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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.

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 tag.

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147 questions
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Do linear independent sets of a central (Lie-)algebra remain linear independent in scalar extensions?

Let K'/K be a field extension, L' a K'-(Lie-)algebra and L a K-(Lie-)algebra, such that $L'\subseteq \K'\otimes_K L$ (by injective embedding). Than we can review $L'=\K'\cdot L$. Consider there is a K-linear independent set $U\subseteq L$, which is…
Isidor Konrad Maier
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How can the height of a non zero prime ideal be $0$?

In my exercise sheet I am supposed to prove that the only prime ideal of height $0$ in an integral domain domain is $(0)$, and to compute the prime ideals of height $0$ in $\mathbb R[x,y]/(xy)$. I seem to be missing something: we defined the height…
Redundant Aunt
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Dimension of vector space of all matrices satisfying AB=BA

Let $A$ be a $55\times 55$ diagonal matrix with characteristic polynomial $(x-c_1)(x-c_2)^2(x-c_3)^3,\ldots ,(x-c_{10})^{10}$, where $c_1,c_2,\ldots ,c_{10}$ are all distinct. Let $V$ be the vector space of all $55\times 55$ matrices $B$ such that…
Sumit Sah
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If $a_1,...,a_r$ is an $M$-regular sequence of maximal length, $M/(a_1,...,a_r)M$ has finite length.

Let $M$ be a finitely generated module over a Noetherian local ring $R$, and $a_1,...,a_r$ be an $M$-regular sequence of maximal length. Then, $M/(a_1,...,a_r)M$ has finite length? I guess it is not true. If the statement is true, by the dimension…
user682705
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Help Understanding Proof about Dimension Theory

I am reading through Atiyah and Macdonald's Dimension theory, chapter, but I can't understand a step in the proof. The relevant definitions are included below. $\lambda$ is hte length function. Why do we get $g(t)$ above? Where does this come from?…
Kind Bubble
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Typo in Introduction to Commutative Algebra by Atiyah-Macdonald?

Proposition 11.4. Let $A$ be a Noetherian local ring, $m$ its maximal ideal, $q$ an $m$-primary ideal, $M$ a finitely generated $A$-module, ($M_n$) a stable $q$-filtration of $M$. Then i) $M/M_n$ is of finite length, for each $n\geqslant0$; ii) for…
user682705
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How to define $\operatorname{dim}(\{0\})$ and $\operatorname{ht}(A)$?

Matsumura's "Commutative Algebra", Chapter 5, Page 72. It follows from the definition that $\operatorname{ht}(\mathfrak p)=\operatorname{dim}(A_{\mathfrak p})\quad (\mathfrak p\in \operatorname{Spec}(A))$, and that, for any ideal $I$ of…
Born to be proud
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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 statement is true? Every open affine subscheme of $X$…
windsheaf
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Prove that the transcendental degree of $k[x_1,\cdots, x_n]/(f_1,f_2,\cdots,f_r)$ is larger than $n-r$.

I am trying to solve the Exercise 1.9 from Hartshorne: Suppose the ideal $\mathfrak a$ in $R$ can be generated by $r$ elements. Show that every irreducible component of $Z(\mathfrak a)$ has dimension $\geq n-r$. I know it is an immediate…
Aolong Li
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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 $\text{Ext}^k_R(M,R) = 0$ for arbitrary $R$-modules $M$ as well? I am trying…
Martin
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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 dimension)?
AIM_BLB
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Finding dimension of a submodule

Let $G= (\mathbb{C}^3, A)$ be the $\mathbb C[x]$-module given by $$ A=\left( \begin{matrix} 0 & 1 & 1 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ \end{matrix}\right). $$ For a vector $v ∈ \mathbb C^3$ let $L(v) :=…
The Problem
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