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 ...
  • 923
20 votes
2 answers
723 views

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 \...
3 votes
1 answer
211 views

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_{...
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3 votes
0 answers
140 views

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
58 views

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 ...
0 votes
0 answers
124 views

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 views

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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0 votes
0 answers
77 views

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 ...
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4 votes
0 answers
148 views

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{...
1 vote
1 answer
174 views

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 ...
4 votes
1 answer
191 views

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 votes
0 answers
49 views

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+...
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4 votes
1 answer
109 views

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 $...
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4 votes
0 answers
106 views

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 ...
2 votes
0 answers
72 views

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)\...
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1 vote
1 answer
394 views

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}...
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2 votes
1 answer
124 views

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 $I$ ...
  • 105
2 votes
0 answers
83 views

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 ...
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2 votes
1 answer
306 views

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{...
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4 votes
1 answer
146 views

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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1 vote
0 answers
71 views

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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1 vote
2 answers
254 views

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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1 vote
0 answers
105 views

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 ...
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3 votes
1 answer
204 views

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 ...
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1 vote
1 answer
188 views

$M$ 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 views

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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-3 votes
1 answer
101 views

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 views

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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1 vote
1 answer
124 views

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 answer
414 views

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
1 answer
220 views

Isomorphism of semidirect products of surface groups

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

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 $\mathcal{C}$ ...
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142 votes
34 answers
19k views

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 votes
0 answers
229 views

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 ...
1 vote
1 answer
265 views

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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1 vote
0 answers
89 views

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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0 votes
0 answers
89 views

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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9 votes
2 answers
615 views

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 votes
0 answers
138 views

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 $\...
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2 votes
0 answers
122 views

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): $$\...
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2 votes
0 answers
54 views

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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5 votes
0 answers
206 views

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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4 votes
1 answer
276 views

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\...
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4 votes
0 answers
98 views

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 $...
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4 votes
1 answer
239 views

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
0 answers
115 views

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 ...
  • 10k
25 votes
2 answers
897 views

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 ...
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0 votes
1 answer
93 views

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 answer
208 views

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 views

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 \...