# Questions tagged [short-exact-sequences]

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

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### Properties on morphism of locally convex vector spaces

Let $K$ be a finite extension of $\mathbb{Q}_p$. Let $U,V,W,W'$ $K$-vector spaces, such that $U$ is a Banach-space and $W,W'$ are finite dimensional. Further we have an (algebraic) short exact ...

20
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2
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### Pair of short exact sequences of groups

Does there exist a pair of finite groups $G$ and $H$ satisfying both of the short exact sequences $1 \rightarrow G \rightarrow H \rightarrow A_4 \rightarrow 1$ and $1 \rightarrow G \rightarrow H \...

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211
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### Derived Hom without injectives nor projectives

I am stuck with the following farce on derived Homs.
I have an abelian category $A$ and I showed that, given any two objects $X$ and $Y$ of $A$, the group of $1$fold extensions $\operatorname{Ext}^1_{...

3
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0
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### For a family of short exact sequences of coherent sheaves, can we define the splitting subscheme?

This question has been asked in SE.
Let $k$
be an algebraically closed field of characteristic zero. Let $X$ be a projective scheme over $k$. We can talk about short exact sequences of coherent ...

1
vote

0
answers

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### A sufficient condition for automorphism of an exact sequence

I asked A sufficient condition for Automorphism of an exact sequence earlier on Math.StackExchange but did not get any response so am posting it here.
I am given the following commutative diagram with ...

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124
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### Exactness of $I$-adic completion in a certain non-finitely generated case

I would like the functor
$$(-\otimes_{\mathbb Z} F)\hat{}: \mathbb{Z}[x_1,\dots,x_r]\text{-Mod}_{\mathrm{f.g.}}\longrightarrow \mathbb{Z}[x_1,\dots,x_r]\text{-Mod}$$
to be exact, where completion is w....

8
votes

1
answer

307
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### Analogue of Bockstein for crossed module extensions and higher Steenrod square

It is well known that in $\mathbb{Z}_2$-valued simplicial cohomology (and other cohomologies)
$$ Sq^1 = \beta\;,$$
where $Sq^1$ is the first Steenrod square and $\beta$ is the Bockstein homomorphism ...

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### Extension of short exact sequence on orthogonal Grassmannians

We work over $\mathbb C$. Let $X=OG(k,V)$ be the orthogonal Grassmannian parametrizing the $k$-dimensional subspaces of $V$, isotropic with respect to a non-degenerate bilinear symmetric form $q$.
As ...

4
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0
answers

148
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### Outer and inner automorphism of $\mathrm{Pin}$ groups

$\DeclareMathOperator\Inn{Inn}\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Out{Out}\DeclareMathOperator\Pin{Pin}\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SO{SO}\DeclareMathOperator\PSO{...

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vote

1
answer

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### Are there nonaffine schemes over which every exact sequence of vector bundles is split?

Is there an example of a non-affine scheme $X$ such that every short exact sequence of vector bundles over $X$ splits? If there are such examples then what if we ask it to be true of all (not ...

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191
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### A Kummer exact sequence involving $\mu_\infty$

Let $k$ be a number field. We have the well-known Kummer exact sequence of etale sheaves on $\mathrm{Spec}\, k$: $$1 \rightarrow \mu_n \rightarrow \mathbb{G}_m \rightarrow \mathbb{G}_m \rightarrow 1.$$...

4
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### Fundamental group of the complement of some quadric cones

cross-posting from MathSE
Problem
Consider the domain
$$\Omega=\mathbb{C}^4\setminus\{z_0(z_1^2+z_2^2+z_3^2)=0\}$$
and the map
$$F:\Omega\to\mathbb{CP}^1\qquad F(z_0,z_1,z_2,z_3)=[z_0^2:z_1^2+z_2^2+...

4
votes

1
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### Linearity of covariant and contravariant $Ext^1$ functors defined via short exact sequences

Let $R$ be a Commutative ring. Let $M,X,Y$ be $R$-modules. Let $f: X \to Y$ be an $R$-linear map.
Then, given an exact sequence $\eta: 0\to X \to Z_{\eta} \to M \to 0$ in $Ext^1(M,X)$, the pushout of $...

4
votes

0
answers

106
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### How can one characterize categories of exact functors?

Does there exist any intrinsic characterization of additive categories equivalent to $\operatorname{Ex}(A,Ab)$, that is, of exact functors from a small abelian category $A$ into abelian groups? Any ...

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### What is the isomorphism from $\operatorname{Ext}^1_{T}(Y,X)$ to $\operatorname{Ext}_T(\mathbf{1},X\otimes Y^{\vee})$?

Let $T$ be an abelian rigid monoidal category and $\mathbf{1}$ be a unit object in $T$. For two objects $X$ and $Y$ in $T$, there is a natural group isomorphism
$$n:\operatorname{Ext}^1_{T}(Y,X)\...

