All Questions
Tagged with local-rings ag.algebraic-geometry
29 questions with no upvoted or accepted answers
8
votes
0
answers
221
views
Finitely generated commutative rings with the same profinite completion
Let $R_1$ and $R_2$ be two finitely generated commutative rings. Assume that their profinite completions are isomorphic: $\widehat{R_1}\cong \widehat{R_2}$.
Suppose that $R_1$ is a domain. Does ...
- 1,418
6
votes
0
answers
478
views
The Zariski Riemann Space, but with Local Rings
The Zariski Riemann space, while an abandoned approach, has lead to later developments and generalizations, including $\text{Spv}$ (the space of valuations) and Huber's work. In studying it, I would ...
user30211
4
votes
0
answers
265
views
Sections of smooth morphisms over henselian rings
Let $(A,\mathfrak m)$ be a henselian local ring. Let $R$ and $S$ be $A$-algebras of finite type and $f\colon R\to S$ be a smooth morphism. Assume that the induced morphism $R/\mathfrak m R\to S/\...
- 1,590
3
votes
0
answers
1k
views
Etale cohomology of regular local rings
Let $R$ be a regular local ring (I am particularly interested in the case when $R$ is the local ring of a point on a smooth scheme of finite type over a field). Let $G$ be the etale fundamental group ...
- 15.6k
2
votes
0
answers
293
views
Local weak factorization
This is a follow-up to question Locally toric resolutions of compactifications, answered by Jason Starr.
In a series of papers (see https://arxiv.org/abs/math/9904076), Jaroslaw Wlodarczyk proves ...
- 7,972
2
votes
0
answers
77
views
Analytic spread of an ideal after reduction
Let $(R,m)$ be a local ring and $I$ an ideal in $R.$ Let $l(I):=\dim \bigoplus_{n\geq 0}(I^n/mI^n)$ and $x\in R\setminus I.$
My question is
what is the relation between $l(I)$ anf $l(I+(x)/(x))?$
- 323
2
votes
0
answers
221
views
Meaning of the statement "$a\in I$ is a general element of $I$"
Suppose $I$ is an ideal in a Noetherian local ring $(R,m)$. In some papers I have seen the following statement:
"$a\in I$ is a general element of $I$".
What is the definition of general element ...
- 1,713
2
votes
0
answers
178
views
Modern dictionary for "old" homological terms
I'm trying to build a little dictionary between old Homological algebra for local rings and the slightly more modern approach via derived functors.
Let $X = SpecA$ be a spectrum of a local ring $(A,...
- 7,799
2
votes
0
answers
140
views
When does $R [x]/I $ have infinitely many idempotents in special case?
At < When does $R [x]/I $ has infinitely many idempotents? >, Er_Ro asked the following question.
Let $R $ be a commutative ring with identity and $R[x] $ its polynomial ring. I am looking for ...
- 21
2
votes
0
answers
327
views
PAC field : Algebraically closed field :: ? : Henselian local ring
I'm wondering if the following exists in the world as a definition. I'll use the word "pseudo-Henselian." I'll restrict to DVRs for simplicity.
I'd want to call a DVR $(R,\mathfrak{m})$ pseudo-...
- 1,499
1
vote
0
answers
187
views
Is it true that monomorphisms of local Artinian $\mathbb{R}$-algebras are regular?
A Weil algebra is a finite-dimensional real algebra, in which each element is the uniquely sum of a scalar and a nilpotent (so nilpotents constitute the only maximal ideal of codimension 1). In other ...
- 6,962
1
vote
0
answers
133
views
A question concerning cancellation of ideals
I am working on a number theory project, and at one stage, I encounter a commutative algebra problem. Vaguely speaking, my hope is to show that two ideals are equal. Now I shall explain the data I am ...
- 11
1
vote
0
answers
217
views
Artin-Winters proof of semi-stable reduction theorem: details
I've been reading through Artin-Winters proof of the semi-stable reduction theorem (Degenerate fibers and stable reduction of curves) and found myself confused about the following detail—
Let $\...
1
vote
0
answers
194
views
Is the following local map unramified?
Let $(R,m)$ and $(S,n)$ be two local rings, $R \subseteq S$, $R$ regular, $S$ a finitely generated and flat $R$-algebra, and $mS=n$.
In comments to this question it was claimed that in such situation ...
- 2,837
1
vote
0
answers
117
views
$A \to B$ with $A$ regular imply that $B$ is CM
The answer to this question says the following:
"The general statement is if $A \to B$ is finite and injective, and $A$ is noetherian and regular, then $B$ is CM if and only if $A \to B$ is flat.
...
- 2,837
1
vote
0
answers
165
views
Structure of Complete Local Rings
Let $X$ be a proper $n$-dimensional $k$-scheme and $x \in X$ a closed point. Consider the stalk $\mathcal{O}_{X,x}$. We consider now it's completion $O_{X,x}^{\wedge}$ wrt it's maximal ideal $m_x$.
