All Questions
7 questions
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Finiteness of Krull dimension of commutative Noetherian ring for which maximal length of regular sequence in maximal ideals have a uniform upper bound
$\DeclareMathOperator\grade{grade}$Let $R$ be a commutative Noetherian ring. For an ideal $I$ of $R$, let $\grade(I,R)$ be the maximal length of an $R$-regular sequence in $I$.
My question is: If $\...
- 413
1
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610
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The Krull dimension of the tensor product of rings
The Krull dimension of a ring $R$ is defined as the length of the longest chain of prime ideals in it. Let $R_i$, for $i\in\mathbb{N}$ denote a sequence of commutative Noetherian rings of Krull ...
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When is the Tor-dimension of $R/(r)$ strictly smaller than that of $R$?
Let $R$ be a ring (commutative with unit) which I assume Noetherian and regular. In particular, the homological dimension of $R$ is the same as its Krull dimension.
I am looking for results in ...
- 1,479
3
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239
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commutative ring satisfying descending chain condition on radical ideals
Let $R$ be a commutative ring with unity which satisfies d.c.c. on radical ideals. then does $R$ satisfy a.c.c. on radical ideals ? If this is not true in general, then what happens if we also assume $...
- 1,209
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269
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Valuation ring satisfying either a.c.c. or d.c.c. on prime ideals
If a commutative ring with unity has finite Krull dimension, then it satisfies a.c.c. and d.c.c. on prime ideals. The converse is not true in general, as can be seen from here An infinite dimensional ...
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2
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165
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Decomposing Noetherian hereditary rings of Krull dimension $1$ into product of hereditary domains (i.e. Dedekind domains)
Let $R$ be a commutative Noetherian hereditary ring (https://en.wikipedia.org/wiki/Hereditary_ring) of Krull dimension $1$. Then is it true that $R$ is a finite direct product of Dedekind domains ?
- 1,209
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Dimension of Ext modules [closed]
Let $(R,m)$ be a noetherian local ring, and $M$ and $N$ be two finitely generated $R$-module. Then is it true that $\dim \text{Ext}^k(M,N)\leq \dim M-k$? If not does the reversed inequality hold?
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