# 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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### Gabber's analogue of Bernstein inequality for category $\mathcal{O}$

Let $\mathsf{k}$ be an algebraically closed field of zero characteristic. It is a well known result due to Gabber that for any finite dimensional algebraic Lie algebra $\mathfrak{g}$ and every ...
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### Does locally DVR implies Dedekind Domain when it is 1-dimensional, semi-local domain but Noetherian not given

Let R be a semi-local integral domain of dimension 1 such that $\forall P \in Spec{R}$ such that $P \ne 0$ we have, $R_P$ to be a Discrete Valuation Ring. Then prove that $R$ is a Dedekind Domain? ...
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(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 $Y \... 0answers 44 views ### How to define$\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{...
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 ...