Questions tagged [krull-dimension]

For questions about or related to the Krull dimension, which counts the length of the longest chain of prime ideals of a ring under inclusion.

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Krull dimension of Rees algebras [closed]

Let $A$ be a commutative ring of Krull dimension $n$. Let $A[It]$ denote the Rees algebra of an ideal $I$ of $A$. Does $\dim A[It]\le2n+1$ hold for all $I$? It is known that $\dim A[t]\le2n+1$ holds (...
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Construct a non-noetherian ring with finite krull dimension [duplicate]

As learning Krull dimension in commutative algebra recently, I cannot successfully construct a non noetherian ring of finite krull dimension. I think the ring must have some special conditions like ...
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Let $A=\oplus_{i \in N \cup \{0\}} A_i$ be a positively graded ring of dimension $d$ with $A_0=k$ and $k$ is a field. If $B$ is a Noetherian graded subring of $A$. Can we say dimension of $B$ cannot ...
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Example of Excellent rings of infinite Krull dimension?

Do there exists Excellent rings of infinite Krull dimension? If I understand correctly, Nagata's famous example of a Noetherian ring of infinite Krull dimension is not excellent...
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Confusion on proof of "dimension of the fiber is an upper semicontinuous function on the source" (From Vakil FOAG, Theorem 12.4.3).

I'm confused on a step of the proof of Theorem 12.4.3(a) in Vakil's FOAG (from the February 21, 2024 version). The theorem states: Suppose $\pi: X \rightarrow Y$ is a morphism of finite type $k$-...
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Krull dimension of modules of tensor products with $\mathbb{Q}$ [closed]

Let $\mathbb{Z}[X_1,...,X_n]$ be the ring of polynomials in $n$ variables over the integers. Let $M$ be a nontrivial finitely generated module over $\mathbb{Z}[X_1,...,X_n]$ which is torsion free as ...
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Confusion about codimension of a subvariety of a scheme

In Eisenbud's and Harris's "3264 & All That", they define the codimension of a subvariety $Y$ of a variety $X$ as $\operatorname{codim}_X(Y)=\dim(X)-\dim(Y)$. This part is fine and also ...
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A monomial ideal with given depth and dimension

Assume we have two integers $a\leq b$. Is there a monomial ideal $I$ in a power series ring $R$ over a field such that $\dim(R/I)=b$ and $\mathrm{depth}(R/I)=a$? Except for depth being at most the ...
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The dimensions of an $A$-algebra

Let $A$ be a commutative ring with unity and $B$ be an $A$-algebra. So, we can consider $B$ as a commutative ring and then calculate its Krull dimension and also we can consider $B$ as an $A$-module ...
132 views

Noether normalization for Laurent polynomial algebras

Let $k$ be a field and $n\geq 1$. Let $A=k[X_1,\dotsc,X_n,X_1^{-1},\dotsc,X_n^{-1}]$ be the algebra of Laurent polynomials in the variables $X_1,\dotsc,X_n$ over $k$. Let $B=A/I$ where $I$ is an ideal ...
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Computing the height of an ideal...?

I hope I'm not overbearing in this site. Yes, I'm still struggling. If you can, I have a question about primary decomposition that still needs help, you can find it in my page. Now I wanted to find ...
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Fiber dimension of finitely presented dominant morphisms of irreducible schemes

This is Exercise 12.4.C in the July 2023 edition of Vakil's algebraic geometry book. We are proving the following theorem (Theorem 12.4.1 in the book): Suppose $\pi:X\to Y$ is a finitely presented ...
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Krull dimension and Lebesgue covering dimension

I'm reading the book Algebraic Geometry, an Introduction (Daniel Perrin) and it introduces a notion of dimension of a topological space as the maximal length of ascending chains of irreducible closed ...
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1 vote
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Finite Krull dimension

If $A$ is a commutative ring and $n=\dim A$ is the Krull dimension of $A$. There exists any criteria for we know that dimension de A is not infinite? For example $A$ is a integral domain and ...
1 vote
134 views

Krull dimension of the local ring at the generic point of a divisor is 1.

Let $X$ be a nonetherian integral separated scheme which is regular in codimension one, i.e. every local ring $\mathscr{O}_x$ of $X$ of dimension one is regular. Let $Y$ be a prime divisor, i.e. a ...
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Tor-dimension of $A/(a)\otimes_k B$, where $A$ and $B$ are Dedekind domains

Let $A$ and $B$ be two Dedekind domains which contain a field $k$ which is algebraically closed in both $A$ and $B$. Let $a$ be a non zero element in $A$. What is the Tor-dimension of $A\otimes_k B$? ...
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