# Tagged Questions

40 views

### Dimension of a topological space [duplicate]

I want to prove this fact that if X is a topological space which is covered by a family of open subsets {U_i} than dimX=supdimU_i One direction I can see that the RHS is less than or equal to the ...
38 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 ...
44 views

### Hochschild dimension

I'm curious; if $A$ ia 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 ...
68 views

### Principal ideal domain is universally catenary …

... actually, even more general statement is true: Theorem. Every regular ring is universally catenary. (see for example Algebraic Geometry by Qing Liu, Corollary 2.16, Chapter 8) Though, the ...
96 views

### Space of global sections for a smooth projective curve of genus $g$

Let $X$ be a smooth projective curve of genus $g$, $T_{X}$ its tangent bundle and $H^{0}(X,T_{X})$ the space of global sections for $X$. What is $\dim H^{0}(X,T_{X})$ and why?
115 views

### algebraic geometry exercise: infinite subset is dense

A hypersurface $C \subset \mathbb{A}^2$ we call a plane curve. Show that any infinite subset of an irreducible plane curve $C \subset \mathbb{A}^2$ is dense in $C.$ Note. We call hypersurface the ...
157 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 ...
185 views

### Which definition of dimension came first?

In my algebraic geometry class, the dimension of an affine variety $X=V(I)$ was defined as the supremum of the length of chains of prime ideals in the coordinate ring $R=k[x_1,\ldots,x_n]/\sqrt{I}$, ...
Suppose $R=\mathbb{C}[x_1,\ldots, x_n]$ is a polynomial ring with $I$ being an ideal of $R$. Let $I'$ be an ideal of $R[t]$. If $R[t]/I'$ is flat as a $\mathbb{C}[t]$-module and over $0$, ...