Question

When an element of the total ring of fractions of a noetherian local ring $R$ is integral over $R$?

score 81 · Answer 1 · 2021-09-19 20:18:06

81

Views

When an element of the total ring of fractions of a noetherian local ring $R$ is integral over $R$?

Published on 19 Sep 2021 - 20:18

score 205 · Answer 2 · 2026-03-27 16:47:46

205

Views

If $\text{Tor}^R_1(M,R/xR)=0$, then is $x$ necessarily $M$-regular?

Published on 27 Mar 2026 - 16:47

score 154 · Answer 3 · 2021-09-25 00:06:56

154

Views

If a module is second syzygy and has no free summand , then it can be defined via a minimal resolution?

Published on 25 Sep 2021 - 0:06

score 94 · Answer 4 · 2026-03-25 12:41:22

94

Views

Does the torsion submodule and the $0$-th local Cohomology module coincide over local Cohen-Macaulay ring?

Published on 25 Mar 2026 - 12:41

score 137 · Answer 5 · 2021-09-28 01:01:43

137

Views

On the natural map $\Phi_M: M\otimes_RM^\rightarrow\text{Hom}_R(M^,M^*),\ x\otimes y\mapsto \left\{f\mapsto f(x)y\right\} $

Published on 28 Sep 2021 - 1:01

score 227 · Answer 6 · 2021-10-05 11:32:42

227

Views

Is there any isomorphism between these two local rings?

Published on 05 Oct 2021 - 11:32

score 56 · Answer 7 · 2026-03-28 03:26:25

56

Views

Understanding the structure of $\mathbb{Z}[[x]]/(x-x^2)$

Published on 28 Mar 2026 - 3:26

score 338 · Answer 8 · 2021-10-17 14:08:40

338

Views

A proof that if A is a local ring, then the set of non-units is an ideal, without using Zorn's Lemma?

Published on 17 Oct 2021 - 14:08

score 200 · Answer 9 · 2026-03-26 02:47:37

200

Views

Elementary proof of injection from $\operatorname{Hom}(R, k[\varepsilon])$ to $\operatorname{Hom}_k(\mathfrak{m}_R /\mathfrak{m}_R^2, k)$

Published on 26 Mar 2026 - 2:47

score 292 · Answer 10 · 2021-11-18 17:25:50

292

Views

Why does a finite module over a Noetherian local ring supported only at the maximal ideal have the residue field as a submodule and a quotient?

Published on 18 Nov 2021 - 17:25

score 77 · Answer 11 · 2026-03-28 00:30:58

77

Views

If $\operatorname{Idem}B\rightarrow \operatorname{Idem}(B/mB)$ is surjective then $B$ is a product of local rings

Published on 28 Mar 2026 - 0:30

score 230 · Answer 12 · 2021-12-13 13:10:02

230

Views

Regular local rings of Krull dimension $1$ are integral domains

Published on 13 Dec 2021 - 13:10

score 222 · Answer 13 · 2021-12-16 10:40:04

222

Views

Can a discrete valuation ring be finitely generated over a field?

Published on 16 Dec 2021 - 10:40

score 67 · Answer 14 · 2026-03-28 06:07:52

67

Views

Computing residue fields of affine schemes

Published on 28 Mar 2026 - 6:07

score 290 · Answer 15 · 2021-12-22 10:18:03

290

Views

For local ring $R$, does funcotor $\operatorname{Hom( Spec}R, X)$ characterize scheme $X$?

Published on 22 Dec 2021 - 10:18

List Question

When an element of the total ring of fractions of a noetherian local ring $R$ is integral over $R$?

If $\text{Tor}^R_1(M,R/xR)=0$, then is $x$ necessarily $M$-regular?

If a module is second syzygy and has no free summand , then it can be defined via a minimal resolution?

Does the torsion submodule and the $0$-th local Cohomology module coincide over local Cohen-Macaulay ring?

On the natural map $\Phi_M: M\otimes_RM^\rightarrow\text{Hom}_R(M^,M^*),\ x\otimes y\mapsto \left\{f\mapsto f(x)y\right\} $

Is there any isomorphism between these two local rings?

Understanding the structure of $\mathbb{Z}[[x]]/(x-x^2)$

A proof that if A is a local ring, then the set of non-units is an ideal, without using Zorn's Lemma?

Elementary proof of injection from $\operatorname{Hom}(R, k[\varepsilon])$ to $\operatorname{Hom}_k(\mathfrak{m}_R /\mathfrak{m}_R^2, k)$

Why does a finite module over a Noetherian local ring supported only at the maximal ideal have the residue field as a submodule and a quotient?

If $\operatorname{Idem}B\rightarrow \operatorname{Idem}(B/mB)$ is surjective then $B$ is a product of local rings

Regular local rings of Krull dimension $1$ are integral domains

Can a discrete valuation ring be finitely generated over a field?

Computing residue fields of affine schemes

For local ring $R$, does funcotor $\operatorname{Hom( Spec}R, X)$ characterize scheme $X$?

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