All Questions
8 questions
8
votes
1
answer
257
views
Minimal resolution of local cohomology module
Let $(R,\mathfrak m)$ be a Noetherian local ring of dimension $d\geq 1.$ Suppose $\lambda(H_{\mathfrak m}^i(R))<\infty$ for some $0\leq i\leq d-1.$
Question Can we say anything about Betti numbers ...
- 1,713
4
votes
1
answer
168
views
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 ...
- 41
2
votes
1
answer
112
views
Example of non injective module over Noetherian local ring with trivial vanishing against residue field?
Is there an example of a module $M$ over a commutative Noetherian local ring $(R,\mathfrak m, k)$ such that $\text{Ext}_R^{>0}(k, M)=0$ but $M$ is not an injective $R$-module?
I know that for such ...
- 480
2
votes
1
answer
511
views
Structure theorem for non-Noetherian local rings
Is there a structure theorem (like Cohen 's structure theorem) for non-Noetherian local rings?
I am adding what I am looking for as someone asked in the comment.
If $R$ is a local domain (not ...
- 271
2
votes
0
answers
66
views
Projective cover (minimal) for (derived)complete modules over Noetherian local rings exist?
Let $(R,\mathfrak m)$ be a commutative Noetherian local ring. Let $M$ be an $R$-module which is $\mathfrak m$-adically derived complete. Then, does there exist a free $R$-module $F$ and a surjective $...
- 412
2
votes
0
answers
221
views
Meaning of the statement "$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
1
vote
2
answers
194
views
The quotient of an algebra with an ideal whose generators are decomposed as the product of irreducible elements
I would like to find reference for the following statement.
I need it only in the particular case when $A=\mathcal{O}_{(\mathbb{C}^n, 0)}$ is the local algebra of holomorphic germs $(\mathbb{C}^n, 0) \...
- 73
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