# Tagged Questions

For questions about short exact sequences in various contexts, including questions on short exact sequences of groups or modules.

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### busby invariant of extensions of $C^*$-algebras

I have a question of an explicit example of a busby invariant of a extension, which can be found in Blackadars book "K-theory for Operator Algebras". Let $0\to B\to E\to A\to 0$ be a short exact ...
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### Short exact sequences for amalgamated free products and HNN Extensions

I asked this question on math stackexchange (see here) but didn't get any answer so I thought I would post it here too: If $A$ and $B$ are groups we have the following short exact sequence:  0 \to [...
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### Proof of Merkurjev's Theorem in “The Algebraic and Geometric Theory of Quadratic Forms”

I just have a little question about the above mentioned proof. I'm thinking for days, but I'm still not getting it. For those who have the book (or want to look it up via google books etc.), it's the ...
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### On (universal) additive functors making a given complex contractible: examples?

Let $M=(M^i)$ be a (cohomological) complex of objects of some additive category $A$ (I am mostly interested in "short" complexes; yet one may also consider an unbounded $M$). I am interested in those ...
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### Exact sequence of groups to exact sequence of sheaves

Disclaimer: This is a cross-listing of a math.stackexchange post. While not research level, after a week of no response, I figured I would ask it here. For a topological group $G$ and a topological ...
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### Is the square diagram of index and exponential maps in $K$-theory of $C^*$-algebras anti-commutative?

Assume we have a $3\times 3$ grid with rows and columns being short exact sequences of $C^*$-algebras. This gives a grid of 6-term exact sequences: 3 "horizontal" sequences and 3 "vertical" sequences,...
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### Splitting lemma for semigroups or monoids

I know there is a splitting lemma for groups, but is there a similar lemma for semigroups or monoids? And do you have the proof of that?
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### Do all exact sequences $0 \rightarrow A \rightarrow A \oplus B \rightarrow B \rightarrow 0$ split for finitely generated abelian groups?

Suppose $A$ and $B$ are finitely generated Abelian groups. Are all exact sequences of the form $0 \rightarrow A \rightarrow A \oplus B \rightarrow B \rightarrow 0$ split? If not, is there an example?
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### Exact sequence of the fundamental group of the general fiber

Let $f\colon X\rightarrow Y$ be a morphism of complex algebraic varieties. Let $y\in Y$ be a general point, then we have a sequence of homomorphisms of fundamental groups induced by the inclusion of ...
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### Origin of exact sequences

I have seen exact sequences appearing a lot in algebraic texts with different purposes. But I've never seen names of the people associated with it. Also I don't understand what's so good about showing ...
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### Canonical sheaf of the fiber of a flat morphism

This is probably a trivial question. While reading the paper R. Elkik, Singularites rationnelles et deformations, Invent. Math. 47 Ž1978., 139147. I came across the following short exact sequence. ...
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### Exact sequence of monoids

What is the right definition of an exact sequence of monoid homomorphisms? I can't seem to find a consistent in my searches; indeed Balmer (Remark 2.6, http://www.math.ucla.edu/~balmer/research/...
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### What is exact sequence in higher categories?

What is the higher categorical generalization of exact sequence (3 terms or $\mathbb{Z}$ terms)? In particularly, consider the simplest cases: chain complexes, $L_\infty$-algebras.
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### Do all exact 1 -> A -> AxB -> B -> 1 split for finite groups?

Let $A$, $B$ be finite groups. Is it true that all short exact sequences $1 \rightarrow A \rightarrow A \times B \rightarrow B \rightarrow 1$ split on the right? In other words, do there exist ...
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### Can Lie algebra cohomology prove Cartan's Semisimplicity Criterion?

Here is what I mean by "Cartan's semisimplicity criterion": Let $\mathfrak g$ be a finite-dimensional Lie algebra over a field of characteristic $0$. Assume that the center of $\mathfrak g$ is ...