All Questions
10 questions
1
vote
1
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96
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On "minimal presentation" of local rings essentially of finite type over a field
Let $k$ be a field of characteristic $0$. Let $(R,\mathfrak m)$ be a local ring essentially of finite type over $k$ (https://stacks.math.columbia.edu/tag/07DR). Then, $R$ is the homomorphic image of ...
- 361
6
votes
0
answers
369
views
Geometric meaning of localization at $(1+I)$?
Let $I\vartriangleleft A$ be an ideal of a commutative ring. Consider the submonoid $1+I\subset A$. What is the geometric interpretation of localization at this submonoid? How does it relate to the ...
- 10.5k
8
votes
1
answer
372
views
Commutative ring $R$ with no nontrivial idempotents, with a localization $R_r$ with infinitely many idempotents
I am looking for a commutative ring $R$ with $1$ such that $R$ has no idempotents and there exists $r\in R$ such that the localization ring $R_r$ has infinitely many idempotents.
- 101
53
votes
2
answers
8k
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Is primary decomposition still important?
On p.50 of Atiyah and Macdonald's Introduction to Commutative Algebra, in the introduction to the chapter on primary decomposition, it says
In the modern treatment, with its
emphasis on ...
- 15.4k
1
vote
1
answer
161
views
Geometric meaning of colocalization of modules?
Let $A$ be a commutative ring and $S\subset A$ a subset. A localization of $A$ at $S$ is defined as a ring morphsim $A\to A[S^{-1}]$ which is initial with respect to inverting $S$. Similarly, a ...
- 10.5k
0
votes
1
answer
314
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Localization and containment in commutative ring
Let $R$ be a commutative ring with identity and $x, y $ be fixed elements of $R$ such that for each maximal ideal $m$ of $R$ we have $\langle \frac{x}{1_m}\rangle\subseteq\langle \frac{y}{1_m}\rangle$ ...
- 11
2
votes
1
answer
154
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Special submodules over almost Dedekind domains
An integral domain $R$ is an almost Dedekind
domain if for each maximal ideal $m$ of $R$, the ring $R_m$ is a Dedekind
domain, where $R_m$ is the localization of $R$ at $m$.
Question: Let $M$ ...
- 51
0
votes
1
answer
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
1
vote
0
answers
172
views
Local cohomology commuting with fiber
Let $A$ be a nice commutative ring (say, $A=k[t_1,\ldots , t_n]$, ring of polynomials over an algebraically closed field $k$).
Let $M$ be an $A[x]$-module, which is finitely generated as an $A$-...
- 5,562
0
votes
0
answers
5
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When localization commutes with arbitrary intersection of ideals
For commutative ring with identity we know that in general localization dose not commute with arbitrary intersection of ideals. I am looking for a paper that consider equivalent condition for rings ...
