What is an exact sequence?

I’ve heard it said that the concept of modules was invented to put ideals of rings on the same footing as quotient rings. Both the inclusion of an ideal I into a ring R and the quotient map R \to R/I are homomorphisms of R-modules. In fact, these two homomorphisms have a special relationship: the elements of R that are sent to zero in R/I are exactly those in the image of I\to R.

In general we say that a composable chain of two homomorphisms A\to B\to C is exact (or exact at B) if the elements of B sent to 0 in C (those in the kernel of B\to C) are exactly those in the image of A\to B.

In the special case that A = 0, this just says that the only element of B sent to 0 in C is 0, i.e. B\to C is injective. Similarly, the chain A \to B \to 0 is exact if and only if A\to B is surjective: everything in B is sent to zero, so the image of A must be all of B.

So given a ring R and an ideal I, we actually have two more examples of exactness: not only is I\to R\to R/I exact, but so are 0 \to I \to R and R \to R/I \to 0. This collection of information is expressed by saying that the sequence

0 \to I \to R \to R/I \to 0

is an exact sequence. In general, a finite or infinite sequence of morphisms \dots\to A_0\to A_1\to A_2\to A_3\to\dots is called an exact sequence if it is exact at each A_i, i.e. each consecutive pair A_{i-1}\to A_i \to A_{i+1} is exact. For example, the  operations \nabla (gradient), \nabla\times (curl), and \nabla \cdot (divergence) on vector fields on \mathbb{R}^3 fit into an exact sequence

\displaystyle C^\infty(\mathbb{R}^3) \mathop{\longrightarrow}^{\nabla} C^\infty(\mathbb{R}^3)^3\mathop{\longrightarrow}^{\nabla\times} C^\infty(\mathbb{R}^3)^3\mathop{\longrightarrow}^{\nabla\cdot}C^\infty(\mathbb{R}^3)

because those smooth vector fields whose curls vanish are exactly the gradients of smooth functions, and those whose divergences vanish are exactly the curls of smooth vector fields.

Incidentally, an exact sequence of the type 0 \to A \to B \to C \to 0 is called a short exact sequence. A short exact sequence is so small that it conveys a very precise relationship between the two homomorphisms A\to B and B\to C, which can be summed up as

A is isomorphic to a submodule of B, and C is isomorphic to the quotient of B by that submodule.

(Incidentally, exact sequences that extend infinitely in one or both directions are called long exact sequences.)

Exact sequences also arise as a way of understanding module presentations. The \mathbb{Z}-module \mathbb{Z}/2\mathbb{Z} has a single generator [1], so we get a surjection \mathbb{Z} \to \mathbb{Z}/2\mathbb{Z} sending 1\mapsto [1]. We can extend this to an exact sequence \displaystyle \mathbb{Z} \mathop{\longrightarrow}^{2\cdot} \mathbb{Z} \to \mathbb{Z}/2\mathbb{Z} \to 0: exactness on the right is surjectivity of \mathbb{Z}\to\mathbb{Z}/2\mathbb{Z}, and exactness on the left says that the integers mapped to 0 are exactly the multiples of 2.

For a more complicated example, say that an R-module M is abstractly presented as \langle e_1, e_2, e_3 \rangle / \langle 2e_1+4e_2\rangle, i.e. it has three generators, e_1, e_2, and e_3, subject only to the relation 2e_1+4e_2 = 0. The three generators give us a surjection R^3\twoheadrightarrow M sending (1,0,0) to e_1 and so on, which we can interpret as an exact sequence R^3 \to M \to 0. We can express the fact that every relation between the elements e_1, e_2, and e_3 is a multiple of 2e_1+4e_2 = 0 by saying that the kernel of R^3 \to M is generated by (2,4,0). In other words, the homomorphism R\to R^3 sending 1 to (2,4,0) makes the sequence R \to R^3 \to M\to 0 exact. In general, if we present a module M with n generators and m relations, we get an exact sequence R^m\to R^n \to M \to 0.

(In the case of \displaystyle \mathbb{Z} \mathop{\longrightarrow}^{2\cdot} \mathbb{Z} \to \mathbb{Z}/2\mathbb{Z} \to 0, we see that we are presenting \mathbb{Z}/2\mathbb{Z} with a single generator modulo a single relation.)

From the exact sequence R^m\to R^n \to M \to 0, all the information defining M is contained in the homomorphism R^m\to R^n; if we know it, then M is its cokernel, the quotient of R^n by the image of R^m in it. In general, if C is the cokernel of a general module homomorphism A\to B, then we get an exact sequence A\to B \to C\to 0. Similarly, if K is the kernel of A\to B then we get an exact sequence 0\to K \to A\to B. In fact, the kernel and cokernel of A\to B are the only modules (up to isomorphism) that fit into the blank spots in 0\to ? \to A\to B\to ? \to 0 to give an exact sequence.

Generalizing 2\in\mathbb{Z} to a general element a\in R gives another useful example: the map a\cdot:R\to R given by multiplication by a has image (a)\subseteq R, thus cokernel R/(a). The kernel of a\cdot is the set \mathrm{Ann}_R(a) of elements whose product with a is zero, an ideal called the annihilator of a. So we get an exact sequence

\displaystyle 0 \to \mathrm{Ann}_R(a) \to R \mathop{\longrightarrow}^{a\cdot} R \to R/(a) \to 0.

Now, the image of a\cdot: R\to R is the ideal (a), which is the kernel of R\to R/(a), so we also get a short exact sequence 0\to (a) \to R \to R/(a)\to 0 as at the top of this post. Less obviously, we also find that (a) is the cokernel of \mathrm{Ann}_R(a)\to R, since the map a\cdot:R\to (a) is surjective with kernel \mathrm{Ann}_R(a). Therefore we have another short exact sequence \displaystyle 0 \to \mathrm{Ann}_R(a)\to R \mathop{\longrightarrow}^{a\cdot} (a) \to 0. The four-term exact sequence above is thus “built” out of two smaller short exact sequences.

In fact, this is a general phenomenon. If

\displaystyle \cdots A_{i-1}\mathop{\longrightarrow}^{f_{i}} A_i\mathop{\longrightarrow}^{f_{i+1}} A_{i+1}\to\cdots

is an exact sequence of any length, then by definition each image \mathrm{im}(f_i) is the same submodule of A_i as the following kernel \mathrm{ker}(f_{i+1}), so we have short exact sequences

0 \to \mathrm{im}(f_i) \to A_i \to \mathrm{im}(f_{i+1}) \to 0

for each i.

This lets us prove facts about general exact sequences from facts about short exact sequences. For example, if 0\to U\to V\to W\to 0 is a short exact sequence of vector spaces, then the rank-nullity theorem says that \mathrm{dim}(V) = \mathrm{dim}(U) + \mathrm{dim}(W), but we can do better. If 0 \to V_1 \to V_2\to \dots \to V_n \to 0 is any exact sequence of vector spaces, with the maps V_{i-1}\to V_i denoted f_i, then \sum_{i\text{ odd}} \mathrm{dim}(V_i) = \sum_{i\text{ even}} \mathrm{dim}(V_i), since both sums are equal to \sum_{\text{all }i}\mathrm{dim}\bigl(\mathrm{im}(f_i)\bigr).

For modules over general rings, there isn’t a notion of dimension that behaves additively in short exact sequences 0\to A \to B \to C \to 0, but the principle of getting information about B from information about A and C appears again and again, making short exact sequences an exceptionally useful concept.

(Incidentally, if you know the origin of the rough “same footing” quotation at the top of this post, please tell me! I’d love to credit its originator.)

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