Questions tagged [zariski-topology]

For questions about the topology of schemes and (classical) algebraic varieties.

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If $f:\operatorname{Spec} A\to\operatorname{Spec} B$ is a map of schemes, how can I show $V(\varphi^{-1}(I)) = \overline{f(V(I))}$ for $I\subset B$?

If $X = \operatorname {Spec}(A)$ and $Y = \operatorname{Spec}(B)$ are affine schemes and I have a morphism $f : X \to Y$, then I am trying to show $$V(\varphi^{-1}(I)) = \overline{f(V(I))}$$ for any ...
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35 views

Maximal spectrum of a ring is quasi-compact in Zariski topology

Let $R$ be a commutative ring with unity. Show that the maximal spectrum of $R$ is quasi-compact under Zariski topology. I tried by taking an open cover $\{ D(I_i) | I \in \Lambda \}$ of $\...
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Closure of $M=\left\{\frac{1}{n}\::\:n\in \mathbb{N}\right\}$ in $\Bbb A^1$

Find the closure of the set $M=\left\{\frac{1}{n}\::\:n\in \mathbb{N}\right\}$ on the line in the Zariski topology. As I understand it, closed sets in the Zariski topology are all one-point sets, but ...
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How can we prove F is irreducible?

This question has been replaced due to errorness in it's original formulation.
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Properties of this set $Z_f = \{(b_1, \dots, b_n) \in \mathbb{A}_k^n : f(x_1, \dots, x_m, b_1, \dots, b_n) \in I(V)\}$

I want to solve the following problem: Let $V \subset \mathbb{A}_k^m$ and $W \subset \mathbb{A}_k^n$ be varieties, with $k$ algebraically closed. Let $f \in k[x_1,\dots,x_m,y_1,\dots,y_n]$. Consider ...
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The topological closure $\overline{X}$ of a subset $X \subset \mathbb{A}_k^n$ is equal to the set $Z(I(\overline{X}))$

My question is on the topological closure of any subset $X$ of the affine space $\mathbb{A}_k^n$. I tried to prove that $\overline{X} = Z(I(X))$. The attempt goes as follows: Notice that we have the ...
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Representability of relative spectrum of a quasi-coherent sheaf of algebras

Let $X$ be a scheme and $\mathscr{R}$ a quasi-coherent $\mathscr{O}_{X}$-algebra. Let $$ F: (Sch/X)^{opp} \rightarrow (Sets), \hspace{1cm} (f:T \rightarrow X) \mapsto \text{Hom}_{(\mathscr{O}_X-Alg)}(\...
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90 views

About the Zariski topology on the spectrum of localization

Let $R$ be a (commutative) ring, $S=\{1,f,f^2,\dots\}$. The exercise I'm working on is asking to show that $Spec(S^{-1}R)$ is an open subset of $Spec R$ and that the Zariski topology on $Spec(S^{-1}R)$...
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Equality of two Zariski closures

Let $X$ be an affine scheme, $R=\Gamma(X,\mathcal O_X)$, $R_0$ a subring of $R$, $Y=\mbox{Spec}(R_0)$, and $\phi:X\rightarrow Y$ the morphism induced by the inclusion of $R_0$ in $R$. Let $\{I_i\}$ be ...
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Counter example of Zariski topology in Ring of polynomials [duplicate]

Let $\mathbb{F}$ be a field and consider the ring of polynomials $\mathbb{F}[x_1,x_2,\cdots x_n]$ in $n$ unknown variables. In $\mathbb{F}^n$ we define the zariski topology given by the open sets of ...
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1answer
85 views

Continuity of Rational Maps in Zariski Topology

Let $X \subseteq \mathbb{C^m}, Y\subseteq \mathbb{C}^n$ be algebraic (not necessarily irreducible), and let $\phi\colon X \to Y$ be a map such that for each $p \in X$, there are there is an open ...
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78 views

Criterium for the representability of a functor

Let $F: (Sch/S)^{op} \to Sets$ be a functor that is both a sheaf in the Zariski topology and has an open covering $(f_i: F_i \to F)_{i \in I}$, where each of the $F_i$ is representable. In theorem 8.9 ...
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Zariski topology

Is there any relation between the Krull intersection theorem and Zariski topology? If it exists? How? I searched for this relationship but I can't find any articles. -Thanks if you can help me.
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Finding radical of an ideal using Nullstellensatz

Find $V(J)=\{(x,y)\in\Bbb{C}^2:x^2-y^7=0, x^4-y^5=0\}$ and $\sqrt{J}$, where $J=\langle x^2-y^7,x^4-y^5\rangle$ My attempt: $(0,0)$ is obviously a solution. Moreover, I can deduce from the first ...
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Density of open sets in finer topology

Let $X$ be any topological space endowed with two topologies $\mathcal{T} \subset \mathcal{T}'$ (the later means that $\mathcal{T}'$ refines $\mathcal{T}$, that is every open subset $U \subset X$ with ...
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105 views

