# Tagged Questions

1answer
83 views

### Niceness of the projection of a closed subscheme of affine space?

Let $k$ be an algebraically closed field, and suppose $C\subseteq \mathbb{A}^{n+m}_k$ is a closed subscheme. What can we say about the image under the projection \$\pi: \mathbb{A}^{n+m}_k\rightarrow ...
1answer
47 views

### studying the topology of a real algebraic set

Let $f_1,\ldots,f_n \in \mathbb{R}[x_1,\ldots,x_m]$ be polynomials with real coefficients and let $I$ be the ideal that they generate. Denote by $V_{\mathbb{R}}(I)$ the corresponding real variety, ...
2answers
55 views

### Looking for a (nonlinear) map from n-dimensional cube to an n-dimensional simplex

I am looking for a (nonlinear) map from n-dimensional cube to an n-dimensional simplex; to make it simple, assume the following figure which is showing a sample transformation for the case when $n=2$. ...
1answer
47 views

### topological properties of an algebraic set in the metric topology

Is there any good strategy of examining whether a given algebraic set is closed or dense in the metric topology of Euclidean space? For example suppose we are in $\mathbb{R}^3$ and consider the set ...
1answer
251 views

### $\mathbb A^n(k)$ and $\mathbb A^n(k)\setminus \{0\}$ are not homeomorphic

Let $k$ be an algebraic closed field. Why $\mathbb A^n(k)$ and $\mathbb A^n(k)\setminus\{0\}$ (for $n>1$) are not homeomorphic with respect to the Zariski topology?
1answer
27 views

### Lie bracket in local coordinates

Can you help for solving this.I have an manıfold exam and ı am working but ı have a problem about lie bracket. And ı am putting what ı did..
0answers
70 views

### Projective closure in the Zariski and Euclidean topologies

In Smith's An Invitation to Algebraic Geometry, following the definition of the projective closure of an affine variety, it was remarked that "the closure may be computed in either the Zariski ...
1answer
56 views

### Looking for a “prime-ish” family of subsets

Is there a nontrivial (what I mean is below) example of a compact Hausdorff space $X$ and a family $\mathscr{F}$ of subsets of $X$ with the following pair of properties? $\mathscr{F}$ is ...
0answers
78 views

### Relation between complex and real sphere

I want to understand relation between complex and real spheres. How to show? $S^1(\mathbb{C}) \approx \mathbb{R} \times S^1$ $S^3(\mathbb{C}) \approx \mathbb{R} \times S^3$ $\approx$ means homotopy ...
0answers
58 views

### Products of sites

Does the category of sites (i.e. small categories equipped with a Grothendieck topology) has products? Is there a connection to the product of locales (as discussed in Johnstone's Stone spaces, ...
1answer
58 views

### Open morphisms are dominant?

This seems very elementary but I haven't been able to prove it: If $f : X \to Y$ is an open map of irreducible topological spaces, then it is dominant (maps generic points to generic points). It ...
1answer
47 views

### Join and Zariski closed sets

A set in $\mathbb{C}^n$ is called Zariski-closed if it can be written as the set of zeroes of some set of polynomial equations  V(f_1,...,f_m) = \left\{ z \in \mathbb{C}^n \mid f_1(z)=...=f_m(z)=0 ...
1answer
80 views

### A new(?) partial order on the set of continuous maps

Let $X,Y$ be topological spaces. Define a partial order on $\hom(Y,X)$ as follows: $f \leq g$ if $f^{-1}(U) \subseteq g^{-1}(U)$ for all open subsets $U \subseteq X$. Equivalently, $f(y)$ is a ...
1answer
189 views

### Equivalent definitions of Noetherian topological space

It is well known that we have many different definitions of noetherianity for rings. Namely, given a ring $R$, the following are equivalent: 1) every ideal of $R$ is finitely generated. 2) $R$ ...
2answers
74 views

### Fibers of the projection of a Zariski dense set are dense?

We will work over an infinite field $\Bbbk$. Let $U\subseteq\Bbbk^m\times\Bbbk^n=\Bbbk^{m+n}$ be a Zariski dense subset and for all $x\in\Bbbk^m$, consider U_x := \{\, y \in \Bbbk^n \mid (x,y)\in U ...
1answer
61 views

### Irreducibility preserved under étale maps?

I remember hearing about this statement once, but cannot remember where or when. If it is true i could make good use of it. Let $\pi: X \rightarrow Y$ be an étale map of (irreducible) algebraic ...
1answer
124 views

