Questions tagged [dimension-theory-algebra]

For questions about notions of dimension, rank, or length used in abstract algebra (e.g. Krull dimension, homological dimensions, composition length, Goldie dimension). Questions about dimension of vector spaces, and rank of linear transformations are better placed under the [linear-algebra] tag.

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32 views

Linear independence of characters implies statement about algebraic independence of regular functions?

Part of proposition 3.1.9 of Geck's Algebraic Geometry and Algebraic Groups has the following setup. Let $G$ be a connected affine algebraic group over $k$ an algebraically closed field. If there ...
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1answer
19 views

Dimension of the orthogonal algebra?

The following is on page 3 of Introduction to Lie Algebras and Representation Theory by Humphreys: Here the author claims that the dimension of the orthogonal algebra is $2l^2+l$; but I think the ...
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45 views

Dimension of product of varieties

I've got this exercise asking me to prove first that the product of quasi-projective varieties $X$ and $Y$ (henceforth just "varieties") is irreducible iff both $X$ and $Y$ are. I managed to solve ...
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47 views

Definition of codimension of variety

Let $X$ be a variety over field $k$. A Weil divisor on $X$ is an integral linear combination of irreducible subvarieties of $X$ of codimension $1$. So I want to know the definition of codimension of a ...
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74 views

Hartshorne II.3.22c following the hint

I know this exercise can be solved in several ways, such as Ravi Vakil's proof or this answer, but I'd like to try to follow the given hint if possible. The exercise says the following: Let $f:X→...
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50 views

Dimension image of morphism of projective varieties

Let $f: \mathbb{P}^n \to \mathbb{P}^m$ be a rational map. Then there exists $U \subset \mathbb{P}^n$ open so that $f_{|U}$ is a morphism. What can we say about the dimension of $\overline{f(U)}$? We ...
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176 views

Hartshorne II-3.22(b)

Let $f:X\to Y$ be a dominant morphism of integral schemes of finite type over a field $k$. Let $e=\dim(X)-\dim(Y)$. For any point $y\in f(X)$, show that every irreducible component of the fibre $...
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1answer
37 views

Dimension of affine affine algebras as a module

Suppose that $A\cong \mathbb{R}[f_1,\dots,f_d]$ is a (commutative) affine $\mathbb{R}$-algebra (with identity); where $f_i$ are polynomials $\mathbb{R}[x_1,\dots,x_N]$. When is $A$ a finite-...
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1answer
46 views

a.c.c. and d.c.c. on radical ideals in commutative ring of dimension zero

Let $R$ be a commutative ring with unity of dimension zero (i.e. every prime ideal is maximal). Does any of the following two conditions imply the other : 1) $R$ satisfies a.c.c. on radical ideals ...
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98 views

Infinite Noetherian ring of dimension $1$ in which distinct non-zero ideals have distinct and finite index

Let $R$ be an infinite commutative ring with unity such that every non-zero ideal has finite index. Then $R$ is Noetherian, every non-zero prime ideal is maximal , and I can also show that $R$ is an ...
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27 views

Does $\operatorname{length}(M/xM) \leq \operatorname{rank}(M) \cdot \operatorname{length}(R/(x))$ hold over non-integral rings $R$?

$\DeclareMathOperator{\len}{length} \DeclareMathOperator{\rk}{rank}$In Eisenbud's book Commutative Algebra with a View towards Algebraic Geometry he says: The basic result of this section expresses ...
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40 views

Axioms that characterize the notion of dimension [duplicate]

Let $\mathcal{C}$ be a class of sets/spaces/structures among which we have a dimension. Namely a map $d:\mathcal{C}\rightarrow \mathbb{N}$ defined in a certain manner that motivated the appellative ...
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81 views

Dimension of a hypersurface of $\mathbb C^n$ / of a cut by a hypersurface

Defining the domain $$\Gamma[V]=\mathbb C [\bar x ]/I(V)$$ for any irreducible variety $V\subset \mathbb C^n$ (by variety, I mean only zero set of a family of polynomials), $\Gamma(V)$ for the field ...
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1answer
99 views

Exercise 1.8 from Hartshorne

(Hartshorne 1.8) Let $Y$ be an affine variety of dimension $r$ in $\mathbf A^n$. Let $H$ be a hypersurface in $\mathbf A^n$, and assume that $Y \nsubseteq H$. Then every irreducible component of $Y \...
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42 views

How to define $\operatorname{dim}(\{0\})$ and $\operatorname{ht}(A)$?

Matsumura's "Commutative Algebra", Chapter 5, Page 72. It follows from the definition that $\operatorname{ht}(\mathfrak p)=\operatorname{dim}(A_{\mathfrak p})\quad (\mathfrak p\in \operatorname{...
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1answer
147 views

Proof of Theorem of Dimension of Fibres

I am following a lecture series on YouTube, and in the series the lecturer skipped the proof of the Theorem on the Dimension of Fibres. I tried to follow the style of the professor's proof but I ...
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1answer
79 views

Extending a morphism to a projective scheme

Let $C$ be a reduced affine Noetherian scheme of pure dimension 1 (all its irreducible components have dimension 1) and $p \in C$ a regular closed point. Suppose we have a morphism $C \backslash p \to ...
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1answer
155 views

What are the discrete valuation rings for the affine plane?