Open, quasi-compact sets in Zariski topology

Question Suppose $A$ is a commutative ring with identity and $U\subseteq Spec(A)$ is open. Show that $U$ is quasi- compact in the Zariski topology if and only if $U = Spec(A)\backslash V(I)$ for some ...
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Zariski topology closure of image of a homomorphism

Just started dealing with Zariski's topology on specta, and encountered the following question: $R,S$ commutative rings with unit and $\phi:R\rightarrow S$ homomorphism. Prove that the induced map $\...
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Points at distance one from an algebraic subset of $\mathbb R^n$

Let $n$ be an integer $\ge2$; let $X_1,\ldots,X_n$ be indeterminates; let $S$ be a subset of the polynomial ring $\mathbb R[X_1,\ldots,X_n]$; let $V(S)$ be the set formed by the points of $\mathbb R^n$...
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closure of polynomial mapping is irreducible

I'm working on an exercise from Kunz' Into to Algebraic Geometry which I've paraphrased: Let $L|K$ be a field extension where $L$ infinite. Given $f_1,\cdots,f_n \in K[t_1,\cdots,t_m]$, let $V$ denote ...
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Why is the Zariski topology coarser than standard topology

I'm trying to learn about the Zariski topology (without prior knowledge of algebraic sets). I'm asked to prove that if $\tau_1$ is the Zariski topology on $\Bbb{C}^2$ and $\tau_2$ is the standard ...
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The natural map $\text{Spec}A_f\to \text{Spec}A$ is a homeomorphism onto $D(f)$ [duplicate]

Let $A$ be a commutative ring with $1$ and $f\in A$ a non-nilpotent element. Then the set $\text{Spec}A_f$ has a natural one-to-one correspondence with the subset $D(f)\subset \text{Spec}A$ consisting ...
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What was the motivation and what is the geometric interpretation of Zariski Topology and Prime Spectrum?

Those are the definitions given to us during our lecture: Zariski Topology We call a Zariski Topology in $\mathbb{K}^n$ a family of complements of $V\left(I\right)=\left\{a\in \mathbb{K}^n\::\:\...
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1answer
69 views

Projection map sends closed sets to closed sets

I am reading Algebraic Geometry, a first course by Joe Harris. In the section on projections, he talks about an application of elimination theory to prove that image of a projection map $\pi: Y \times ...
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1answer
29 views

Composition of polynomials and Zariski-topology

I am learning algebraic geometry and have a rather vague question about the the correspondence between polynomial ideals and closed sets in the Zariski-topology in the context of composition of ...
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1answer
98 views

Generic smoothness in the analytic setting.

If $X, Y$ are non-singular algebraic varieties over an algebraically closed field of characteristic 0, and $f: X \to Y$ is a morphism, then there exists a Zariski-open, non-empty subset $U \subset Y$, ...
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128 views

Irreducibility of $\operatorname{Spec}$ $\iff$ primality of nilradical

The following is an exercise (Problem 1.19) in Atiyah and Macdonald's Introduction to Commutative Algebra text : A topological space $X$ is said to be irreducible if $X \neq \emptyset$ and if every ...
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48 views

Accumulation points with respect to Zariski Topology on $\mathbb{R}$.

Consider the topology on $\mathbb{R}$ : the closed sets are given by the finite sets and $\mathbb{R}$ (basically the Zariski Topology on $\mathbb{R}$). Let $(x_n)_{n \in \mathbb{N} }$ be a sequence ...
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1answer
97 views

Properties of Map: Morphisms, isomorphism and Zariski continuous

I am self studying Commutative algebra out of Kemper's text. I came across an exercise in his text that is meant to better solidify the idea of mappings between affine varieties. As I am very new to ...
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107 views

About Zariski closure

Let $A=\left\{\left(a^{2}+1, a^{3}+a\right) \mid a \in \mathbb{R}\right\} \subset \mathbb{R}^{2}$. Find Zariski closure of $A$ . I think Zariski closure of $A$ is $\mathbb{R}^{2}$ because let $ u=a^{...
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A ring is Jacobson iff $Spec_{max}(R)$ is dense in any closed topological subspace of $Spec(R)$

I am trying to prove that $R$ is a Jacobson ring iff for any $Y \subseteq Spec(R)$ closed in the Zariski topology, one has that the closure of $Spec_{max}(R) \cap Y$ is $Y$ itself. I denote for any ...
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about closed set of $\mathbf{A}_{k}^{1}$

Theorem : $A$ regular function is continuous, when $k$ is identified with $\mathbf{A}_{k}^{1}$ in its Zariski topology. PROOF. It is enough to show that $f^{-1}$ of a closed set is closed. A closed ...
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Zariski closure of an algebraic linear group

Let $G$ and $H$ algebraic linear groups and $\phi : G \to H$ a regular group homomorphism. I wonder if $\overline{\phi(G)}$ (the Zariski closure of $\phi(G)$) is again a subgroup and how this could be ...
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I am looking for an example of an open set in $\operatorname{Spec}(R)$ equipped with the Zariski topology which is not compact.

