Short exact sequences every mathematician should know

Question

I'd like to have a big-list of "great" short exact sequences that capture some vital phenomena. I'm learning module theory, so I'd like to get a good stock of examples to think about. An elementary example I have in mind is the SES:

$$ 0 \rightarrow I \cap J \rightarrow I \oplus J \rightarrow I + J \rightarrow 0 $$

from which one can recover the rank-nullity theorem for vector spaces and the Chinese remainder theorem. I'm wondering what other 'bang-for-buck' short exact sequences exist which satisfy one of the criteria:

• They portray some deep relationship between the objects in the sequence that is non-obvious, or
• They describe an interesting relationship that is obvious, but is of important consequence.

Answer 1

I don't know whether this is a SES that every mathematician should know but it does satisfy the first sentence of the body of your question, since one could say it captures triangulability:

$$ 0 \to \text{ker}f \to \Theta_{3}^{H} \overset{f}{\to} \mathbb{Z}/2 \to 0 $$

where:

• the abelian group $\Theta_3^H$ is the cobordism group of oriented homology three spheres modulo binding an acylic PL/smooth 4-manifold.
• f is the Rokhlin homomorphism, which is 1/8th the signature of a compact, smooth spin(4) manifold that the integral homology sphere bounds.

Galewski, Stern and Matumoto showed in the 1980s that the non-splitting of this SES is equivalent to there being non-triangulable manifolds in every dimension 5 and above. Whereas, Manolescu recently showed that the SES does not in fact split.

Answer 2

A mathematician specialising in $C^*$-algebra theory should probably know the following short exact sequences. These play a central role in the $K$-theory of $C^*$-algebras.

• Let $A$ be a $C^*$-algebra with unitisation $A'= A \oplus \mathbb{C}$. There is an obvious split short exact sequence $$0 \to A \to A' \to \mathbb{C}\to 0$$ Hence, $K_i(A') \cong K_i(A)\oplus K_i(\mathbb{C})$.

• Let $\mathbb{A}$ be the Toeplitz algebra and $\mathbb{T}$ be the unit circle and $H^2$ be the Hardy space. Then we have a short exact sequence $$0 \to B_0(H^2) \to \mathbb{A}\to C(\mathbb{T})\to 0$$ This can be used to give a proof of the Bott-periodicity theorem.

• As mentioned in another answer, given a Hilbert space $H$, there is the short exact sequence $$0 \to B_0(H)\to B(H)\to B(H)/B_0(H)\to 0$$ which allows us to determine the $K$-theory of the Calkin algebra (via the six-terms exact sequence).

• If $\mathbb{S}=\{f\in C(\mathbb{T}): f(1)=0\}$, there is also a trivial split short exact sequence $$0 \to \mathbb{S}\to C(\mathbb{T})\to \mathbb{C}\to 0$$ which allows us to determine the $K$-theory of the $n$-dimensional torus $\mathbb{T}^n$, i.e. we can determine the $K$-theory of $C(\mathbb{T}^n)$.

Answer 3

Given an abelian category $\mathcal{A}$, and a diagram

$$\require{AMScd}\begin{CD} W @>f>> Y \\ @VgVV @VVhV \\ X @>>k> Z \end{CD}$$

we can form an associated sequence

$$0 \to W \xrightarrow{(g,f)} X \oplus Y \xrightarrow{(k,-h)} Z \to 0.$$

Then the diagram is:

0. Commutative iff the sequence is a complex,
1. Cartesian iff the sequence is a left exact complex,
2. co-Cartesian iff the sequence is right exact complex,
3. semi-Cartesian iff the sequence is a middle exact complex.

$\hphantom{0}$

Standard disclaimer: maybe not every mathematician, etc. etc., but it would behoove anyone working with commutative & homological algebra to be aware of this. It would be quite hoove indeed.

Answer 4

From tensor of vector spaces, I find this exact sequence useful: $$0\rightarrow U\otimes V'+U'\otimes V\rightarrow U\otimes V \rightarrow (U/U')\otimes(V/V')\rightarrow 0. $$

For a commutative ring R and an element $x\in R$, denote $(0:x)=\{y\in R\colon yx=0\}$, I find this really good: $$0\rightarrow (0:x) \rightarrow R \rightarrow Rx\rightarrow 0 $$ It can be used to show that, in a local ring, we have: $$\mathrm{pd}_R(Rx)\leqslant 1 \iff Rx \text{ is projective } \iff x \text{ is not a zero-divisor} $$

Answer 5

The defining short exact sequence for Milnor's $K_2(R)$ ($R$ any ring) is

$$0\rightarrow K_2(R)\rightarrow St(R)\rightarrow E(R)\rightarrow 0$$

where $St(R)$ and $E(R)$ are the Steinberg and elementary groups. If we cheat a little on the definition of "short" this extends to

$$0\rightarrow K_2(R)\rightarrow St(R)\rightarrow GL(R)\rightarrow K_1(R)\rightarrow 0$$

Answer 6

An explicit example that is my favourite is below which is in Humphreys book on Linear Algebraic Groups.