Let $X = \mathbb{A}^2_k$ be the affine plane over an algebraically closed field $k$, and let $K = k(x,y)$ be the field of rational functions over $X$. How can one describe all discrete valuation rings ...
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1answer
71 views

Valuation ring of finite Krull dimension whose every non-maximal ideal is principal

Let $(R, \mathfrak m)$ be a Valuation ring of finite Krull dimension such that every non-maximal ideal i e. every ideal which is not $\mathfrak m$, is principal. Then is $R$ Noetherian i.e. a discrete ...
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60 views

Connected components of an open subset

Suppose we have an (affine) irreducible algebraic variety $X$ over $\mathbb{R}$, and we are given one polynomial $f$. Consider the closed subset $C$ where this is zero. What can we say about the ...
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57 views

Irreducibility of “almost” flag variety

Suppose that $k_1, k_2, k_3, n$ are integers such that $k_1<k_2<n$, $k_1<k_3<n$. Consider the variety $\mathcal I(k_1,k_2,k_3,n)$ of vector subspaces $$ V_1\subset V_2 \subset V, \quad ...
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108 views

Proof of Krull's Height Theorem for irreducible affine varieties

I'm trying to solve exercise 11.3.H in Vakil's notes (http://math.stanford.edu/~vakil/216blog/FOAGnov1817public.pdf). We want to show that if $X = \mathrm{Spec} A$ is an irreducible affine $k$-variety ...
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339 views

Proving that irreducible components of Zariski closed subsets are minimal prime ideals

So the question is to prove that if $X = Z(\mathfrak{U})$ is a Zariski-closed subset in $A^n$, then $Y = Z(\mathfrak{P})$ is an irreducible component of $X$ if and only if $\mathfrak{P}$ is a minimal ...
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82 views

When do open affine subschemes of equidimensional schemes are again equidimensional?

Let $X$ be an equidimensional scheme satisfying the properties $P_1,\ldots,P_n$. Could someone please give me an example (with a reference or proof) of $P_1,\ldots, P_n$ such that the following ...
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215 views

About Hartshorne's Proposition 1.10

This is from Hartshorne's Algebraic Geometry: 1) I don't understand how he concludes that $\text{height } m=n$. I agree that $\overline{Z_0}\subset...\subset\overline{Z_n}$ is maximal and gives rise ...
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2answers
204 views

Final step of exercise 11.7 from Atiyah-Macdonald ($\dim A[x]=\dim A+1$)

Ex. 11.7 from Atiyah-Macdonald is basically to prove $\dim A[x]=\dim A+1$ for $A$ noetherian. From exercise 11.6, we get $\dim A[x]\geq\dim A+1$, so we are left to prove "$\leq$". I've followed the ...
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1answer
106 views

Vanishing of $\text{Ext}(M,R)$ for Noetherian $R$

Let $R$ be a Noetherian ring and $n \in \mathbb{Z}$ such that for any f.g. $R$-module $M$ and $k > n$ we know that $\text{Ext}^k_R(M,R) = 0$. Does it follow from this that $\text{Ext}^k_R(M,R) = 0$ ...
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124 views

Dimension of localization

Let $R$ be a Noetherian ring and let $𝔭=(h_1, ..., h_n)$ be a prime ideal in $R$. Suppose that $R$ is a finitely generated $A$-module, $A=k[h_1, ..., h_n]$. What is the Krull dimension of $R_𝔭$ as ...
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403 views

Criterion for a locally factorial domain

Let $R$ be a Noetherian domain. Recall that $R$ is said to be locally factorial if $R_\mathfrak p$ is factorial for all prime ideals $\mathfrak p\subset R$. How do I show that $R$ is locally factorial ...
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109 views

Dimension of a Variety defined by the Weierstrass Equation

I want to see why the following is true: A Variety described by the Weierstrass equation has dimension 1. Let $K$ be a field. An elliptic curve over $K$ is defined by the set of solutions in $\...
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1answer
349 views

Codimension 1 points

Today I was reading a proof of the following Lemma from Liu's "Algebraic Geometry and Arithmetic Curves" Recall: A a point $x \in X$ is called a codimension 1 point if $ \overline{ \{x \}}$ has ...
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1answer
197 views

Lower bound on dimension of fibres of a dominant mophism of irreducible affine varieties

Whilst doing exercise $11.4.B$ of Ravi Vakil's "Foundations of Algebraic Geometry", I got stuck with the following problem (although I think that many of the hypotheses are unnecessary and a more ...
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1answer
299 views

Codimension and height of prime ideals

Definition: Let $Z$ be an irreducible closed subset of $X$. Then the codimension $\textrm{codim} (Z,X)$ is the supremum of integers $n$ such that there exists a chain $$ Z = Z_0 < Z_1 < \dots &...
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92 views

