# Questions tagged [local-rings]

In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or prime.

345 questions
Filter by
Sorted by
Tagged with
1 vote
20 views

### Ring of total fractions of the strict henselization of a non-normal local ring

Let $A$ be an integral Noetherian local ring of dimension 1 which is not normal and with residue field a finite field $\mathbb{F}_q$ (so typically some local ring of a singular curve on a finite field)...
• 340
27 views

### Subset of $\operatorname{End}_A(M)$ faithful implies $M$ is faithful.

Let $A$ be a complete intersection ring, $M$ be a finite $A$-module of positive depth (over the maximal ideal), and $B$ be the image of $A$ in $\operatorname{End}_A(M)$. It is easy to show that $B$ ...
• 467
113 views

### Reduced noetherian local ring of depth zero is artinian?

It is well-known that a local ring $A$ with maximal ideal $\mathfrak{m}$ of depth zero is not necessarily artinian (e.g. $k[x,y]/(xy, x^2)$ localised at the origin), but what if we further require ...
• 353
217 views

### Codimension inequality of prime ideal in a regular local ring

The following is an exercise (#6) of Eisenbud's commutative algebra, chapter 10: Exercise. We mentioned that if $P$ is a prime ideal in a regular local ring $R$ and if $R\to S$ is a map of local rings,...
• 2,034
29 views

• 11
86 views

39 views

• 363
52 views

### Two Properties of Perfect Rings

I came across Kunz's theorem about the characterization of regular rings in characteristic $p$. In the paper that I am reading, the author uses perfect rings to prove this result. Perfect rings $R$ ...
• 31
48 views

### Are uniformizers of DVR's unique?

I just started learing about discrete valuation rings, so I don't know a lot of examples and I can't find any counterexamples. So consider a DVR $(R,v)$ with valuation group $\Gamma_v \neq \{0\}$. A ...
138 views

### Why are local rings called local?

I gather that rings of germs of functions at a point $p$ on a manifold/variety/etc. are local with the maximal ideal containing exactly the germs of functions which vanish at $p$. So in some sense, ...
• 9,988
1 vote
96 views

### Question about a proof involving local rings: $R$ has exactly $3$ ideals, show that if $a,b\in I$ then $ab=0$

I have a question regarding the following thread: Commutative unitary ring with exactly three ideals. I believe I've put the pieces together, but I am, for whatever reason, feeling uncomfortable. So, ...
• 2,432
84 views

### What does finding a "free local ring" have to do with finding the spectrum of a ring?

In Tierney's 1976 paper On the Spectrum of a Ringed Topos (which you can find here) at the top of section 2 we read Let $A$ be a commutative ring in [a topos] $\mathbf{E}$. We look at the problem of ...
• 25.7k
1 vote
73 views

### Support of module and faithfully flat base change

Let $R \subseteq S$ be a faithfully flat extension of Noetherian local rings. Let $M$ be a finitely generated $R$-module such that $\operatorname{Supp}_R(M)=\operatorname{Spec}(R)$. Then, is it true ...
• 129
111 views

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

Let $\bf{Sch, Sets, Ring}$ be a category of schemes, sets, commutative rings. By Yoneda's lemma, scheme $X$ is characterized by contravariant functor $$\operatorname{Hom}(*, X): \bf{Sch}^{op}\to Sets$$...
• 936
11 views

### Computing residue fields of affine schemes

I have taken a course on schemes, so I am familiar with the basic definitions, but I'm very rusty and I've forgotten how to do this (if I ever knew). Basically, I want to compute the residue fields of ...
• 3,293
71 views

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

In my homework of schemes, the professor proposed the following exercise: "Let $X$ be a scheme of finite type over a field $k$ and $f \in \mathcal{O}_X (X)$ a global section. Show that $f$...
51 views

• 986
137 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?

I am reading the book “Fourier-Mulkai transforms in algebraic geometry” by Daniel Huybrechts. In the proof of Lemma 4.5, in page 92, it is written that if $M$ is a finite module over a local ...
• 1,220
1 vote
64 views

• 2,126
29 views

### A basic question on commutative finite local rings

Let $R$ be a commutative finite local ring of order $p^n$ ($p$ is a prime and $1\in R$). I'm struggling with the following two basic questions: (a) Is it true that $x^n=0$ for every non-unit $x\in R$ ?...
• 4,316
38 views

### surjection of complete noetherian local rings -reference help

I am looking Lemma 1.1: $\widehat{C}$ is the category of complete local noetherian rings. Is there a reference to the last two lines of the above argument ? i.e. : associated graded rings being ...
• 8,954
152 views

### Noetherian local ring of depth $1$ whose localization at some prime ideal has higher depth? [closed]

Let $(R,\mathfrak m)$ be a Noetherian local ring of depth $1$. Then, is it possible that depth$(R_P)\ge 2$ for some prime ideal $P$ of $R$? Of course such an example has to be non-Cohen-Macaulay, i.e. ...
• 497
105 views

• 3,583
1 vote