1
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1
answer

394
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### Computing Ext sheaves over complex projective plane

Let $X:=\mathbb{P}^2_K$ with $K$ algebraically closed field. Take $p\in X$ a point and $\mathcal{I}_p$ its ideal sheaf. One can prove (using Serre Duality and the exact sequence defining $\mathcal{I}...

2
votes

1
answer

124
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### About nuclear-by-exact extensions

I know that in general exact-by-exact extensions of $C^*$-algebras need not be exact. Is it true that, if we have a short exact sequence of $C^*$-algebras
$$0 \to I \to A \to B \to 0$$
such that $I$ ...

2
votes

0
answers

83
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### Splitting of short exact sequence of strongly-semistable sheaves

Does short exact sequence of strongly semi-stable bundles (torsion-free sheaves) of the same slope split after applying few Frobenius pullbacks? Strongly semi-stable means that pullback under ...

2
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1
answer

306
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### The left exactness of conormal sequence when $X$ is singular

When $X$ is a nonsingular variety over a field $k$ and Z is a closed nonsingular subvariety, it is known that the conormal sequence
$$
0\to\mathscr{I}/\mathscr{I}^2\to \mathscr{O}_Z \otimes_{\mathscr{...

4
votes

1
answer

146
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### An explicit description of $\operatorname{Isom}(\widetilde{\operatorname{Sl}_2})$

$\DeclareMathOperator\Sl{Sl}\DeclareMathOperator\PSl{PSl}\DeclareMathOperator\Isom{Isom}$Let $\widetilde{\Sl_2}$ be the Thurson geometry that can either be described as the universal cover of $\PSl(2,\...

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0
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### Two questions regarding double short exact sequences

Two short exact sequences on the same objects is called double short exact sequence. The morphism of double short exact sequences is defined in the same way you'd expect, it is a morphism of the ...

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2
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254
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### When splitting of short exact sequence preserves the kernels

This is a problem that I thought at first was obvious but that became less clear the more I thought about it. Assume we have a finitely generated algebra $A$ over a field $k$, and a short exact ...

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0
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105
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### Extending automorphism from an affine

Given a projective variety $X$ and an open affine $U$ in $X$. Is there a way to decide whether a given automorphism of a vector bundle $E$ on $U$, is the restriction of automorphism of some coherent ...

3
votes

1
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204
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### Resolution of short exact sequences by the split ones

Given a short exact sequence of vector bundles on a projective variety, after tensoring with an $\mathcal{O}(n)$ with high $n$ that makes all terms globally generated (so that taking global sections ...

1
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1
answer

188
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### $M$ comodule if and only if $N$ and $L$ comodules

Let $k$ be a field, $C$ a $k$-coalgebra, and $M$ a left $C$-comodule. Then, for a short exact sequence
$$
0 \rightarrow N \rightarrow M \rightarrow L \rightarrow 0
$$
of vector spaces, we have that $N$...

5
votes

1
answer

158
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### Colimits of short exact sequences of C*-algebras

Assume I have an inductive system of short exact sequences of $C^{\ast}$-algebras (i.e., short exact sequences $0 \to A_n \to B_n \to C_n \to 0$ together with transformations from the $n$-th to the $(...

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votes

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answer

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### Exact sequence of sheaves that generates an exact sequence of Abelian groups [closed]

Let $X$ be a smooth manifold of dimension $n > 1$. Let us denote by $\underline{\mathbb{S}}^{1}$ the sheaf of the smooth functions over circle, $C^{\infty}$ the sheaf of the smooth functions over $\...

2
votes

0
answers

59
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### On exactness of associating smooth representation-functor $(\,)^\infty$

Let $G$ be a locally profinite group, e.g. reductive group over $\mathbb{Q}_p$. For a (abstract) representation $(\pi,V)$ of $G$ and $K\subset G$ compact open subgroup denote by $V^K\subset V$ the $\...

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vote

1
answer

124
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### Self-map of short exact sequences

Consider the commutative diagram of finite abelian groups
$\require{AMScd}$
\begin{CD}
0@>>> A @>i>> B@>\pi>> C@>>> 0\\
\ @VV 0 V@VVfV@VV 0 V\\
0@>>>A @&...

4
votes

1
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### Short exact sequence $0\to \mathbb Z\to A \to \mathbb R \to 0$

Does every short exact sequence $0\to \mathbb Z\to A \to \mathbb R \to 0$ split in the category of Abelian groups?

4
votes

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### Isomorphism of semidirect products of surface groups

Recall that the fundamental group of a closed Riemann surface of genus $h$ has the presentation $$\Pi_h= \langle a_1, \,b_1, \ldots, a_h,\, b_h \; | \; [a_1, \, b_1]\ldots [a_h, \, b_h]=1 \rangle.$$ ...

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### additivity of trace with respect to short exact sequences

Let $\mathcal{C}$ be an abelian rigid symmetric monoidal category over a field $K$. Assume that the endomorphism ring of the tensor unit in $\mathcal{C}$ is $K$. If $X$ is an object in $\mathcal{C}$ ...

142
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34
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### 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 ...

6
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0
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### How much does Ext tell me about isomorphisms?