For a field $k$, let $T(n,k)$ denote the group of $n\times n$ non-singular upper triangular group and $D(n,k)$ the non-singular diagonal matrices and $U(n,k)$ upper triangular matrices with 1's in the diagonal.

$1\to U(n,k)\to T(n,k)\to D(n,k)\to 1$ is actually a split-sequence. This can be generalized to any connected solvable group leading to its structure theorem as semi-direct product of maximal unipotent normal subgroup and maximal torus subgroup.

Answer 7

An example that might be useful in virtually all branches of mathematics: If $V$ and $W$ are vector spaces (over the same field $\mathbb{F}$) and if $U \subseteq V$ is a subspace, then the obvious exact sequence $$ 0 \longrightarrow U \longrightarrow V \longrightarrow V/U \longrightarrow 0 $$ turns into the non-trivial exact sequence $$ 0 \longrightarrow U \mathbin{\otimes} W \longrightarrow V \mathbin{\otimes} W \longrightarrow (V/U) \mathbin{\otimes} W \longrightarrow 0. \tag*{$(*)$} $$ (I rarely need exact sequences in my work, but simple manipulations like this make quotients and subspaces of tensor products much easier to deal with.)

Answer 8

$\DeclareMathOperator\Aut{Aut}$Wells's exact sequence: if $G$ is a group with normal subgroup $N$, it relates $\Aut(G)$ to $\Aut(N)$ and $\Aut(G/N)$. I'll state it, then explain it; for those who haven't seen it, the notation may make it seem scary, but hopefully my explanation below makes it less scary. It starts with:

$$1 \to Z^1_\alpha(G/N, Z(N)) \to \Aut(G; N) \xrightarrow{f_1} \Aut(N) \times \Aut(G/N).$$

The idea is this: first, we want to restrict our attention to automorphisms of $G$ that send $N$ to itself — this is precisely what $\Aut(G; N)$ is. If we don't do that, things get…complicated. (When $N$ is characteristic, $\Aut(G; N) = \Aut(G)$, so this is a particularly nice case. And if you apply this inductively in a finite group, your base case is characteristically simple groups, whose automorphism groups are easily described.)

Once we have that, we find that any automorphism $\varphi$ also induces an automorphism of the quotient $G/N$, so we certainly get a map from $\Aut(G; N) \to \Aut(N) \times \Aut(G/N)$. This is $f_1$.

Now, what is the kernel of $f_1$? It's exactly the automorphisms of $G$ that fix $N$ pointwise and that induce the trivial automorphism of $G/N$; in other words, all they do is they move around the elements within their respective $N$-cosets. Thus, any such automorphism $\varphi$ is given by a function $c \colon G/N \to N$ such that $\varphi(g) = g \cdot c(\pi(g))$, where $\pi \colon G \to G/N$ is the natural quotient map, and $c(1) = 1$. For $\varphi$ to be a homomorphism (hence isomorphism), it is necessary and sufficient that $$c(\pi(g) \pi(h)) = h^{-1} c(\pi(g)) h c(\pi(h))$$

for all $g,h \in G$. Applying this equation to arbitrary $h \in N$, we find that we must have $\operatorname{Image}(c) \subseteq Z(N)$, and the equation above is precisely the equation of a "Derivation of $G/N$ into $Z(N)$ consistent with the given action of $G/N$ on $Z(N)$ (by conjugation)". In the notation above, $\alpha$ is the "outer" action of $G/N$ on $N$, which induces an actual action on $Z(N)$, and the notation $Z^1_\alpha(G/N, Z(N))$ is precisely this set of derivations.

What Wells actually did was to show that we can replace $\Aut(N) \times \Aut(G/N)$ with its subgroup $C$ of "compatible pairs" (the notion of compatibility just depends on the action $\alpha$), and then extend this exact sequence with an additional map $C \to H^2_\alpha(G/N, Z(N))$. Malfait further expanded this into a 27-term commutative cube diagram. There's a really great summary of work on this exact sequence and its extensions by Jill Dietz, that also highlights the role of Buckley's group action in this (and related) sequences.

Answer 9

The sequence $$0 \rightarrow \Psi^{\mu-1}(\Omega) \overset{\iota}{\rightarrow} \Psi^\mu(\Omega) \overset{\sigma_\mu}{\rightarrow}S_{h}^{\mu}(\Omega \times (\mathbb{R}^{n} \setminus 0)) \rightarrow 0$$ is exact, where $S_{h}^{\mu}(\Omega \times (\mathbb{R}^{n} \setminus 0))$ is the space of the functions $b:\Omega \times (\mathbb{R}^{n}\setminus 0) \rightarrow \mathbb{C}$ positively homogeneous of degree $\mu$, $\iota$ is the inclusion operator and $\sigma_\mu$ is the principal symbol of a pseudodifferential operador $B=op(b)$ with $b \in S^{\mu}(\Omega \times \mathbb{R}^{N})$.