User Jian

6 votes

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Let $f(x)$ be an irreducible polynomial over $F$ of degree $n$, and let $K$ be a field extension of $F$ with $[K:F]=m$

answered May 4, 2018 at 6:14

5 votes

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pullback is preserved by left exact covariant functor $T : \mathsf{Mod}_R \to \mathsf{Ab}$

answered Jun 12, 2017 at 8:40

4 votes

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Why are epis stable under pullback in an abelian category ?

answered May 24, 2018 at 9:58

4 votes

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Is there a nonzero module $M$ over an artinian local ring $(R,\mathfrak m)$ such that $\mathfrak mM=M\ ?$

answered Oct 6, 2018 at 11:37

3 votes

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How do I understand the module structure on Yoneda Ext?

answered Nov 20, 2018 at 1:48

3 votes

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Yoneda Lemma (again)

answered Aug 18, 2017 at 6:38

3 votes

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A relation between prime ideals and ring of fraction.

answered Jan 19, 2018 at 13:21

2 votes

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Prime Ideal Properly Contained in principal Ideal.

answered May 3, 2018 at 23:32

2 votes

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Homotopy category of chain complexes and a example of a null complex

answered Aug 30, 2017 at 2:56

2 votes

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Question about definition of injective module

answered Oct 2, 2017 at 15:11

2 votes

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how to prove a graded version of Nakayama's lemma

answered Oct 25, 2018 at 12:39

2 votes

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$A_1\cdots A_s=0$, prove $\sum_{i=1}^s \operatorname{rank}(A_i) \leq (s-1)n$

answered May 5, 2018 at 12:51

2 votes

For $d\mid m$, $\mathbb{Z}/d\mathbb{Z}$ is not an injective $\mathbb{Z}/m\mathbb{Z}$-module when some prime divides $d$ and $\frac{m}{d}$

answered May 11, 2018 at 9:09

2 votes

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confirmation on direct/inverse limits

answered May 4, 2018 at 15:07

2 votes

Understanding surjective morphism of sheaves

answered Apr 27, 2019 at 4:33

2 votes

Over which Noetherian local rings are any second syzygy module reflexive?

answered Dec 20, 2019 at 23:43

2 votes

Support of module and faithfully flat base change

answered Dec 26, 2021 at 5:07

1 vote

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Proposition 3.6 of Yoshino's book--Characterization of maximal Cohen-Macaulay modules

answered Apr 23, 2022 at 9:56

1 vote

Finite product of semilocal rings is semilocal

answered Nov 21, 2019 at 15:07

1 vote

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Calculation of length of modules

answered Jan 13, 2020 at 0:19

1 vote

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Local Cohen-Macaulay ring over which every finitely generated module of finite injective dimension also has finite projective dimension

answered Nov 14, 2019 at 16:43

1 vote

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Equivalence for rings with localization property

answered May 16, 2019 at 14:44

1 vote

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Any $R$-module is a quotient of some free module. Do we need "finitely generated" condition?

answered May 5, 2018 at 12:23

1 vote

Prove that $(g ◦ f)_∗ = g_∗ ◦ f_∗ : H_∗(C_∗) → H_∗(E_∗)$. Also show that $(Id_{C_∗})_∗ = Id_{H∗(C∗)}: H_∗(C_∗) → H_∗(C_∗)$.

answered May 22, 2018 at 2:25

1 vote

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injective dimension and Ext

answered Jun 12, 2018 at 13:19

1 vote

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Let $R$ be a Noetherian ring and $M$ a finite $R$-module. Show that $\ell(M)<\infty$ if and only if $\text{Supp}(M)\subset m\text{-Spec}(R)$

answered Oct 31, 2018 at 15:33

1 vote

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length of minimal projective resolution and projective dimension

answered Nov 6, 2018 at 11:53

1 vote

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Prove that for distinct prime ideals $P, Q$ of PID $R$, $P^n + Q^m = R$.

answered Nov 6, 2018 at 12:35

1 vote

Computing projective resolutions over quotients of polynomial rings

answered Nov 19, 2018 at 1:18

1 vote

Accepted

A completely reducible module is Artinian if and only if it is Noetherian.

answered Nov 23, 2018 at 10:31

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51 Answers

Let $f(x)$ be an irreducible polynomial over $F$ of degree $n$, and let $K$ be a field extension of $F$ with $[K:F]=m$

pullback is preserved by left exact covariant functor $T : \mathsf{Mod}_R \to \mathsf{Ab}$

Why are epis stable under pullback in an abelian category ?

Is there a nonzero module $M$ over an artinian local ring $(R,\mathfrak m)$ such that $\mathfrak mM=M\ ?$

How do I understand the module structure on Yoneda Ext?

Yoneda Lemma (again)

A relation between prime ideals and ring of fraction.

Prime Ideal Properly Contained in principal Ideal.

Homotopy category of chain complexes and a example of a null complex

Question about definition of injective module

how to prove a graded version of Nakayama's lemma

$A_1\cdots A_s=0$, prove $\sum_{i=1}^s \operatorname{rank}(A_i) \leq (s-1)n$

For $d\mid m$, $\mathbb{Z}/d\mathbb{Z}$ is not an injective $\mathbb{Z}/m\mathbb{Z}$-module when some prime divides $d$ and $\frac{m}{d}$

confirmation on direct/inverse limits

Understanding surjective morphism of sheaves

Over which Noetherian local rings are any second syzygy module reflexive?

Support of module and faithfully flat base change

Proposition 3.6 of Yoshino's book--Characterization of maximal Cohen-Macaulay modules

Finite product of semilocal rings is semilocal

Calculation of length of modules

Local Cohen-Macaulay ring over which every finitely generated module of finite injective dimension also has finite projective dimension

Equivalence for rings with localization property

Any $R$-module is a quotient of some free module. Do we need "finitely generated" condition?

Prove that $(g ◦ f)_∗ = g_∗ ◦ f_∗ : H_∗(C_∗) → H_∗(E_∗)$. Also show that $(Id_{C_∗})_∗ = Id_{H∗(C∗)}: H_∗(C_∗) → H_∗(C_∗)$.

injective dimension and Ext

Let $R$ be a Noetherian ring and $M$ a finite $R$-module. Show that $\ell(M)<\infty$ if and only if $\text{Supp}(M)\subset m\text{-Spec}(R)$

length of minimal projective resolution and projective dimension

Prove that for distinct prime ideals $P, Q$ of PID $R$, $P^n + Q^m = R$.

Computing projective resolutions over quotients of polynomial rings

A completely reducible module is Artinian if and only if it is Noetherian.