### Zariski topology analogue for non-algebraically closed fields

Let $k$ be a field and $\bar{k}$ its algebraic closure. The set $X$ of $n$-tuples over $\bar{k}$ can be given the Zariski topology in which the closed sets are the sets of zeros of sets of polynomials ...
1answer
64 views

### Can a closed subset of an affine scheme have empty interior?

I have an inclusion of closed subsets $V(J) \subset V(I)$ in an affine scheme $Spec(R)$ with the property that $V(I) = V(J) \cup \partial V(I)$. I would like to conclude that $V(J)=V(I)$. (Here ...
1answer
89 views

### Metric tensor of complex numbers & Hamiltonian Mechanics

The Euclidean $\mathbb{R}^2$ geometric space can be mapped onto $\mathbb{C}$. In other words I see it like this \vec{v} = x\vec{x}+y\vec{y} = x\vec{1}+y\vec{i}= \begin{bmatrix}x \\y\end{bmatrix} ...
1answer
182 views

### Fundamental group of multiplicative group in Zariski topology

What is the fundamental group of the multiplicative group of the complex numbers $\mathbb{G}_m(\mathbb{C})$ with respect to the Zariski topology. More precisely, what are the homotopy classes of ...
1answer
115 views

### Decomposition of Noetherian space into irreducible subsets

I am trying to relate two (maybe not) different decompositions of a noetherian topological space into irreducible subsets, given in Ravi Vakil's notes on algebraic geometry. Exercise 4.6.N : Let ...
2answers
129 views

### Sheafs and closed immersion

Let $f:X \rightarrow Y$ be a continuous map of topological spaces, such that it is closed immersion. Let $\mathfrak{F}$ and $\mathfrak{G}$ be sheafs on $X$ and $Y$ respectively. How to show, that ...
2answers
510 views

### What is algebraic geometry?

I am a second year physics undergrad, loooking to explore some areas of pure mathematics. A word that often pops up on the internet is algebraic geometry. What is this algebraic geometry exactly? ...
2answers
78 views

### About the proof of the existence of a decomposition of subset of $\mathbb{A}^n$

The following is the proposition 7.4.11 in "Advanced Topics in Linear Algebra" by Kevin O'meara et.al. Proposition 7.4.11 Every subset $X$ of $\mathbb{A}^n$ has a decomposition \$X = X_1 ...
2answers
62 views

### Closed sets of a curve

I am guessing that the closed sets of a curve (i.e. an algebraic variety of dimension 1) are finite . How would one go about proving this? Is it trivial?! Thanks for any help.
2answers
57 views

### Equivalence conditions on induced Zariski topology

This is a homework problem and I am looking for clarification of some of my doubts. (not solutions) Let $X\subset A^n$ or $P^n$, where $X$ is a non-empty algebraic set. Open sets of $X$ are given ...
1answer
78 views

### Is this a surjection of rings? What am I doing wrong?

Let $Z\newcommand{\df}{:=}\df\newcommand{\C}{\mathbb C}\C$ and $T\df\C^\times$. Then, the coordinate ring of $Z$ is $\C[z]$ and that of $T$ is $\C[t,t^{-1}]$. Consider another copy of $T$ with ...
2answers
67 views

### What do I need to know to understand the completion of the field of rational functions of a non-singular projective curve?

So the title gives the jist of my question. Specifically, let $X$ be a non-singular projective curve, $P$ a point on $X$, $v_P$ the discrete valuation associated to the ring $\mathcal{O}_P$. Then I ...
2answers
239 views

### Irreducibility of an Affine Variety and its Projective Closure

Volume I of Shaferevich's Basic Algebraic Geometry has the following as an exercise: Show that the affine variety $U$ is irreducible if and only if its closure $\bar U$ in a projective space is ...
0answers
71 views

### Continuous choice of basis for subspaces

Consider the flag variety (or flag manifold, depending on who you are) $V=\mathrm {Fl} (3,\mathbb C)$ of complete flags of subspaces of $\mathbb C^3$. That is, an element of M is a tuple (L , P) ...
0answers
61 views

### Euler characteristic of structure sheaf of symmetric product

I recently asked about calculating the Euler characteristic of the symmetric square of a space. There we determined that for a sufficiently well-behaved space $X$ there is a formula \chi(X \times ...
1answer
106 views

### Euler characteristic of a quotient space

I have a question relating to an answer on MathOverflow.net. The cited answer says: Let $X$ be a topological space for which [the Euler characteristic] $\chi(X)$ is defined and behaves in the ...
3answers
521 views

### Zariski Open Sets are Dense?

Is it true than any nonempty open set is dense in the Zariski topology on $\mathbb{A}^n$? I'm pretty sure it is, but I can't think of a proof! Could someone possibly point me in the right direction? ...
0answers
110 views

