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Questions tagged [zariski-topology]

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

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Using equivalence of quantifiers to show $\bigcap_{f\in {\bigcup_{\alpha\in I}S_{\alpha}}}V(f)=\bigcap_{\alpha\in I}\bigcap_{f\in S_{\alpha}}V(f)$

Background Quantifier distribution and negation laws $\forall x(P(x)\wedge Q(x))=\forall xP(x)\wedge \forall xQ(x)$ $\exists y(P(y)\vee Q(y))=\exists yP(y) \vee \exists yQ(y)$ $\sim (\forall x P(x))=\...
Seth's user avatar
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proof: projecting a projective variety onto the first component remains a projective variety

I have recently begun studying algebraic geometry and have decided to start with Harris' classic, supplementing it with these online notes from a lecture, based on his book, given by Harris himself at ...
mathrandom's user avatar
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1 answer
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dimension 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 ...
Alessandro Vagni's user avatar
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Isomorphism locus of a morphism of objects in the derived category is Zariski open?

Let $R$ be a commutative Noetherian ring and $\text{Mod} R$ be the category of $R$-modules. Let $M,N\in \mathcal D(\text{Mod } R)$ with finitely generated homologies. Let $f: M\to N$ be a morphism in $...
Alex's user avatar
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Problem 6.8 in Fulton's Algebraic Curves: the nonvanishing locus of a regular function is open

Question: Let $U$ be an open subset of a variety $V$, $z\in k(V)$. Suppose $z\in O_p(V)$ for all $P\in U$. Show that $U_z = \lbrace P\in U \mid z(P) \ne 0 \rbrace $ is open. Attempt: Initially I want ...
leo's user avatar
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Proving certain subset of the product of the affine line and the Grassmannian is closed in the Zariski topology

Let $k$ be an algebraically closed field of characteristic $0$, $0 < d < n$ be integers,$\mathrm{Gr}_k(d,n)$ be the Grassmannian parametrizing all linear subvarieties of dimension $d$ contained ...
buky miao's user avatar
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1 answer
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Closure of quasi-projective variety: if $X=V(I)\setminus V(J)$, must $\overline{X}=V(I)$?

Let $X=V(I)\setminus V(J)$ in a complex projective space $\Bbb P^n$, where $I,J$ are ideals of complex polynomials in $n+1$ variables and $V(\ldots)$ their common zeros. I mean $X$ is locally closed ...
Alessandro Vagni's user avatar
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1 answer
99 views

Zariski topology on $\mathbb{Z}$

I'm trying to understand the Zariski topology on $\text{Spec}(\mathbb{Z})$. I've just learned about this new concept and I wanted to compute this topology for a more concrete example to see how it ...
clem2005's user avatar
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Zariski-density on almost diagonal embedding

It is not hard to see that the Gaussian integers $\mathbb{Z}[i]$ are Zariski-dense inside $\mathbb{C}$, seen as an affine space over $\mathbb{C}$. Consider now the set $$D = \{(z,\overline{z}) \in \...
Henrique Augusto Souza's user avatar
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Is there a name for the (Zariski like) topology whose closed sets are generated by degree $1$ polynomials?

Given a field $K$, if we consider affine $n$ space over $K$, denoted $\mathbb{A}^n_K$. Now lets consider $K[z_1,z_2,...,z_n]$, given some ideal $I\subset K[z_1,z_2,...,z_n]$, we associate the ...
Steven Creech's user avatar
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Is $V \to \text{Spec}(k[V])$ a topological embedding for non-algebraically closed field?

Let $k$ be an arbitrary field and $V\subseteq \mathbb{A}^n(k)$ be a Zariski-closed subset and $\Psi: V \to \text{Spec}(k[V]), x \mapsto m_x$, where $m_x \subseteq k[V]$ is the maximal ideal in the ...
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Are polynomials open maps w.r.t. the Zariski topology? [duplicate]

I am super new to algebraic geometry, so I have the following simple question. Are polynomial maps between affine varieties open with regards to the Zariski topology? Many thanks in advance.
Ogawa's user avatar
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Hausdorff-ness of irreducible affine algebraic sets [closed]

In what case is $X$, an irreducible affine algebraic set not a Hausdorff topological space with respect to the Zariski topology? My understanding is that this would only occur if $X$ is infinite, ...
Jeff's user avatar
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Prove $S = \{ (a,b) \in \mathbb{A}^2 \ | \ a \overline{a} + b \overline{b} = 1\}$ is not Zariski closed

I'm working on a problem where I have to prove that Consider the subset $S$ of $\mathbb{A}^2$, the affine space over $\mathbb{C}$, defined by $$ S = \{ (a,b) \in \mathbb{A}^2 \mid a \overline{a} + ...
Oopsilon's user avatar
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Question about exc. $15.2.21.$b in Dummit and Foote.

