Short exact sequences every mathematician should know

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.
• Should every mathematician know what a short exact sequence is? Jun 21, 2020 at 16:56
• @VilleSalo I was under the impression every mathematician does know what a short exact sequence is :) I'm a computer science student though, so I wouldn't know :D Jun 21, 2020 at 17:15
• @SiddharthBhat I definitely do not think every mathematician knows what a short exact sequence is. While this topic probably appears in most first-year PhD courses, if a mathematician goes into, say PDEs, there is no real reason they would remember this topic. Just like in computer science, if someone goes into systems they might forget the Ford-Fulkerson algorithm, and a theoretical computer scientist might forget the soldering they learned in computer architecture. Jun 21, 2020 at 17:26
• I once met a mathematician who had never heard of a homeomorphism. Maybe they were not a true Scotsman though. Jun 21, 2020 at 18:39
• @VilleSalo : I once heard the following (probably apocryphal) story about an oral exam. Professor: "Are $M_1$ and $M_2$ homeomorphic?" Student: "$M_1$ is, but $M_2$ isn't." Jun 21, 2020 at 19:23

34 Answers

A useful short exact sequence of sheaves in algebraic geometry is

$$0 \rightarrow \mathcal{O}_C(K_C) \rightarrow \mathcal{O}_C(K_C + P) \rightarrow Q \rightarrow 0,$$

where $$Q$$ is the quotient (a skyscraper sheaf with support in $$P$$).

Considering the long exact cohomology sequence associated to this short exact sequence leads to a standard proof of the Riemann-Roch formula for algebraic curves, since $$L(C,K_C)$$ is the space of global sections of the sheaf $$\mathcal{O}_C(K_C)$$ (see the textbook on Riemann surfaces by Otto Forster).

• Up to a twist, this is a special case of the ideal sheaf sequence $I_Z \rightarrow O_X \rightarrow O_Z$ for a closed subvariety $Z$ in a variety $X$. That sequence and its twists are absolutely fundamental tools in projective algebraic geometry.
– Pop
Jun 24, 2020 at 9:59

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

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

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 $$b \in S^{\mu}(\Omega \times \mathbb{R}^{N})$$.

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.)

• Can you please describe why $(\star)$ is a very useful exact sequence? Jun 23, 2020 at 17:32
• @SiddharthBhat hm, yeah, very useful is a little too strong (so I changed it). I guess my main point is that manipulations of exact sequences make life a lot easier when dealing with subspaces and/or quotients of tensor products. Jun 23, 2020 at 17:38