List questions from exact-sequence

830

Views

Showing that $0 \to \mathrm{Hom}(M, M_1) \to \mathrm{Hom}_R(M, M_2) \to \mathrm{Hom}(M, M_3)$ is exact

Published on 11 Dec 2014 - 9:23

202

Views

Exactness and Naturality

Published on 17 Dec 2014 - 8:49

861

Views

Finding all abelian groups such that there exists certain short exact sequence.

Published on 25 Dec 2014 - 18:12

3.3k

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Example of a non-splitting exact sequence $0 → M → M\oplus N → N → 0$

Published on 27 Dec 2014 - 10:36

120

Views

Equivalence of short exact sequences

Published on 28 Dec 2014 - 0:33

606

Views

Exactness of Hom Functor

Published on 28 Dec 2014 - 1:17

225

Views

Splitting of an exact sequence

Published on 02 Jan 2015 - 4:05

276

Views

Non exact sequence of quotients by torsion subgroups

Published on 02 Jan 2015 - 14:34

241

Views

Is the four lemma wrong? Do the rows actually need to be exact?

Published on 02 Oct 2019 - 18:14

48

Views

(shorter) exact sequences

Published on 03 Oct 2019 - 12:44

91

Views

Isomorphism between free abelian groups that fit into a short exact sequence

Published on 03 Oct 2019 - 15:40

651

Views

$Q$ is an injective object in an abelian category iff $\text{Hom}_C(\cdot, Q)$ is exact, the $\Rightarrow$ direction.

Published on 23 Mar 2026 - 18:28

897

Views

Show splitting lemma for short exact sequences

Published on 04 Oct 2019 - 12:23

587

Views

Injective module characterisation

Published on 26 Mar 2026 - 2:51

247

Views

Injective homomorphism of artinian modules is surjective.

Published on 05 Oct 2019 - 12:13

List Question

Showing that $0 \to \mathrm{Hom}(M, M_1) \to \mathrm{Hom}_R(M, M_2) \to \mathrm{Hom}(M, M_3)$ is exact

Exactness and Naturality

Finding all abelian groups such that there exists certain short exact sequence.

Example of a non-splitting exact sequence $0 → M → M\oplus N → N → 0$

Equivalence of short exact sequences

Exactness of Hom Functor

Splitting of an exact sequence

Non exact sequence of quotients by torsion subgroups

Is the four lemma wrong? Do the rows actually need to be exact?

(shorter) exact sequences

Isomorphism between free abelian groups that fit into a short exact sequence

$Q$ is an injective object in an abelian category iff $\text{Hom}_C(\cdot, Q)$ is exact, the $\Rightarrow$ direction.

Show splitting lemma for short exact sequences

Injective module characterisation

Injective homomorphism of artinian modules is surjective.

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