135
votes

### Short exact sequences every mathematician should know

There is one obvious sequence that underlies all vector analysis and a lot that builds up on it, no matter if its applied analysis, PDE, physics or the original foundations of algebraic topology. Yet ...

Community wiki

76
votes

### Short exact sequences every mathematician should know

This might be very basic, but the short exact sequence
$$
0 \to \mathbb{Z} \to \mathbb{Q} \to \mathbb{Q}/\mathbb{Z} \to 0
$$
is both an injective resolution of $\mathbb{Z}$, and a flat resolution of $\...

Community wiki

74
votes

### Short exact sequences every mathematician should know

The exponential sheaf sequence:
$$0\to 2\pi i\,\mathbb Z \to \mathcal O_M {\buildrel\exp\over\to}\mathcal O_M^*\to 0$$
where $\mathcal O_M$ is the sheaf of holomorphic functions on the complex ...

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68
votes

### Short exact sequences every mathematician should know

I find it hard to believe that three days have gone by and no one has explicitly mentioned
$$
0 \to \Bbb Z \to \Bbb R \to \Bbb S^1 \to 0
$$

61
votes

### Short exact sequences every mathematician should know

I think a short exact sequence that every teacher should know is
$$ 0 \to \mathbb R^d \to \mathrm{Isom}(\mathbb R^d) \to \mathrm{O}(\mathbb R^d) \to 0, $$
maybe for $d=2$ or $d=3$. Better still, ...

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47
votes

### Short exact sequences every mathematician should know

The short exact sequence
$$ 0 \to \mathrm{rad}({\mathfrak g}) \to {\mathfrak g} \to {\mathfrak g}/\mathrm{rad}({\mathfrak g}) \to 0$$
separates a Lie algebra ${\mathfrak g}$ into its solvable radical $...

46
votes

### Short exact sequences every mathematician should know

"Every mathematician should know" is too much to ask, but I do think the following is a great short exact sequence that captures a vital phenomenon:
$$0 \to K(H) \to B(H) \to Q(H) \to 0.$$
$...

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45
votes

### Short exact sequences every mathematician should know

I strongly doubt there is any short exact sequence that every mathematician should know, but I certainly wish that those of them who know that for a (co)chain complex $(C,d)$
$$
0\to\operatorname{Im}(...

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41
votes

### Short exact sequences every mathematician should know

An example of a short exact sequence satisfying your first desiderata, but one which you probably won't fully understand till you are further along in homological algebra, is the Universal Coefficient ...

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35
votes

### Short exact sequences every mathematician should know

Within the category of Banach spaces and bounded linear maps,
$$0\to c_0 \to \ell_\infty \to \ell_\infty / c_0 \to 0$$
is a paradigm example of a short exact sequence that does not split, contrary to ...

Community wiki

29
votes

### Short exact sequences every mathematician should know

How about the short exact sequence that expresses that every group can be expressed in terms of generators and relators? For any group $G$, there is a short exact sequence (in fact many) of the form $$...

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28
votes

### Short exact sequences every mathematician should know

For any abelian group $A$, there is a short-exact sequence $$0 \to T(A) \to A \to A/T(A) \to 0,$$
where $T(A)$ is the torsion subgroup of $A$, and $A/T(A)$ is torsion-free.

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27
votes

### Short exact sequences every mathematician should know

The kernel-cokernel exact sequence: in an abelian category, given $A \xrightarrow{f} B \xrightarrow{g} C$, the following sequence is exact
$$ 0 \to \ker f \to \ker gf \to \ker g \to \text{coker } f \...

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26
votes

### Short exact sequences every mathematician should know

Given a finitely generated module $M$ over a commutative Noetherian ring $R$, there is a short exact sequence $$0\to M_1 \to R^n \to M\to 0$$
where you map $1$ in each $R$ to a generator of $M$ and $...

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26
votes

Accepted

### Pair of short exact sequences of groups

Call two finite groups $Q_1$ and $Q_2$ compatible if there exists a finite group $G$ with two isomorphic normal subgroups $N_1$ and $N_2$ such that $G/N_i\cong Q_i$.
One can show the following:
...

