# Tag Info

### 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 ...

### 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 ...

### 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 ...

### 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 ...

### 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 ...
• 5,775

### 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 $P$-...

### 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 ...

### 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 $[a,b]=aba^{-1}b^{-1}$ ...
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 ...
• 4,659

### 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:...
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 ...
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)\...