All Questions
Tagged with ag.algebraic-geometry local-rings
73 questions
0
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Analytic spread of localization of an ideal
Let $J$ be an ideal in a Noetherian local ring $(R,m)$. It is well known that for any prime ideal $p\in Spec(R)$, $l(J_p)\leq l(J)$, where $l(J)$ is the analytic spread of $J$.
Q) Are there ...
- 2,773
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112
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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
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300
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Behaviour of length function under faithfully flat extension
Let $(R,m)$ and $(S,n)$ be local Noetherian rings such that $S$ is a faithfully flat extension of $R$. Let $J\subsetneq I $ ideals of $R$.
Can we relate $l_R(I/J)$ and $l_S(IS/JS)$?
PS: Here $l(-)$ ...
- 28.8k
1
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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 $\...
- 63.9k
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2
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524
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Almost complete intersection ideal and $d$-sequence
In a Noetherian local ring $R$, an ideal $I$ is called an almost complete intersection ideal if $\mu(I)=\text{ht}(I)+1$.
Q) Is it true that $I$ is generated by a $d$-sequence?
- 591
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Properties of d-sequence
Let $x_1,\ldots,x_n$ is a sequence in a Noetherian local ring $R$. We say $x_1,\ldots,x_n$ is a $d$-sequence if
1) $x_i\notin (x_1,\ldots,\hat{x_i},\ldots,x_n),$
2) for all $k\geq i+1$ and all $i\...
- 7,249
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0
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178
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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
3
votes
1
answer
358
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uniqueness of uniformizers
Let R be a noetherian normal domain (if it makes any difference, I'm happy to assume R is also local).
If $p$ is a height one prime, then the localization $R_p$ is a dvr, hence the maximal ideal $...
- 1,889
1
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2
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748
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Krull-dimension of local domain
Let $(R,{\frak m}_R)$ be a local domain (not necessarily Noetherian). That is, $R$ is integral and ${\frak m}_R$ is the unique maximal ideal of $R$. Suppose that ${\frak m}_R$ is finitely generated.
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- 28.8k
4
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1
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Can K[[T_1,...,T_∞]] be embedded into K[[X,Y]]?
In the MathOverflow question about common false beliefs, the following answer teaches us that there is an embedding $\iota_n \colon K[[T_1,...,T_n]] \hookrightarrow K[[X,Y]]$. Now let us define the ...
CommunityBot
- 1
4
votes
1
answer
168
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Reference on the classification of (low rank) Gorenstein rings over $\mathbb{C}$
I am interested in the question of the classification of (low rank) Gorenstein rings over $\mathbb{C}$. The socle of a local algebra is the annihilator of its maximal ideal. A commutative local ring ...
- 6,237
2
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0
answers
140
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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 ...
CommunityBot
- 1
2
votes
1
answer
191
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what are the possible approximations for ideals
(Fix some local ring $(R,\mathfrak{m})$ over a field of zero characteristic.)
Suppose an ideal $J$ is defined by some complicated formula/procedure. And there is no hope of computing it/or writing ...
CommunityBot
- 1
7
votes
1
answer
621
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automorphisms of local rings vs local change of coordinates
Let $R$ be a local (commutative, associative) ring over a field of zero characteristic. (My typical examples are $k[[x_1,..,x_p]]/I$, $k\{x_1,.,x_p\}/I$, $C^\infty(\Bbb{R}^p,0)$. If it helps one can ...
- 25.3k
2
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1
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771
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Does the normalization morphism induce isomorphism on residue fields?
The question is basically coming from the following situation:
Let $C$ be an integral curve over a field $k$ (EDIT and assume that $k$ is not algebraically closed) and let $\phi\colon C^N\to C$ be the ...
- 165
3
votes
1
answer
422
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Automorphisms of complete discrete valuation ring
Let ${\Bbb F}_2[[T]]$ be a c.d.v.r over ${\Bbb F}_2$. We consider the automorphism $\sigma$ of ${\Bbb F}_2[[T]]$ such that $\sigma \colon T \mapsto T + c_2T^2 + c_3T^3 + \cdots$ with $c_i \in {\Bbb F}...
CommunityBot
- 1
1
vote
0
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155
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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
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327
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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
10
votes
1
answer
2k
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Etale cohomology of the completion of a Henselian local ring
Let $\pi: R\to S$ be a local morphism of Henselian local rings. Let $f: R \to \hat{R}$ and $g: S \to \hat{S}$ be their completions. Let $\mathcal F$ be a constructible $l$-adic sheaf on $\operatorname ...
- 375
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1
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575
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Can you suggest a good name for a local homomorphism φ:(R,m)->(S,n) of local rings with the property that φ(m)S is n-primary?
Can you suggest a good name for a local homomorphism $(R,\mathfrak{m})\stackrel{\varphi}{\rightarrow}(S,\mathfrak{n})$ of local rings with the property that $\varphi(m)S$ is $\mathfrak{n}$-primary?
...
- 11.1k
5
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1
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On the functoriality of scalar extensions of local rings (edited)
Note. I have edited my question to make it more transparent, following some very good comments that I received. I am sorry if it is a bit long.
A local homomorphism of local rings $(A,\mathfrak{m})\...
- 3,908
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1
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320
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covers of complete regular local rings
It is well-known that if one assumes algebraic closedness and characteristic 0 of the residue field then finite covers of complete DVRs are all of the form $A[x]/(x^m-a)$ for some $a \in A$ (direct ...
- 22.6k
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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