Questions tagged [zariski-topology]

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

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Name of topology analogous to Zariski topology for an arbitrary model of a first-order theory

In the Zariski topology, a set of points in $A^n$, $w$, is closed by definition when there exists a set of polynomials in $n$ variables, $S$, such that the following holds. $$ \forall u \in A^n \...
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empty set in zariski topology

I stumbled across the following assertion: Let $A$ be a commutative ring and Spec($A$) given with the Zariski topology. In this topology all closed sets are of the form: $V(\mathfrak{a}):=$ {$\...
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well-definedness of closed set in Zariski topology

Let $A$ be a commutative ring and $\mathfrak{p} \subset$ Spec($A$). Then for an ideal $\mathfrak{a} \subset A$, we have a closed set in the Zariski topology defined by $V(\mathfrak{a})$:={$\...
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$f$ maps Zariski closed set to another Zariski closed set

$f:X\rightarrow Y$ is a polynomial map between two algebraic varieties. $X$ and $Y$. $f_{*}:k[Y]\rightarrow k[X]$ is the corresponding algebraic homomorphism between the coordinate ring. Assume $f_{*}$...
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Prove that for $k$ a finite field every subset is closed (and open) in Zariski topology

The affine algebraic sets define a topology on the affine $n$-space, which is called the Zariski topology. I think that the algebraic sets are the closed sets in the Zariski topology, so I am not very ...
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On showing a set is Zariski open subset of $\mathfrak m/\mathfrak m^2$

Let $(R, \mathfrak m,k)$ be a Noetherian local ring such that the residue field $k$ is infinite. Let $n=\mu(\mathfrak m)$. Then $n=\dim_k(\mathfrak m/\mathfrak m^2)$ . By fixing $x_1,...,x_n \in \...
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Atiyah MacDonald's exercise 1.21 on Zariski topology

The exercise is the following: Let $\phi:A\longrightarrow B$ be a ring homomorphism. Let $X = Spec(A)$ and $Y = Spec(B)$. If $\mathfrak{q} \in Y$, then $\phi^{-1}(\mathfrak{q})$ is a prime ideal of ...
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Showing that there is a closed point in the preimage

I have asked a question here Applying Chevalley's Theorem to Elimination of quantifiers And now I'm having some trouble showing that $(im \pi) \cap \bar{k}^m =Y = im{\pi}^{cl}$ What I have ...
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65 views

Zariski topology and product topology

I'm pretty new to algebraic geometry; I have not clear what does it mean that the Zariski topology and the product topology over the affine space $\mathbb A^m × \mathbb A^n $ are different. I'm ...
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On The Fundamental Theorem of Elimination theory in Vakil's FOAG

On page 221 of Vakil's FOAG, he states The Fundamental Theorem of Elimination theroy as follows: The morphism $\pi : \mathbb{P}_A^n \to \mathrm{Spec}A $ is closed.(sends closed sets to closed ...
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61 views

Non integral domain locus of a commutative Noetherian ring

Let $R$ be a commutative Noetherian ring. Then is it true that the set $$\{P\in \mathrm{Spec}(R): R_P \text{ is not an integral domain} \}$$ is a closed subset of $\mathrm{Spec}(R)$ under Zariski ...
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51 views

Why is the set of all full rank $\mathbb{R}^{m \times n}$ (rectangular) matrices a Zariski open set?

Note: this question is almost certainly a duplicate. I have spent more than an hour on Google and searching for the original question, but can't find it. Please feel free to close this question when ...
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Is my proof that every set in Zariski topology of $\mathbb{A}^n$ is compact correct?

Proof that every set in $\mathbb{A}^n$ is compact in the Zariski topology. Let $\{ U_\alpha \}$ be a collection of open sets. Let's first pick a numerable collection of them $U_1, U_2, \dots$, so we ...
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45 views

Why is there a basic open, everywhere dense neighborhood of every closed point on a reduced affine scheme?

Let $R$ be a reduced, noetherian ring with minimal prime ideals $P_1,\ldots,P_m$. Let $M$ be a maximal ideal of $R$. Does there exist a regular element of $R$, i.e. $a \in R \setminus \bigcup_{i=1}^m ...
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37 views

Can the Zariski topology range over infinite sets of polynomials?

I need someone to sanity check my understanding: I believe when we define the closed sets of the Zariski topology as: $$ C \in \mathbb R^n~\text{is closed} \iff \exists S, f_{s \in S}: \mathbb R^n \...
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64 views

Striking examples to show the non-hausdorffness of the Zariski toplogy?

We are told that the Zariski topology is not Hausdorff, but I have rarely seen "concrete examples" of the dramatic failures this can induce. A concrete example I have in mind: For example, one way ...
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Using the comparison between Euclidean and Zariski topology to prove that subsets are not affine varieties

As in the title I was wondering if this very simple approach is correct or not, in general. I am adopting the definition of affine variety as the zero locus of systems of polynomials. For instance ...
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How to understand the statement of Exercise 6.19 on Fulton's Algebraic Curves

I think that I am not fully understanding what the exercise asks me to show. Here it is: (Fulton, Algebraic Curves, 2008) By definition, a variety $X$ is an open subset of $V$, where $V$ is an ...
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48 views

Fulton's exercise 6.11: varieties are quasi-compact

I am dealing with the following exercise from Fulton's Algebraic Curves, 2008. I have read several posts such as: Closed affine sets are quasi-compact Show that affine varieties are quasi-compact. ...
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On a special map $ \operatorname {Spec}(\mathbb C[x,z]) \to \operatorname {Spec}(\mathbb C[x,y])$

