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

  1. 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 ?

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