Search Results

I think there is a neat answer to this question. Lemma: Let $S$ be regular local of dimension $d$, $M$ a f.g $S$-module. Then: $$\text{depth}(M)\geq n \Longleftrightarrow \text{Ext}^i(M,S)=0\ …

Interesting question! Here is a generalization which includes the Dedekind domain case: Proposition: Let $R$ be a Noetherian regular domain with $n=\dim R$ and $I\subset R$ an ideal such that $R/ …

I have to admire your persistence, perhaps you really want an answer (-: In general, the answer to your first question (second paragraph) is NO, it is not $S$, even for monomial ideals in a polynomi …

For your question in the second paragraph, any semiring can be embedded in one with non-trivial derivation.

UPDATE 10/04/10: Here are some partial results in the graded case. Let $R=\oplus_0^s R_i$ be a graded Gorenstein algebra over $k=R_0$ ($s$ is the socle degree). $R$ is said to have strong Lefschetz p …

This question is a bit vague, but I will try my best. Equivalent conditions involving only vanishing of Tor or Ext over $R$ are unlikely to exist, as they tend to be able to detect only projective dim …

You may assume that $R,S$ are complete and $S$ is a domain. Now take the integral closure $T$ of $S$, which is $S$-finite. Since we are in dimension $2$, $T$ is maximal Cohen-Macaulay module over $R$, …

The purpose of this separate answer is to address the other part of the question, when does equality happen? (also, my first answer is getting too long). I. As FC pointed out, equality happens if $J …

May be I am missing something, but the adjoint isomorphism gives: $$ \text{Hom}_R(M, \text{Hom}_A(R,N)) \cong \text{Hom}_A(R\otimes_R M, N) = \text{Hom}_A( M, N)$$ Since $R$ is Azumaya, $R$ is a pro …

EDIT: here is a counter-example for the question in general. Let $P \subset R= \mathbb C[[x,y,z,a,b,c]]$ be generated by the $2$ by $2$ minors of the obvious $2$ by $3$ matrix. Then the local cohomol …

Let me offer a somewhat elementary version of what others have written. PID and Dedekind domains were studied because people were interested in factorization properties. In non-domains factorization b …

This is not true even over $\mathbb C$. Take $\mathbb C[x,y]$ and $x^2, y^2$. You need general combination of a regular sequence of length at least $3$. Search for "local Bertini theorem" and "Flenner …

This is only an observation, but perhaps it would be helpful. The problem is equivalent to the case $M$ being a cyclic prime module, i.e $M=S/P $ for some prime ideal $P$. That is because if we take …

Let $R=k[x,y]$, $I$ be any height $2$ ideal with at least 3 generators. Then $I$ has a resolution: $0 \to R^a \to R^b \to I \to 0$ Counting ranks gives $b=a+1$. Dualizing the above sequence, notin …

Here is just some sanity check: We may as well work on the local case. Suppose $R=k$ local and $S=R/m$, $m$ is the maximal ideal of $R$. I will also assume $L,M$ finitely generated. Then the LHS is …