23
votes

### Short exact sequences every mathematician should know

I suppose many algebraic topologists would agree that the short exact sequence
$$0\longrightarrow \mathbb Z/p \longrightarrow \mathbb Z/p^2 \longrightarrow \mathbb Z/p\longrightarrow 0$$
giving rise ...

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21
votes

### Short exact sequences every mathematician should know

The Tate extension. Let $k$ be a field, and let $V$ be the space $k((t))$ be the space of Laurent series with coefficients in $k$, considered as a topological vector space. If we write $\operatorname{...

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20
votes

### Short exact sequences every mathematician should know

I saw $0\to \mathbb Z_p\to \mathbb Z_{p^2}\to \mathbb Z_p\to 0$ as an answer, with $p$ prime, but I will add with $p$ not prime and the particular choice $p=10$,
$$0\to \mathbb Z_{10}\to \mathbb Z_{...

Community wiki

17
votes

### Short exact sequences every mathematician should know

I guess the quintessential example, satisfying your second desiderata, is
$$
0 \rightarrow A \stackrel{f}{\rightarrow} B \rightarrow B/f(A) \rightarrow 0.
$$
For example, if $f = \mu_n: \mathbb{Z} \to ...

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15
votes

### Short exact sequences every mathematician should know

Despite it being frequently used implicitly in papers (a classical example being Milnor's '56 paper about exotic spheres), I have never seen the following spelled out anywhere, so this might be a good ...

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14
votes

### Short exact sequences every mathematician should know

This one is just too much fun to leave out.
Write the braid group on $n$ strands as $B_n$.
By following the strands of a braid $\sigma\in B_n$ we construct a permutation of $n$ items, which we write ...

Community wiki

13
votes

### Short exact sequences every mathematician should know

Short exact sequences form a bridge of sorts between homological algebra and representation theory. For example, Maschke's theorem is the statement that, if $G$ is a finite group and $k$ is a field ...

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13
votes

### Another notion of exactness: how to refine it, and where does it fit?

I'd like to argue that the current definition is too minimal to allow for much theory development, since the only substantial axiom is the pasting condition. In particular, it would be possible to ...

12
votes

### Short exact sequences every mathematician should know

Another fundamental (half) short exact sequence is the Jacobi--Zariski sequence. For algebras over operads, for example, it takes the following form: for a triple $C\to B\to A$ of maps of $P$-...

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12
votes

### Short exact sequences every mathematician should know

Decided to turn into an answer my comment to another answer here.
The Atiyah class $\alpha_E\in\operatorname{Ext}^1(E,\Omega^1\otimes E)$ of a holomorphic vector bundle $E$ is the class of the short ...

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11
votes

### Short exact sequences every mathematician should know

For a free product $A*B$ of groups $A$ and $B$, there is the exact sequence
$1 \to [A,B] \to A*B \to A \times B \to 1$
where $[A,B]$ is the subgroup generated by all elements $[a,b]=aba^{-1}b^{-1}$ ...

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11
votes

Accepted

### Analogue of Bockstein for crossed module extensions and higher Steenrod square

$\DeclareMathOperator{\Sq}{Sq}\newcommand{\Z}{\mathbb{Z}}$The short version is that every cohomology operation can be interpreted as a Bockstein operator for an "exact sequence" (read: fiber ...

10
votes

### Short exact sequences every mathematician should know

A starting point in anabelian geometry (a "thème central de la géométrie algébrique anabélienne", as Grothendieck writes in his Esquisse d'un Programme) can be considered to be the following:...

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10
votes

### Does the cohomology Bockstein homomorphism map to the homology Bockstein homomorphism under Poincarè duality?

I'm not sure I quite understand your formulation of the question (what, for example, is a section $s\colon A_3 \to A_1$?) but the following is probably what you're looking for. Given a closed oriented ...

9
votes

### Short exact sequences every mathematician should know

Let $M$ be a smooth manifold and $x:M\rightarrow \mathbb{R}$ a smooth function with $0$ as regular value, such that $X=\{x=0\}\subset M$ is a smooth submanifold. Then
$$
0\rightarrow x C^\infty(M)\...

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