring $\mathbb{Z}[x]$, the sequence $2,x$.
3. In the ring $\mathbb{Z}$, the sequence $4$

So a regular sequence in a given ring $R$ will be a finite sequence of elements in it. Because of the usefulness of module-theoretic and homological methods in this area, it's useful to define a regular sequence with respect to an $R$-module: if $A$ is an $R$-module, then a regular sequence on $A$ is a finite sequence of elements $r_1,\dots,r_k\in R$ such that $(r_1,\dots,r_k)A\not= A$ and for each $i$, the element $r_i$ is not a zero divisor on $A/(r_1,\dots,r_{i-1})A$. (For any $R$-module $A$, the element $x\in R$ is called a zero divisor on $A$ if $xa = 0$ for some nonzero $a\in A$.)

It's not too hard to find regular sequences in some specific cases in rings. For instance, in a polynomial ring in $n$ variables, the variables themselves form a regular sequence. It's also true that if $I = (a,b)$ is a proper ideal in a GCD domain (e.g. UFD domains such as $\mathbb{Z}[x]$) then $a,b$ is a regular sequence if and only if $a$ and $b$ are relatively prime. So for instance, in $\mathbb{Z}[x]$, the sequences $4,x$ and $x,x-1$ are both regular sequencess.

What Can You Do With Regular Sequences?

The above three examples are regular sequences on the $R$-module $R$ itself. Now that you know what a regular sequence is, the next obvious question is: what can we do with a regular sequence? Here is a quick answer: given a Noetherian ring $R$ and an ideal $I\subseteq R$, there is an upper bound to the length of a regular sequence of elements in $I$. The length of the longest regular sequence is called the grade of $I$, and is denoted $G(I)$. It is an invariant of the ideal $I$, and in fact it is a number less than the rank of $I$ (if $I$ is any prime ideal, we define the rank of $I$ to be the minimum of the set $\{ \mathrm{rank}(P) : P\supseteq I\}$. If the grade and rank are equal for every ideal then such a ring is called Macaulay. These are very special kinds of rings in algebraic geometry. Polynomial rings are examples of such rings.

However, this might be a bit unconvincing, especially for those more interested in immediate algebraic applications. So consider a nonzero divisor $x\in R$. Then $0\to R\xrightarrow{x} R\to R/x\to 0$ is a free resolution of $R/x$ because multiplication by $x$ is injective.

It turns out that we can turbocharge this observation into something called the Koszul complex. Given any commutative ring $R$ and a regular sequence $x_1,\dots,x_n\in R$, the Koszul complex gives a finite free $R$-resolution of the $R$-module $R/(x_1,\dots,x_n)$. Another bonus is that this free resolution is very easy to construct. Here is a recipe: if $(x_1,\dots,x_n)$ is an $R$-sequence, let $N = R^n$ and consider the chain complex

$0 \to R\to N\to \wedge^2N\to \dots\to \wedge^nN\to 0$.

The differentials are all the same: they send $y$ to $x\wedge y$. It turns out that this sequence is exact, except that the cokernel of $\wedge^{n-1}N\to\wedge^nN$ is exactly $R/I$ (just check where the generators of $\wedge^{n-1}N\cong R^n$ go as an exercise). Hence this gives a free resolution of $R/I$ as an $R$-module. This implies for instance that $R/I$, as an $R$-module, has projective dimension at most $n$.

1 Comment

• esra says:

Did you know the relation between regular sequence and McCoy's theorem?
If a non zerodivisor exists in the ideal how we know the regular sequence's length ?