### Computing the hypercohomology of a complex of acyclic sheaves

Let $K^{\bullet}$ be a cochain complex of sheaves of finite-dimensional vector spaces, I wanted to compute $\mathbb{H}^{\bullet}(X,K^{\bullet})$ = the hypercohomology of the complex $K^{\bullet}$, the ...
2answers
379 views

### The prime spectrum of a Dedekind Domain

Let $A$ be a Dedekind Domain, let $X = \operatorname{Spec}(A)$. Are all open sets in $X$ basic open sets? Thinking about the Zariski topology (in the classical sense) of a non-singular affine curve, ...
0answers
56 views

### Recovering the topology of an affine scheme from the specialization preorder

Let $A$ be a commutative ring. The specialization preorder on $\mathrm{Spec}(R)$ is given by \$\mathfrak{p} \prec \mathfrak{q} \Leftrightarrow \mathfrak{p} \in \overline{\{\mathfrak{q}\}} ...
5answers
858 views

### Why Zariski topology?

Why in algebraic geometry we usually consider the Zariski topology on $\mathbb A^n_k$? Ultimately it seems a not very interesting topology, infact the open sets are very large and it doesn't satisfy ...
1answer
377 views

### An interesting topological space with $4$ elements

There is an interesting topological space $X$ with just four elements $\eta,\eta',x,x'$ whose nontrivial open subsets are $\{\eta\},\{\eta'\},\{\eta,\eta'\}, \{\eta,x,\eta'\}, \{\eta,x',\eta'\}$. This ...
0answers
87 views

### Algebraic varieties and Hausdorff spaces

Let $(X,\mathcal O_X)$ be an algebraic prevariety, by definition, it is an algebraic variety iff the diagonal $\Delta(X)$ is closed in the product $X\times X$. The above property is equivalent to the ...
1answer
270 views

### How do mathematicians think about high dimensional geometry?

Many ideas and algorithms come from imagining points on 2d and 3d spaces. Be it in function analysis, machine learning, pattern matching and many more. How do mathematicians think about higher ...
2answers
122 views

### The notion of a germ in singularity theory

I quote from my lecture: Let $X$ be a topological space (think of $X=\mathbb{C}^n$ with the classical topology), $p\in X$, $A,B\subseteq X$. Then $A\sim B$ if there exists an open subset ...
0answers
55 views

### Continuity of a map of a topological space to a pro-topological space

Let $(X_i)$ be a projective system of topological spaces. Let $X$ be the projective limit of $X_i$. Let $G$ be a topological space. What does it mean for $G\to X$ to be continuous? My guess is that ...
2answers
271 views

### Zariski Topology question

Could you please give a hint how to show that the zariski topology on $\mathbb{A}^2$ is not the product topology on $\mathbb{A}^1\times\mathbb{A}^1$
3answers
178 views

### Zariski topology in the complex plane: an example

I want to find the closure under the zariski topology, of this set $\left\{ {\left( {x,y} \right) \in {\Bbb C}^2 ;\left| x \right| + \left| y \right| = 1} \right\}$ I have no idea what I can do
5answers
537 views

### Every subspace of a compact space is compact?

It seems as if every subspace of a compact topological space (equipped with its relative topology) had ought to be compact as well. Is this true in general? And in particular, I want to use the fact ...
2answers
215 views

### Explaining the motivation behind two different definitions of a generic point

This question is primarily regarding the definition of a generic point of a topological space that I came across in Qing Liu's Algebraic Geometry and Arithmetic Curves. First I will give the ...
3answers
179 views

### $\operatorname{Spec} (A)$ as a topological space satisfying the $T_0$ axiom

I have been spending a few days now proving the last bit of the following problem of Atiyah Macdonald: Prove that $X = \operatorname{Spec}(A)$ as a topological space with the Zariski Topology ...
1answer
132 views

### Compact Sets in Projective Space

Consider the projective space ${\mathbb P}^{n}_{k}$ with field $k$. We can naturally give this the Zariski topology. Question: What are the (proper) compact sets in this space? Motivation: I ...
0answers
159 views

### homework problem about the projective real space

Sorry for ask this problem, but I am very complicated with this problem :/ . My course it´s of topology, the teacher said that we only need the definition of the quotient topology and of  P_R^2 ...
1answer
91 views

### Curves on the projective plane

I have two little questions, I'm learning this, and I'm not accustomed yet )=. The questions are so simple. First define on $$R^3 - \left\{ {\left( {0,0,0} \right)} \right\}$$ the topology given ...
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