This is an excerpt (paraphrased) from Dummit and Foote (I believe 3rd edition): Let $V \subset \mathbb{A}^n$ be an algebraic set and let $f \in k[V]$. If $J$ is the ideal generated by $I(V)$ and $x_{...
Ben123's user avatar
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On the finiteness of certain Zariski closed subsets of the prime spectrum of commutative Noetherian local rings

Let $(R,\mathfrak m)$ be a Noetherian local ring. Let $S,T$ be Zariski closed subsets of $\text{Spec}(R)$ such that if $\mathfrak p\in S, \mathfrak q \in \text{Spec}(R)$ and $\mathfrak p \subsetneq \...
uno's user avatar
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Image of a projective variety is a projective variety

I am using Karen E. Smith et al.'s Invitation to Algebraic Geometry, and was wondering the following: if $$\phi : V \subseteq \mathbb{P}^n \to W \subseteq \mathbb{P}^m $$ is a morphism of ...
Neckverse Herdman's user avatar
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Is open always Zariski open in the context of Algebraic geometry, unless otherwise mentioned?

Going through Smith et al.'s Invitation to Algebraic Geometry I sometimes find myself wondering whenever they use the word open, do they mean open in the usual sense on $\mathbb{A}^1$ for example, or ...
Neckverse Herdman's user avatar
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1 answer
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All the open affine subsets of $\mathbb{A}^1 \subseteq \mathbb{P}^1$. Are open affine subsets always Zariski open?

Let the projective line $\mathbb{P}^1$ be the set of all points $[x:y]$ and suppose $\mathbb{A}^1$ is the subset of all those points with $y \neq 0$. So we may write $\mathbb{A}^1 = \{ [x:1] \mid x \...
Neckverse Herdman's user avatar
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Zariski topology on $\mathbb{P^n}$ [duplicate]

I want to prove or disprove that the open sets of the Zariski topology on $\mathbb{P^n}$ are path-connected. Here $\mathbb{P^n}$ is the complex projective n-space. Thanks for any hint or answer.
100nanoFarad's user avatar
3 votes
1 answer
330 views

The complement of an algebraic set in $\mathbb{P}^n$ is path-connected

I'm studying the marvellous book "Algebraic Curves and Riemann Surfaces" of Rick Miranda and I've found this problem (III.5.Q pag.103). Actually it's not related to Riemann surfaces, but I ...
100nanoFarad's user avatar
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1 answer
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how to prove that a set is not algebraic

Consider the Zariski topology on $\mathbb{C}^2$. How can I prove that a set is not an algebraic variety? For example let $Y=\{(z_1,z_2)\in \mathbb{C}^2| \text{Im}(z_1) \cdot \text{Im}(z_2)>0\}$ be ...
Mario's user avatar
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Proof that the image of $(x,y) \mapsto (x, xy)$ is neither open, closed nor locally closed.

Let $k$ be an algebraically closed field and $f: \mathbb{A}^2_k \rightarrow \mathbb{A}^2_k$, $(x,y) \mapsto (x,xy)$. I understand what the image looks like, i.e. $\mathbb{A^2_k} - (\mathbb{A}^1-0)$. ...
Absent mind's user avatar
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Zariski Topology on $\mathbb{A}^1$

Let $\mathbb{A}^1$ be the one dimensional $\mathbb{C}$ affine space, and equip it with the Zariski topology. I know the topology is non-Hausdorff, but I am trying to understand if my reasoning is ...
Chris's user avatar
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Need help in competing the proof of $V_K(I_K (E)) = \overline{E}$

I am taking a course on Algebraic geometry this semester and this proof was left as an exercise. I am struck in one argument of the proof, so I request you to help me. Statement: Let E be a subset of ...
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$Spec(R)$ finite and discrete implies $R_{red}$ Artinian proof verification

My proof feels weird to me because I on the way prove that an Artinian ring with trivial nilradical is a product of division rings, and I can't find this result anywhere, so even though I am quite ...
DevVorb's user avatar
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Subbasis of Zariski topology on $\mathbb R^n$.

