Articles of exact sequence

Exactness of short exact sequences

Let $A$ be a ring and let $P$ be a projective $A$-module. Then, the exactness of the sequence: $$0\longrightarrow M_1 \overset{f}{\longrightarrow}M_2\overset{g}{\longrightarrow}M_3\longrightarrow 0 \tag{1}$$ implies the exactness of the induced $\mathbb{Z}$-module sequence: $$0\longrightarrow \text{Hom}_A(P,\,M_1) \overset{f_\bullet}{\longrightarrow} \text{Hom}_A(P,\,M_2)\overset{g_\bullet}{\longrightarrow} \text{Hom}_A(P,\,M_3)\longrightarrow 0 \tag{2}$$ Does the exactness of (2) imply that $g$ is an epimorphism? Let $x\in M_3$. Suppose that there […]

$M' \to M \to M'' \to 0$ exact $\implies 0\to \text{Hom}(M'',N) \to \text{Hom}(M,N) \to \text{Hom}(M',N)$ is exact.

Let $M’, M”, M, N$ be $A$-modules. If $$M’ \stackrel{u}{\to} M \stackrel{v}{\to} M” \to 0$$ is exact, then $$0\to \text{Hom}(M”,N) \stackrel{\bar{v}}{\to} \text{Hom}(M,N) \stackrel{\bar{u}}{\to} \text{Hom}(M’,N)$$ is exact, where $\bar{u} (f) = f \circ u$ and similarly for $v$. I’ve shown that $\ker (\bar{u}) \supseteq \text{im} (\bar{v})$ by starting with $v\circ u = 0$. How do I […]

Property of short exact sequences.

This question already has an answer here: Example of a non-splitting exact sequence $0 → M → M\oplus N → N → 0$ 1 answer

Why any short exact sequence of vector spaces may be seen as a direct sum?

This is actually the first time I have worked with short exact sequences and I have no clue why the following assertion is true: Any short exact sequence of vector spaces $$ 0 \longrightarrow U \overset{A}{\longrightarrow} V \overset{B}{\longrightarrow} W \longrightarrow 0 $$ reduces to a decomposition $V \simeq U \oplus W$. Can anyone give a […]

Construction of an exact sequence $1 \to N_{16} \to G_{64} \to \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z} \to 1$

Let us consider the following possible exact sequence $$1 \to N_{16} \to G_{64} \to \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z} \to 1$$ $\bullet$ The full group is a non-Abelian finite group $G_{64}$ with 64 group elements. $\bullet$ The normal subgroup is a non-Abelian finite group $N_{16}$ with 16 group elements. $\bullet$ The quotient group is a Abelian finite […]

Can $\frac{SO(5)}{N}= SU(2) \times SO(2)$ be true?

Inspired by this, consider the unknown short exact sequence, $$ 1 \to N \to SO(5) \to SU(2) \times SO(2) \to 1 $$ What is the normal subgroup $N$ here so that $SU(2) \times U(1)$ is a quotient group for the total group $SO(50$, and $SO(5)/N= SU(2) \times SO(2)$? Is it an allowed short exact sequence? […]

Are there any non-trivial group extensions of $SU(N)$?

Are there any non-trivial group extensions of $SU(N)$? If not, can one show/prove there are no non-trivial group extensions of $SU(N)$? It is possibly partial related to the homotopy group property. Or one can try to argue from the exact sequence. Proof/Show: Let us call $Q=SU(N)$. If the above claim is true, namely, we cannot […]

Cover and extension of a Lie group

We know that $SU(2)$ is a double cover of $SO(3)$, such that $$SU(2)/Z_2=SO(3),$$ through a finite extension $N=Z_2$. For other examples of simply-connected Lie groups such as $SU(2)$, $SU(N)$ or $E_8$, (1) are there nontrivial extensions of $SU(2)$, $SU(3)$, $SU(N)$ or $E_8$? (2) are there nontrivial finite extensions of $SU(2)$, $SU(3)$, $SU(N)$ or $E_8$? Please […]

Prove $\ln(2^5)-\pi=8\sum_{n=1}^{\infty}\frac{1}{n}\left(\frac{1}{e^{n\pi}+1}-\frac{1}{e^{2n\pi}+1}\right)$

We took this idea from Simon Plouffe see here $$\ln(2^5)-\pi=8\sum_{n=1}^{\infty}\frac{1}{n}\left(\frac{1}{e^{n\pi}+1}-\frac{1}{e^{2n\pi}+1}\right)$$ Can anyone prove this identiy? We found this identity via a sum calculator by varying Simon Plouffe identities

Splitness of quotient sequence

Let $A, B, C$ be holomorphic vector bundles over some complex manifold $X$. Let $A’, B’, C’$ be sub bundles, respectively. Suppose that we have short exact sequences: $$0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$$ $$0 \rightarrow A’ \rightarrow B’ \rightarrow C’ \rightarrow 0$$ Let’s also suppose that the quotient sequence $$0 \rightarrow […]