All Questions
61 questions
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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
1
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0
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133
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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 $\...
- 781
1
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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
0
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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?
- 1,713
0
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1
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295
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Depth of almost complete intersection rings
Let $R$ be a regular local ring and let $I \subset R$ be an almost complete intersection ideal, that is, $\mu(I)=\text{ht}(I)+1$ where $\mu(I)$ is the number of minimal generators of $I$ and $ht(I)=\...
0
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1
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300
views
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 $l(-)$ ...
- 1,713
0
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1
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260
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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 ...
- 1,713
0
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0
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94
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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
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177
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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 $k$-...
- 6,028
0
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53
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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
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0
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287
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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