(quasi)coherent rings for which $\dim R[T]\neq \dim R+1$

What are some some examples of (quasi)coherent rings for which $\dim R[T]\neq \dim R+1$? Why (hopefully geometrically) should we not always have equality? Notation. Let $I,J$ be two ideals of ...
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1answer
381 views

Rank of a matrix over a principal ideal domain

I apologize if my question is stupid but I'm not very familiar with matrices over a principal ideal domain $R$ (For example, $R=\mathbb{Z}$ or $R=\mathbb{R}[X]$). I was wondering how to define the ...
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42 views

Short proof of the coincidence of left dom.dim., right dom.dim. and relative dom.dim. for semi-primary left QF-3 rings?

is there a short and/or elementary proof of the following fact (which is taken from theorem 7.7 from H. Tachikawa, ‘‘Quasi-Frobenius Rings and Generalizations’’, Springer Lecture Notes in Mathematics):...
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2answers
500 views

Dimension of local rings on scheme of finite type over a field.

In chapter III Hartshorne seems to be using without proof or mention a theorem on the dimensions of local rings of schemes of finite type over a field. I know that for an integral scheme of finite ...
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112 views

Prove that this condition is true on an Zariski open set

Why is there an Zariski open set of $P\in GL(n,\mathbb{R})$ such that $P\,\text{diag}(1,\dots,1,-1)P^{-1}$ can be conjugated by a diagonal matrix $D$ to get an orthogonal matrix? Note that $M=P\,\...
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1answer
58 views

Global dimension of the center

Let $R$ be a ring. Must the global dimension of the centre $Z(R)$ of the ring $R$ always be atmost that of $R$ itself? I mean is it generally true that: $D(Z(R)) \leq D(R)$ (where D is the global ...
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94 views

Depth and kernel of multiplication map

If $A$ is a commutative $k$-algebra, $\mu: A\otimes_k A\rightarrow A$ is its multplication map and $I$ is the kernel of that map (viewed as an $A\otimes_k A$-module map) then what is the relationship ...
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578 views

Krull dimension of localization

If $R$ is a commutative ring and $m$ a maximal ideal therein, then what are the conditions for the Krull dimension of $R$ equaling to the Krull dimension of $R_m$?
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1answer
125 views

Hochschild dimension

I'm curious; if $A$ is a commutative $k$-algebra over a field $k$ of global dimension $n$, then is its $A^e$-projective dimension $2n$ (this is also sometimes called the Hochschild cohomological ...
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188 views

prove $\dim\mathbb{Z}[X_1,X_2]=3$ from first principles

Since $\dim R[X] =\dim R+1$ for any Noetherian ring $R$, the ring $\mathbb{Z}[X_1,X_2]$ must have dimension 3. But how can this be proved 'from first principles', i.e. without using any big theorems ...
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1answer
295 views

Equivalence of definitions of Krull dimension of a module

I've seen two definitions of Krull dimension of a module $M$ over a (commutative) ring $R$, and their equivalence does not seem obvious: Matsumura on page 31 of his book Commutative Ring Theory ...
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3answers
129 views

Converse of a dimension lemma

Consider the following lemma. It comes from the Stacks Project. Lemma 9.59.11. Suppose that $R$ is a Noetherian local ring and $x\in\mathfrak m$ an element of its maximal ideal. Then $\dim R\le ...
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1answer
197 views

Krull dimension of a $\mathbb Q$-algebra

I'm trying to find the Krull dimension of $\mathbb{Q}[X,Y,Z]/(X^{2}-Y,Z^{2})$. My professor said that I have to consider that $\mathbb{Q}[X,Y,Z]/(X^{2}-Y,Z^{2})$ is a $\mathbb{Q}$-algebra, but I don'...
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1answer
174 views

Defining the Rank of a Projective Module

I am trying to understand the definition of rank for a projective module over a noncommutative ring. The definition I am using is: A sufficient condition for the rank of a free module over a ring $...
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36k views

A short proof for $\dim(R[T])=\dim(R)+1$?

If $R$ is a commutative ring, it is easy to prove $\dim(R[T]) \geq \dim(R)+1$. For noetherian $R$, we have equality. Every proof I'm aware of uses quite a bit of commutative algebra and non-trivial ...
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362 views

Hilbert (polynomial) dimension and dimension of a support of a module

$\newcommand{\Supp}{\mathrm{Supp}}$ $\newcommand{\Ann}{\mathrm{Ann}}$ Let $X$ be an affine algebraic variety (over a field $K$, can assume it is algebraicaly closed), $M$ a finitely generated $\...
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1answer
125 views

Injective endomorphism and Hilbert dimension

Let $\mathcal{O}$ be a finitely-generated $K$-algebra where $K$ is a field and let $M$ be a finitely-generated $\mathcal{O}$-module. For every good filtration $0 = M_0 \subset M_1 \subset M_2 \subset ...