I know that in $\mathbb R^n$, the collection of open sets $\{(Z(\{f\}))^C: f\in k[x_1,x_2,\dots, x_n]\}$ is a basis for the Zariski topology in $\mathbb R^n$. When $n=1$, I could find a subbasis for ...
Runyang Wang's user avatar
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1 answer
16 views

How to calculate the Zariski closure of $Z(A)B$?

Suppose $A$ is a $n\times n$ matrix, and $Z(A)$ is the centralizer of $A$ in $M_{n\times n}$. If $B\in M_{n\times m}$, then how to calculate the Zariski closure of $Z(A)B$ in $M_{n\times m}$?
Richard's user avatar
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2 votes
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Is a closed, nonempty subset of a (linear) algebraic group $G$ that is closed under taking products a subgroup of $G$? [duplicate]

This is a question based on Exercise 7.6.5 of Humphreys', "Linear Algebraic Groups". For a solution to that particular problem, see: A closed subset of an algebraic group which contains $e$ ...
Shaun's user avatar
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3 votes
1 answer
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Let $\mathfrak{A}$ be a finite dimensional $k$-algebra for alg. closed $k$. Prove ${\rm Aut}(\mathfrak{A})$ is a closed subgroup of $GL(\mathfrak{A})$

This is Exercise 7.6.3 of Humphreys', "Linear Algebraic Groups". The Question: Let $\mathfrak{A}$ be a finite dimensional $k$-algebra for algebraically closed field $k$. Prove that ${\rm ...
Shaun's user avatar
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Computing the Annihilator and Zariski closure of $Z = \lbrace (t, t^2, t^3) \mid t \in \mathbb{C} \rbrace \subseteq \mathbb{A}^3(\mathbb{C})$

I want to compute the Zariski closure of $Z = \lbrace (t, t^2, t^3) \mid t \in \mathbb{C} \rbrace \subseteq \mathbb{A}^3(\mathbb{C})$. I know that for this, it suffices to compute $V(\operatorname{Ann}...
tolUene's user avatar
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1 vote
1 answer
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The torsion subgroup of a diagonalisable linear algebraic group $G$ with ${\rm char}(k)=p$ (alg. closed $k$) is dense in $G$

This is Exercise 3.2.10(5b) of Springer's, "Linear Algebraic Groups (Second Edition)". The Question: Let $p$ be the characteristic exponent of an algebraically closed field $k$. Let $G$ be ...
Shaun's user avatar
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0 votes
1 answer
120 views

Compute the Zariski closure of a set

I have to compute the Zariski closure of the image of the following rational map: $f:{P}^2 \rightarrow \mathbb{P}^4$ $[x_0:x_1:x_2]\rightarrow [x_0x_1:x_0x_2:x_1^2:x_1x_2:x_2^2]$ I have already proved ...
Aron's user avatar
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2 answers
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Why is the graph of polynomial closed in Zariski-Topology?

The official wikipedia entry to Zariski-Topology states that: "In the Zariski topology on the affine plane, this graph of a polynomial is closed." I guess the argumentation would be the ...
Zedssad's user avatar
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3 votes
1 answer
248 views

Salvaging Exercise 3.2.10(2) of Springer's, "Linear Algebraic Groups (Second Edition)".

This uses the soft-question tag because there might be more than one valid answer, and it's a matter of guesswork to some extent; but there is a right answer (in theory). Thoughts and Motivation: As a ...
Shaun's user avatar
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1 answer
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How exactly does Zariski-Topology relate to Standard topology on $\Bbb{C}$?

I'm currently self-studying algebraic geometry and I tried to construct some proofs / counterexamples for relations between open / closed sets in Zariski and Standard Topology on $\mathbb{C}$ but I'm ...
Zedssad's user avatar
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1 vote
1 answer
74 views

Let $G$ be a linear algebraic group and $C_G(x)$ be the centralizer of $x$. Show that $C_G(x)$ is a closed subgroup for all $x\in G$.

Let $G$ be a linear algebraic group and $C_G(x)$ be the centralizer of $x$. Show that $C_G(x)$ is a closed subgroup for all $x\in G$. I want to enstablish a homomorphism from $G$ to $G$, and the ...
Shi Chen's user avatar
1 vote
1 answer
174 views

Can $\operatorname{Spec}(R)$ be not homeomorphic to $\operatorname{Spec}(S)$ but be isomorphic as a poset?

