# 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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### 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 ...
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### 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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### $\mathbb{Z}[x,y]/(xy-7)$ is regular

I have to prove that $R=\mathbb{Z}[x,y]/(xy-7)$ is regular. So I have to prove that $R$ is local, Noetherian and that $$dim R= dim_{R \setminus P} P/P²$$ where $P$ is the maximal ideal of $R$. $R$ is ...
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### Equality of transcendence degree and local dimension for non-algebraically closed fields

In Atiyah-Macdonald, the authors prove that if $V$ is an irreducible variety over an algebraically closed field $k$, then the local dimension of $V$ (i.e. the Krull dimension of the localization of ...
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### Questions about Theorem 11.4.1 and Exercise 11.4.C in Vakil's FOAG

Background. I am trying to solve Exercise 11.4.C in Vakil's Foundations of Algebraic Geometry (November 18, 2017 draft) (Exercise 11.4.C) Suppose $\pi: X \to Y$ is a proper morphism to an irreducible ...
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### Dimension property of certain rings involving localization and quotient by primes?

Following this answer, let us make the following definition: Definition: We say a comm ring $R$ has the "DIM property" iff for every prime ideal $p \subset R$, we have  \mathrm{dim}(R_p) +...
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### How can I compute $m^k/m^{k+1}$ for the ideal $m=(X,Y,Z)$ in $R=\Bbb{C}[[X,Y,Z]]$?

Let $R=\Bbb{C}[[X,Y,Z]]$ then I want to compute the Krull dimension of $R$. My idea was to compute the Samuel function and bring it into a polynomial "form" then we immediately know that ...
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### Krull dimension question - Show that it has a unique primary decomposition

PROBLEM: Let $A$ be a Noetherian domain that is an integral extension of $\mathbb{Z}[x]$ and $J$ be a non-zero ideal of $A$. Show that if $I=J\cap \mathbb{Z}[x]$ has height 2, then $J$ has a single ... 62 views

### Commutative Algebra - Krull Dimension and Artinian Ring

Suppose $A = K[x_1, ..., x_n]$ ($n \geq 2$) is the ring of polynomials in $n$ variables over the field $K$. Let $I$ be a proper ideal of $A$. Show that if $A/I$ is Artinian and $I = (f_1, ..., f_n)$ ... Let $A = K[x_1, ..., x_n]$ a ring of polynomials over a field $K$ and $I$ a principal (non-zero) ideal of A. Show that $dim(A/I) = n - 1$. attempt: By the Principal Ideal Theorem it is easy to see ... 