What is the $\mathbb{F}_p$ dimension of $\mathbb{F}_p[G]$?
I heard recently that the $p$-rank of a finite abelian group (the number of cyclic components of size $p^n$) is given by $\dim_{\mathbb{F}_p}(\mathbb{F}_p \otimes_\mathbb{Z} \mathbb{Z}[G])$, which is a...
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Question about dimension of a vector space
I'm reading Exercises in Classical Ring Theory (T.Y.Lam) and there is a exercise:I'm not sure how to determine the left (right) dimension of the vector spaces (red underline in the above).My though is...
View ArticleFind subset of vectors which form basis
QuestionLet W be the subspace of $R^5$ spanned by$ u_1= (1, 2, –1, 3, 4)\\u_2 = (2, 4, –2, 6, 8) \\u_3 = (1, 3, 2, 2, 6)\\ u_4 = (1, 4, 5, 1, 8)\\u_5 = (2, 7, 3, 3, 9)$Find a subset of the vectors...
View ArticleIs $\operatorname{height} \mathfrak{p} + \dim A / \mathfrak{p} = \dim A$ true?
Let $A$ be an integral domain of finite Krull dimension. Let $\mathfrak{p}$ be a prime ideal. Is it true that $$\operatorname{height} \mathfrak{p} + \dim A / \mathfrak{p} = \dim A$$ where $\dim$ refers...
View ArticleVC dimension of indicator functions is equal to pseudo dimension
I am reading the "Foundation of machine learning" by Mehryar Mohri (https://cs.nyu.edu/~mohri/mlbook/). In the proof of Theorem 11.8, it said the following statement, which I can not understand.Let $H$...
View ArticleWhat is the dimension and nature of this variety?
Let $1 < N \in \mathbb{N}$ and $x, a \in \mathbb{C}^N$ with $a$ fixed; also, let $b \in \mathbb{R}_{\ge 0}^N$ be fixed (this last bit can be weakened to the extent it makes no difference). For $n...
View ArticleSuppose U is a subspace of V such that V/U is finite dimensional. Can we say...
Suppose U is a subspace of V such that V/U is finite dimensional. V/U is the quotient sapce, namely the set of all affine subsets of V parallel to U.I think we cannot show that V is finite dimensional,...
View ArticleRank and dimension of matrix in a linear map
Let's say I map a $3 \times 1$ vector $\underline v=(x, y, z)$ by multiplying it with a $3 \times 3$ matrix of rank $2$. Would I be correct in thinking that it transforms all points in 3D space into a...
View ArticleDimension of a Matrix subspace
What is the dimension and the number of basis vectors for a subspace of 3×3 symmetric matrices?Earlier my professor told us that the dimension and the number of basis vectors for a subspace are the...
View ArticleA proof of $\dim(R[T])=\dim(R)+1$ without prime ideals?
Please read this first before answering. This question is only concerned with a proof of the dimension formula using the Coquand-Lombardi characterization below. If you post something that doesn't...
View ArticleIs the dimension of this subspace 1?
"Let $V=M_{2\times2}(\mathbb{R})$ denote the vectors space of all $2\times2$ matrices with real number entries. Determine which of the following subsets are subspaces of $V$. If it is a subspace, find...
View ArticleDimension of $\mathcal{O}_X(X-\{P\})$.
Let $X$ be a smooth projective and irreducible curve and $P\in X$. I am asked to show that the dimension (as a $k$-vector space where $k$ is a algebraically closed field) of $\mathcal{O}_X(X-P)$ is...
View ArticleClassical Krull Dimension of Commutative Rings
I've been looking at the extension of Krull dimension to non-commutative rings as defined, for example, in On the Krull-Dimension of Left Noetherian Left Matlis-Rings [Krause, Mathematische Zeitschrift...
View ArticleTensor product of simple $sl_2$ modules
I am working on the following problem: Let $M(n)$ be the finite-dimensional, simple $\mathfrak{sl}_2(\mathbb{C})$-module with highest weight $n\in\mathbb{N}_0$. Show that the module...
View ArticleIs the inequality always true: $\dim_R (U_1 + U_2 + U_3)\leq...
If V is the linear space over the field R of real numbers, and $U_1, U_2, U_3$ are subspaces of this space. Is the inequality always true: $\dim_R (U_1 + U_2 + U_3)\leq \dim_RU_1+\dim_RU_2+\dim_RU_3$I...
View ArticleTo calculate the dimension of a vector space
Let $E$ and $F$ be two subspaces of $\mathbb{R}^n$, and let $$G = \{\begin{pmatrix} X \\ Y \end{pmatrix}\in \mathbb R^{2n} \mid X+Y \in E, Y \in F\}$$. I am trying to calculate the dimension of $G$,...
View Articlemisunderstanding on real algebraic varieties
Bochnak-Coste-Roy's book "Real Algebraic Geometry" (1998) is probably the main reference on this subject. I am probably misunderstanding something very fundamental as I can apparently find easy...
View ArticleReal-valued dimension
Let $\overline{\mathbb{R}}_{\geq 0} = \mathbb{R}_{\geq 0} \cup \{\infty\}$.Does there exist an example of the following?A commutative ring with unity $R$A mapping $\operatorname{d}:...
View ArticleComputing the height of an ideal...?
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 the...
View ArticleProof of Proposition 11.20 of Atiyah-Macdonald
I struggle with verifying the pole order inequality asserted in the proof of proposition 11.20. (Full statement and proof of the proposition can be found here: Atiyah-Macdonald 11.20 and 11.21)My...
View ArticleA semi-algebraic set in $\mathbb{R}^d$ has dimension $d$ if and only if its...
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 emptyThe dimension here has to be understood in the semi...
View Articledimension of intersection of algebraic variety
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...
View ArticleConfusion about codimension of a subvariety of a scheme
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 agrees with...
View ArticleHow to combine the $4$-dimensions of spacetime into 1 dimension?
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...
View ArticleDoes the set of antiderivatives of $tan(x)$ have uncountable dimension?
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...
View ArticleKodaira dimension
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...
View ArticleA prime ideal of height $\geq 2$ in a Noetherian ring contains infinitely...
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, respectively,...
View ArticleContraction of principal ideal in integral extension
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...
View ArticleQuestions about the proof of Theorem 13.4 in Matsumura's Commutative Ring Theory
I'm reading the Matsumura's Commutative Ring Theory and I'm trying to understand theorem 13.4.Suppose $A$ is Noetherian semilocal ring, $M$ is a finite $A$-module, $\mathfrak{m}$ is the Jacobson...
View ArticleIs $\dim (M/xM) = \dim M - 1$ for some $x \in m$ implies $x$ is an...
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 equal...
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