...
- 6,006
1
vote
0
answers
138
views
Power series ring $R[[X_1,\ldots,X_d]]$ over a domain $R$
Let $R$ be a domain and
\begin{align*}
T \,\colon= R[[X_1,\ldots,X_d]].
\end{align*}
Suppose that we have $d$ elements $f_1,\ldots,f_d \in T$ and let us consider an ideal $J$ of $T$ such that $(f_1,\...
- 563
1
vote
0
answers
78
views
Relation between lifts of simple roots and lifts of idempotents (Henselian property)
Let $f:A\to B$ be a morphism of commutative rings. Given a monic $\varphi\in A[x]$ write $Z(A,\varphi)$ for the set of simple roots of $\varphi$ in $A$. Consider the following properties of $f:A\to B$....
- 10.5k
1
vote
0
answers
46
views
Integral closure of lexsegment ideal
Let $R=k[x_1,\ldots,x_d]$ where $k$ is a field and $I$ be a lexsegment ideal of $R$ and $l(I)=d$ (where $l(I)$ is analytic spread of $I$).
Is $I$ integrally closed?
If I is generated by elements ...
- 1,713
1
vote
0
answers
294
views
Is it true that the functor of completion of a module over a local ring is injective on isomorphism classes?
Let $A$ be a commutative Noetherian local ring and $\hat A$ be its completion. Then we have the functor of completion from the category of finitely generated $A$-modules to the category of finitely ...
- 1,484
1
vote
0
answers
285
views
Analytic spread of an ideal
How to calculate analytic spread of the ideal $I=\left<xyw^2,xyz^2,xw^2+yz^2\right>$ in $\mathbb Q[x,y,z,w]?$
I think it is 3.
- 1,713
1
vote
0
answers
112
views
Asymptotic stability of prime divisors
Suppose $I$ is an ideal in a formally equidimensional local ring $R.$ Let $A(I)$ and $\overline A(I)$ denote Ass$R/I^n$ and Ass$R/\overline{I^n}$ for all large $n$ respectively.
My question is
What ...
- 1,713
1
vote
0
answers
133
views
Intersections of Noetherian regular local rings of finite Krull dimensions
Let us consider Noetherian regular local rings $R_i$ of finite Krull-dimensions for each $i \geq 1$ such that
\begin{equation*}
R_1 \supset R_2 \supset \cdots
\end{equation*}
Suppose each embedding $\...
- 781
1
vote
0
answers
155
views
Universally catenary and all its formal fibers over minimal members are Cohen-Macaulay but it has a nonCohen-Macaulay formal fiber
Please help me to find a Noetherian local ring $R$ such that: $R$ is universally catenary and all its formal fibers over minimal members of $Spec(R)$ are Cohen-Macaulay but $R$ has a nonCohen-Macaulay ...
- 89
0
votes
0
answers
96
views
Length of generic intersection in local ring
Let $(R, \mathfrak{m})$ be a regular local ring and let $I\subset R$ be an ideal of coheight 1. Let $a \in \mathfrak{m}\setminus \mathfrak{m}^2$.
If $a$ that is not a zero divisor of $R/I$ we have ...
0
votes
0
answers
178
views
Finite monomorphism $A \to B$ with reduced $A$ and special fiber implies $B$ reduced
I have a question about correctness of following statement claimed here in $\boxed{2} \ $:
Let $k$ arbitrary field, let $f : X \longrightarrow Y$ be a finite dominant morphism between finite type $k$-...
- 6,006
0
votes
0
answers
53
views
When a given set of primes of height 1 is a set associated primes of an element
Let $R$ be a Noetherian local ring of dimension $\geq 3$ and $\{p_1,\ldots , p_n\}$ be a collection of prime ideals of height $1$. Does there exist an element $x\in R$ such that $Ass(R/xR)=\{p_1,\...
- 1,713
0
votes
0
answers
440
views
A question about strict henselian local rings
Let $f: X\to Y$ be a finite morphism of schemes, let $y\in Y, Y(\bar{y}):={\rm Spec}(\mathscr{O}_{Y, \bar{y}}), X_{\bar{y}}:=X\times_Y{\rm Spec}(\kappa_y^s)=X_y\otimes_{\kappa_y}\kappa_y^s, X({\bar{y}}...
- 1,906
0
votes
0
answers
287
views
On the product in the power series ring
Let $A_n \colon= K[[X_1,\ldots,X_n,Y_1,\ldots,Y_n]]$ be a power series ring over a field $K$ in $2n$ variables and ${\frak m}_{A_n}$ be the unique maximal ideal of $A_n$.
Suppose we have two ...
- 563