Questions tagged [dimension-theory-algebra]

Question 1

Let $R$ be a Jacobson ring, meaning that every prime ideal is an intersection of maximal ideals, and let $S$ be a finitely generated $R$-algebra. I am interested in proving the following. (I sadly ...

Question 2

The following is 2.3 in Krick and Logar (1991). (ISBN: 978-3-540-38436-6) Let $I \leq k[X_1, \dots X_n]$ be of dimension $d$, and w.l.o.g assume that $I \cap k[X_1, \dots, X_d] = (0)$. We say that $...

Question 3

Question Let $k$ be an algebraically closed field, and give $\mathbb{A}^n_k$ the Zariski topology. Let $Y$ be an irreducible closed subset of $\mathbb{A}^n$, and let $Z:=V(F)\subseteq \mathbb{A}^n$ ...

Question 4

Let $R$ be a commutative ring, and let $p_0 \subset p_1$ be prime ideals in $R[x]$ such that $p_0 \neq p_1$ and $p_0 \cap R = p_1 \cap R$. Prove that the chain $p_0 \subset p_1$ is saturated. ( A ...

Question 5

Let $A$ an integrally closed domain and $B$ a commutative ring extension of $A$ that is finitely generated as an $A$-module. For $f\in B$ is it true that there exists $a_f\in A$ s.t. $\sqrt{(f)\cap A}=...

Question 6

Let $R$ be a Noetherian local ring with maximal ideal $m$, and let $M$ be a finitely generated $R$-module. It is known that if $x \in m$ is an $M$-regular element, then the dimension of $M/xM$ is ...

Question 7

I am trying to prove this statement by deducing a contradiction to Krull's Principal ideal theorem. Here is my attempt: Denote $R$ and $P$ the ring and the prime ideal under consideration, ...

Question 8

Let $C$ be a smooth projective curve over an algebraic field $k$. I suppose the Kodaira dimension of $C$ is $0$. Why the genus must be $1$? If $C$ is a smooth projective surface, and if I suppose the ...

Question 9

On a connected domain, the set of antiderivatives of a continuous function $f$ is $1$-dimensional. However, for a punctured domain like $\mathbb{R} - \{0\}$, the set of antiderivatives becomes $2$-...

Question 10

I have been thinking about the possibility of representing all points in a $4$-dimensional spacetime coordinate system $\mathbb{R}^{1,4}$, as points on one line $P$ (or axis of a $1$-dimensional ...

Question 11

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 ...

Question 12

I know there are similar questions, but everyone uses different approaches and it's complicated to change proofs. In my algebraic geometry course, we're dealing with algebraic variety (topological ...

Question 13

I would like to prove the following result : A semi-algebraic set in $\mathbb{R}^d$ has dimension $d$ if and only if its interior is non empty The dimension here has to be understood in the semi ...

Question 14

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 ...

Question 15

Let $\overline{\mathbb{R}}_{\geq 0} = \mathbb{R}_{\geq 0} \cup \{\infty\}$. Given a commutative ring with unity $R$, $R\operatorname{-Mod}$ denotes the category of $R$-modules. Given $R$-modules $A$ ...

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Questions tagged [dimension-theory-algebra]

Contraction of Zero-dimensional Ideal in Finite Extension of Jacobson Ring is Zero-dimensional

Minimal Primes of Highest Dimension of a Polynomial Ideal in Noether Position

Part of Affine Dimension Theorem

Chain of Prime Ideals in Polynomial Ring $R[x]$ ($R$ is commutative) is saturated When Their Intersection Over $R$ is Equal

Contraction of principal ideal in integral extension

Is $\dim (M/xM) = \dim M - 1$ for some $x \in m$ implies $x$ is an $M$-regular element ? where $m$ is the maximal ideal of a local ring $R$.

A prime ideal of height $\geq 2$ in a Noetherian ring contains infinitely many prime ideals of height 1

Kodaira dimension

Does the set of antiderivatives of $tan(x)$ have uncountable dimension?

How to combine the $4$-dimensions of spacetime into 1 dimension?

Confusion about codimension of a subvariety of a scheme

dimension of intersection of algebraic variety

A semi-algebraic set in $\mathbb{R}^d$ has dimension $d$ if and only if its interior is non empty

Computing the height of an ideal...?

Real-valued dimension

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