Looking at the definition of the spectrum of a ring made me wonder when/whether is its topology determined by the poset of prime ideals of the ring. Let $R$ and $S$ be commutative unital rings. Can ...
Carla_'s user avatar
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1 answer
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If the product of closed sets is closed in the product topology shouldn't {(0,0)} be closed in $A^1 \times A^1$ zariski topology.

I am not sure what is incorrect about the statement. If the product of closed sets is closed in the product topology shouldn't {(0,0)} be closed in $A^1 \times A^1$ zariski topology, i.e singletons ...
ben huni's user avatar
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1 vote
1 answer
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Is "Zariski connected" equivalent to "ring connected"?

Let A be a ring, in class we defined the following two definitions of connected: A is Zariski connected if Spec A is connected with the Zariski topology. A is ring connected if $$a+b=1, ab=0\...
DevVorb's user avatar
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3 votes
1 answer
93 views

Commutative algebra: Irreducible components of $\operatorname{Spec}(A)$

Let $k$ be a field of characteristic other than $2$. Consider the ring $$A:= k[X,Y]/(X(Y+1),X(Y+X^2)).$$ Describe all the irreducible components of $\operatorname{Spec}(A)$. It does not look that ...
Nik1987's user avatar
  • 33
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1 answer
159 views

Curves homeomorphic under Zariski topology

Prove any two curves over some field $k$ are homeomorphic, where $k$ might not be algebraically closed. Curves are defined to be varieties (integral separated scheme of finite type) of dimension $1$. ...
user avatar
0 votes
1 answer
106 views

When the Zariski topology is $T_1$ (depending on $k \ge 1)$.

Please explain when the Zariski topology is $T_1$ (depending on $k \ge 1$). Let $X$ be a topological space and let $x$ and $y$ be points in $X$. We say that $x$ and $y$ are separated if each lies in a ...
ppppp's user avatar
  • 21
1 vote
1 answer
120 views

The Lie algebra of zariski closure of an algebraic group.

I think this question should be pretty straightforward but through a combination of being self-taught in group theory and being awful at geometry, something is escaping me. I am trying to understand ...
Yushi MuGiwara's user avatar
0 votes
1 answer
237 views

The Zariski topology on $\mathbb{K}^n $ is Noetherian.

A topological space is Noetherian if there are no infinite descending chains of closed sets. Show that the Zariski topology on $\mathbb{K}^n $ is Noetherian. I know and understand what is topological ...
Artur111's user avatar
1 vote
2 answers
115 views

Let $f:A→B$ be a ring morphism and $φ : \operatorname{Spec}B→\operatorname{Spec}A$ its induced continuous map, then prove that $\rm im\ φ ⊆ V(\ker f)$

Today I was asked this question by my teacher: Let $A$,$\ B$ be two conmutative rings with unity lef $f :A \rightarrow B$ be a ring morphism and $φ : \operatorname{Spec}(B) \rightarrow \operatorname{...
Pablo Borrego's user avatar
1 vote
0 answers
62 views

Dimension of Zariski closure of an infinite discrete group

Let $ \Gamma $ be a subgroup of $ G:=GL(n,\mathbb{C}) $ which is discrete with respect to the manifold topology on $ G $. Let $ \overline{\Gamma} $ be the Zariski closure of $ \Gamma $. Suppose that $ ...
Ian Gershon Teixeira's user avatar
2 votes
1 answer
76 views

${\rm Spec}(A/\mathfrak{a})={\rm Spec}(A)$ if and only if $\mathfrak{a}$ is generated by nilpotent elements

I have been introduced to the Zariski topology and I cant solve this problem: Let $A$ be a commutative ring with unity and $\mathfrak{a}$ an ideal of $A$, we define ${\rm Spec}(A) $ as the set of ...
Pablo Borrego's user avatar
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1 answer
150 views

Showing the linear algebraic subgroup $\Bbb U_n$ of $\Bbb{GL}_n(k)$ is closed.

This is part of Exercise 2.1.5(2) of Springer's, "Linear Algebraic Groups (Second Edition)". It is likely to be a simple question. Fix an algebraically closed field $k$. Let $n\in \Bbb N$. ...
Shaun's user avatar
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6 votes
3 answers
344 views

Zariski topology and morphisms that coincide on a dense set

In Miles Reid Undergraduate Algebraic Geometry book it is stated informally, about the Zariski topology: (1) two morphisms which coincide on a dense open set coincide everywhere I am suprised that ...
Weier's user avatar
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