Tagged Questions

5
votes
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 ...
4
votes
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, ...
4
votes
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 $n=2$. ...
3
votes
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 ...
13
votes
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?
0
votes
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..
3
votes
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 ...
1
vote
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 ...
3
votes
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 ...
4
votes
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, ...
3
votes
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 ...
4
votes
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 ...
5
votes
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 ...
12
votes
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) $R$ ...
2
votes
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 ...
4
votes
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 ...
8
votes
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 ...
3
votes
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 ...
2
votes
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} ...
2
votes
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 ...
1
vote
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 ...
1
vote
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 ...
17
votes
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? ...
1
vote
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 ...
1
vote
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.
1
vote
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 ...
5
votes
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 ...
0
votes
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 ...
4
votes
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 ...
5
votes
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) ...
1
vote
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 ...
4
votes
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 ...
6
votes
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? ...
4
votes
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 ...
16
votes
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, ...
2
votes
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}\}} ...
16
votes
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 ...
19
votes
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 ...
0
votes
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 ...
6
votes
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 ...
2
votes
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 ...
0
votes
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 ...
2
votes
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 $\mathbb{A}^1\times\mathbb{A}^1$
8
votes
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
1
vote
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 ...
8
votes
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 ...
6
votes
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 ...
2
votes
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 ...
1
vote
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 ...
1
vote
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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