So this was a question I sort of stumbled on and realised I was quite stumped. Suppose we have two finitely generated $R$-modules $M, N$ (I have the group ring $R=\mathbb{Z}[G]$ in mind) which appear ...

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1
answer

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### Splitting of short exact sequence in the category of finitely generated modules over a commutative Noetherian ring

In the category of finitely generated modules over a commutative Noetherian ring, the splitting of a short exact sequence can be checked locally at the maximal ideals of the ring. One reference for ...

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0
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### Exactness of a certain sequence

Let $R$ be a commutative unitary ring and $I_1,..., I_n$ ideals in $R$. For each $p\in\{0,...,n-1\}$ consider the direct sums $\bigoplus_{i_0<...<i_p} I_{i_0}\cap...\cap I_{i_p}$ and define an $...

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### Increasing the number of ideals in an exact sequence

In Broadmann and Sharp's book, Local Cohomology: An Algebraic Introduction with Geometric Applications, the exercise $3.2.4$ is about an exact sequence of the form $\DeclareMathOperator{\Hom}{Hom}$
$...

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2
answers

615
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### The geometry of the action of the semidirect product

I'm going by the maxim
Groups, like men, are known by their actions
This naturally leads one to ask "given groups $G, H$ which act on sets $S, T$ and the semidirect product $G \rtimes H$, how does ...

2
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0
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### To understand the description of relative group homology $H_{*}(G,H;\mathbb{Z})$ in terms of free $G$-resolution

Let $G$ be a group and $H$ its subgroup ($H$ need not to be normal). Consider a chain complex $(C_{*}(G), \partial)$ where $C_n(G)$ is the free abelian group generated over the set $G^{n+1}$ and $\...

2
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0
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### Exactness of sequences preserved under resolution of singularities

Let $X$ be a noetherian, affine, normal, isolated singularity and $\pi:\widetilde{X} \to X$ be a resolution of singularities. Suppose, we have an exact sequence (not necessarily short exact):
$$\...

2
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0
answers

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### direct limit in locally convex modules and continuous map

Let we have short exact sequences of LCM over LC algebra $A$ with continuous linear maps
$$
0\to B_j\;{\xrightarrow {\ f_j\ }}\;C_j\;{\xrightarrow {\ g_j\ }}\;D_j\to 0.
$$
We can take inductive limit (...

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answers

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### Atiyah class and coboundary map

Let $L$ be a line bundle on a smooth algebraic variety $X$. Let $\sigma_i:U_i \times \mathbb{C} \to L_{|U_i} $
be its local trivializaations and $u_{ij}$ the transition functions satisfying $\sigma_j=...

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276
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### Extensions of compact Lie groups

Let $G$, $H$ be two compact Lie groups (possibly disconnected). Two short exact sequences of compact Lie groups
$$
0\rightarrow G\rightarrow M_1 \rightarrow H\rightarrow 0,
$$
$$
0\rightarrow G\...

4
votes

0
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### SQ-universality and short exact sequence

Let $G$ be a finitely generated group. Assume that $G$ decomposes as a short exact sequence
$$1 \to N \to G \to A \to 1$$
where $A$ is free abelian and $N$ SQ-universal. Is $G$ SQ-universal?
A group $...

4
votes

1
answer

239
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### The computation of $d_2$ in the Hochschild-Serre spectral sequence

I'm trying to understand the Hochschild-Serre spectral sequence by an example. Consider the short exact sequence of groups:
$1\to N\to G\to G/N\to 1$
where $G\cong \mathbb{Z}_4$, $N\cong\mathbb{Z}_2$.
...

6
votes

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### Bordism groups and a short exact sequence

Let us consider a short exact sequence:
$$
1\to N\to G\to Q \to 1,
$$
where $N$, $Q$, and $G$ can be continuous Lie groups in general (or finite groups).
Suppose I have the data and the computations ...

25
votes

2
answers

897
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### Another notion of exactness: how to refine it, and where does it fit?

There are many notions of "exactness" in category theory, algebraic geometry, etc. Here I offer another that generalizes the category of frames, the notion of valuation (from probability theory), and ...

0
votes

1
answer

93
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### Stabilizer of two short exact sequences at the same time

For two short exact sequences of say, finitely generated modules of some ring, $0\rightarrow N\xrightarrow{a} R\xrightarrow{b} M\rightarrow0, 0\rightarrow K\xrightarrow{a'}R\xrightarrow{b'}L\...

2
votes

1
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208
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### Profinite extension of a Lie group

Let $H,G,K$ be three topological groups, we say that $G$ is an extension of $K$ by $H$ if the following short sequence
$$0\rightarrow H\rightarrow G\rightarrow K\rightarrow 0$$
is exact. (If $H$ is a ...

8
votes

2
answers

440
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### Exact sequence of $n$th powers of abelian groups

Let $A,B,C$ be finitely generated abelian groups. Assume that there is an exact sequence $$0 \to C \to A^n \to B^n \to 0,$$where $A^n = A \oplus \dotsc \oplus A$ as usual. It is not assumed that $A^n \...