Question

How to prove that there exists an $R$-module $E$ and an exact sequence?

score 70 · Answer 1 · 2023-10-05 13:03:52

70

Views

How to prove that there exists an $R$-module $E$ and an exact sequence?

Published on 05 Oct 2023 - 13:03

score 71 · Answer 2 · 2026-03-25 10:55:39

71

Views

Commutative diagrams of exact rows.

Published on 25 Mar 2026 - 10:55

score 51 · Answer 3 · 2026-03-25 10:53:43

51

Views

Follow up question on diagram-chasing/short exact sequences.

Published on 25 Mar 2026 - 10:53

score 89 · Answer 4 · 2023-10-06 23:41:54

89

Views

Schanuel's Lemma

Published on 06 Oct 2023 - 23:41

score 64 · Answer 5 · 2023-10-09 08:56:17

64

Views

$0\rightarrow N \hookrightarrow M \twoheadrightarrow M/N \rightarrow 0$ is split exact when $M/N$ is free (again!)

Published on 09 Oct 2023 - 8:56

score 89 · Answer 6 · 2026-03-25 11:15:57

89

Views

Relative cohomology of open Mobius strip

Published on 25 Mar 2026 - 11:15

score 42 · Answer 7 · 2023-10-12 10:23:00

42

Views

Is this Proof of Well-definedness Correct?

Published on 12 Oct 2023 - 10:23

score 130 · Answer 8 · 2023-10-14 01:28:35

130

Views

What does it mean by "$\chi$ is trivial"?

Published on 14 Oct 2023 - 1:28

score 28 · Answer 9 · 2023-10-14 02:19:50

28

Views

How to show $\text{Im } i=\text{ker }\pi$ for sequece $0\xrightarrow{}X'\xrightarrow{i}X\xrightarrow{\pi}X/X'\xrightarrow{}0?$

Published on 14 Oct 2023 - 2:19

score 33 · Answer 10 · 2023-10-14 02:21:54

33

Views

Meaning of the phrase "$X'$ can be identified with a subspace of $X$"?

Published on 14 Oct 2023 - 2:21

score 91 · Answer 11 · 2026-03-26 19:01:51

91

Views

A module $J$ is injective iff every short exact sequence of the form $0\to J\to A\to B\to 0$ splits.

Published on 26 Mar 2026 - 19:01

score 70 · Answer 12 · 2023-10-15 02:10:51

70

Views

Assumptions for $\text{Ker}(p)=\text{Im}(i)$ to hold so that $0\xrightarrow{}N\xrightarrow{i}M\xrightarrow{p}P\xrightarrow{}0$ is exact?

Published on 15 Oct 2023 - 2:10

score 45 · Answer 13 · 2023-10-15 02:59:59

Interpreting $0\xrightarrow{}\text{Im }f, 0\xrightarrow{}\text{Ker }f$ in examples of exact sequences for showing $\text{Im }f_{n+1}=\text{Ker }f_n.$

score 51 · Answer 14 · 2026-03-27 03:49:52

51

Views

Why does an injective $R$-module homomorphism $M\to N$ preserve submodules?

Published on 27 Mar 2026 - 3:49

score 37 · Answer 15 · 2023-10-20 19:09:17

37

Views

Let $0\to N\to M\to P\to 0$ be a split exact sequence. Show that $\varphi:N\oplus P\to M$ given by $\varphi(n,p)=f(n)+h_1(p)$ is an isomorphism.

Published on 20 Oct 2023 - 19:09

List Question

How to prove that there exists an $R$-module $E$ and an exact sequence?

Commutative diagrams of exact rows.

Follow up question on diagram-chasing/short exact sequences.

Schanuel's Lemma

$0\rightarrow N \hookrightarrow M \twoheadrightarrow M/N \rightarrow 0$ is split exact when $M/N$ is free (again!)

Relative cohomology of open Mobius strip

Is this Proof of Well-definedness Correct?

What does it mean by "$\chi$ is trivial"?

How to show $\text{Im } i=\text{ker }\pi$ for sequece $0\xrightarrow{}X'\xrightarrow{i}X\xrightarrow{\pi}X/X'\xrightarrow{}0?$

Meaning of the phrase "$X'$ can be identified with a subspace of $X$"?

A module $J$ is injective iff every short exact sequence of the form $0\to J\to A\to B\to 0$ splits.

Assumptions for $\text{Ker}(p)=\text{Im}(i)$ to hold so that $0\xrightarrow{}N\xrightarrow{i}M\xrightarrow{p}P\xrightarrow{}0$ is exact?

Interpreting $0\xrightarrow{}\text{Im }f, 0\xrightarrow{}\text{Ker }f$ in examples of exact sequences for showing $\text{Im }f_{n+1}=\text{Ker }f_n.$

Why does an injective $R$-module homomorphism $M\to N$ preserve submodules?

Let $0\to N\to M\to P\to 0$ be a split exact sequence. Show that $\varphi:N\oplus P\to M$ given by $\varphi(n,p)=f(n)+h_1(p)$ is an isomorphism.

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