I have learned the basic open sets i.e. the sets of the form $\{P\in\operatorname{Spec}(R): x\notin P\}$ where $x$ is a fixed element of $R$ are compact. Now any open set is union of such basic open ...
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63 views

Zariski and Euclidean topologies

I've shown that every open set in the Zariski topology is open in the Euclidean topology, but I wonder why they are not equivalent. I'm searching for an open set in the Euclidean topology that is not ...
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1answer
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Open Zariski subsets of $\mathbb{P}^1$

Are open subsets of $\mathbb{P^1}$ connected? I believe yes because they are complement of finite sets. Is this right?
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Morphisms of locally ringed spaces and functors of points

I'm learning algebraic geometry and I'm trying to reconcile the locally ringed space and functor of points perspectives. Often when one defines a morphism $X \to Y$ of schemes thought of as locally ...
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1answer
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Are the number of maximal opens sets in a Noetherian topological space finite?

I know that a Noetherian topological space (such as Zariski topological space on Affine space) has ascending chain condition on open sets (descending chain condition on closed sets). This means that ...
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126 views

Zariski closure of infinite subset of the circle is the circle

Let $S^1 \subseteq \mathbb{R}^2$ be the unit circle circle $\mathbb{V}(X^2+Y^2-1)$. Let $S$ be an infinite subset of $S^1$. I want to show that $$\overline{S}^{\text{Zariski}}= \mathbb{V}(\mathbb{I}(S)...
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Geometric quotient of $\mathbb{A}^2$ modulo $G$

I have the following exercise: (This exercise deals with the action of a group $G$ on an affine closed.) Let $G=\langle w \rangle$ be a cyclic group of order $n$ (in multiplicative notation). ...
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Zariski topology & analytic topology of complex varieties

Let $X \subset \mathbb{P}_{\mathbb{C}}^n$ a complex smooth projective variety, $x \in X$ a closed point and $S \subset X$ a constructible subset containing $x$. Recall, on $X$ we can consider two ...
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Topological and algebraic groups

I learnt that an algebraic group $G$ is not a topological group because Zariski topology on $G$ x $G$ is not the product topology. Are there any other topologies, other than Zariski, that do not ...
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1answer
241 views

How to reduce to affine case to determine whether a given functor is representable

[Definition] These are the contents of Gortz and Wedhorn , Algebraic Geometry. $\widehat{(Sch)}$ is the category of functors $(Sch)^{opp} \rightarrow (Sets)$ For scheme $X$, define the functor $h_X :...
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Zariski density of a matrix semigroup generated by Jordan blocks

In the field of random matrix products, it seems a lot of theorems which give nice statistical properties (central limit theorem, large deviation, etc.) assume—among other things—a Zariski density ...
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Showing that the map $t\mapsto(t,t^2,t^3)$ is continuous in Zariski Topology. Twisted Cubic.

Consider the affine space $\mathbb{A}^n$ where $\mathbb{A}^n$ is the set of $n$-tuples $(a_1,\dots,a_n)$ with $a_i\in k$, and $k$ is an algebraically closed field. Next, consider the map $f:\mathbb{A}^...
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Clarification question about the definition of irreducible topological space.

We say that a topological space $X$ is reducible if $X$ can be written as a union of two proper non-empty closed subsets $X_1$ and $X_2$ i.e. $$X=X_1\cup X_2.$$ Is there an equivalent definition of an ...
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1answer
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$\mathbb{R}^n$ with the Zariski topology is not $T_2$

Regarding the Proof of the problem mentioned in the title, I followed the Proof included in this post: When is the Zariski topology $T_2$? We know that $\mathbb{R}$ with the Zariski topology is not T2....
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1answer
71 views

Zariski topology is not first countable on $\mathbb{R}$

Prove that the Zariski topology is not first countable on $\mathbb{R}$. All I'm able to show right now is that all the one-point sets $\left(\{a\}\subset \mathbb{R}^n\right)$ are closed as every ...
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1answer
58 views

An example of $I(X_1\cap X_2)\neq I(X_1)+I(X_2)$ where $X_i$ is an algebraic set.

Let $X_i$ be an algebraic set of an affine space $\mathbb{A}^2$ with Zariski topology with $k=\mathbb{C}$. I want to show that $I(X_1\cap X_2)\neq I(X_1)+I(X_2)$ for certain choice of $X_1$ and $X_2$ ...
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1answer
31 views

$Z(P_1P_2)=Z(P_1)\cup Z(P_2)$ where $Z(P)$ is a vanishing set of $P\subset k[x_1,\dots,x_n]$.

Let $k$ be any field. If $P_1,P_2\subset k[x_1,\dots, x_n]$ are subsets of polynomials, then I want to show that $$Z(P_1P_2)=Z(P_1)\cup Z(P_2)$$ where $Z(P)$ is a vanishing set of $P\subset k[x_1,\...
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2answers
148 views

affin line $\mathbb A_{R}^{1}$ is not homeomorphic to real line $\mathbb{R}$

I want to show affin line $\mathbb A_{R}^{1}$ is not homeomorphic to real line $\mathbb{R}$(the topology is euclidean topology). I think the topology on $\mathbb A_{R}^{1}$ is just cofinite topology, ...

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