This is a continuation of this question On the flatness of a particular ring map on two variable polynomial ring . Given the $\mathbb C$-algebra map $f: \mathbb C[x,y]\to \mathbb C[x,z]$ defined by $...
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Showing a map from projective space can't be extended

Let $\ U = \big\{ \ [x:y]\in\mathbb{P}^1 \big|\ x\neq0\big\}$ and $f:U\rightarrow \mathbb{A}^1$ be the map $$f([x:y])=\frac{y}{x}$$ Show this is a morphism $U\rightarrow \mathbb{A}^1$ and that it ...
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Application of the Nullstellensatz

The question is as follows Let $k\subset K$ be algebraically closed fields. And $I \leq k[x_1,...x_n]$ an ideal. Show that if $f \in K[x_1 ,...x_n]$ vanishes on $Z(I)$ it vanishes on $Z_K(I)$. ...
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49 views

Algebraic Varieties and Zariski Topology

Given a set $S \subset F[x_1, \dotsc ,x_n]$ of polynomials, an affine variety is defined by $S$ is the set $$V(S) := \{a \in A^n \mid f(a) = 0\ \ \ \forall f \in S\}$$ And the zariski topology ...
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Projective varieties are compactifications of affine varieties

This question has been asked here and here, though no answers have been given. It is exercise 3.2.1 in Smith's Invitation. Showing $\mathbb{P}^n$ is compact boils down to showing it is the ...
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Zerodimensional, compact space is homeomorphic to spectrum of some ring — elementary attempt

Let $X$ be a zerodimensional (X has a base with clopen sets) and compact (quasicompact and Haussdorf) topological space and I would like to prove that $X\cong Spec(A)$ of some ring $A.$ I don't want ...
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Is the Zariski topology on the spectrum of a ring defined for a fixed ideal, or all ideals?

Initially this question was meant to ask how to prove that closed points of $\text{Spec}(R)$ correspond to maximal ideals of $R$ (This question had been asked previously in various forms, e.g. here ...
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56 views

Spectrum of a ring in which $a^p=a$ for all $a$ and prime number $p$

It is known the result that spectrum of a boolean ring is zerodimensional and compact topological space, e.g. has a base with clopen sets and satisfies Haussdorf condition. It is asked if this result ...
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183 views

Matrices with eigenvalue of multiplicity $k$ are algebraic subset of special linear group.

In the previous exercise, I have showed that the special linear group $SL_n$ is a closed subvariety of $Mat(n,K)$ where $K$ is an algebraically closed field with characteristic zero. Now, I have to ...
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28 views

How to check if a subset is open in Zariski

I'm having troubles determining if a given subset of $\operatorname{Spec}A$ is open or not. The contest is not trivial. I have to consider a morphism of finitely generated $k$-algebras $A\rightarrow B$...
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Example where supp(M) is not Zariski-closed [duplicate]

I found on a book that there exists R-modules M such that supp(M) is not Zariski-closed. I already know that if M is finitely generated then it is Zariski closed but I can't find an example of a non ...
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Arbitrary Unions of Closed Sets in Zariski Topology

I'm aware that, for an arbitrary topology $\tau\subset \mathscr{P}[U]$ on a set $U$, the collection $\tau^c$ of closed sets is only guaranteed to be closed under finite union. In the Zariski ...
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78 views

Affine Algebraic Sets: Open and Closed Sets?

Ladies and gentlemen! I was hoping you'd might be able to clear something that should be minor out for me since I've been getting a bit confused over seemingly contradictory statements as to whether ...
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44 views

Functions Agreeing on Zariski Dense Sets

Let $X$ be an affine variety and $U\subset X$ Zariski open, Then, $U$ is dense in $X$ (i.e., the smallest set containing $U$ which is also the zero locus of a set of polynomials is $X$). Let $\...
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$X=Z(xy-1)\subset \mathbb{A}_k^2$ is not isomorphic to $\mathbb{A}^1_k$ [duplicate]

I need to prove that $X=Z(xy-1)\subset \mathbb{A}_k^2$ is not isomorphic to $\mathbb{A}^1_k$. I solved an exercise where I proved that, for instance, some $X\subset \mathbb{A}_k^3$ is isomorphic to $\...
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Zariski topology - closed sets of $\mathbb{A}_\mathbb{C}^2$

I have to prove that: The closed sets in Zariski topology of $\mathbb{A}_\mathbb{C}^2$ are the sets of form $Z(g)\cup F$, where $g\in \mathbb{C}[x,y]$ and $F$ is a finite set. So, I thinked the ...
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Can the zero set of an irreducible polynomial contain a non-empty Zariski open subset?

Let $k$ be an algebraically closed field and $f \in k[x_1,...,x_n]$ be an irreducible polynomial. Is it possible that $Z(f)$, the zero set of $f$, contains a non-empty Zariski open subset of $\...
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Radical Ideals and Set Difference of Ideals

I'm looking at the 1.4.3 exercise of Gathmann's notes on Algebraic Geometry. Here is a link: https://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2002/alggeom-2002.pdf The question is on page 19. ...
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1answer
49 views

Proof that the spectrum of a ring is Kolmogorov

Let $A$ be a commutative ring and consider its spectrum $\operatorname{Spec}A$ equipped with the Zariski topology. Wikipedia claims that $\operatorname{Spec}A$ satisfies the separation axiom $\mathbf{...
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Open Set in $\mathbb{Q}_p$-rational points of a torus is Zariski-dense in the torus.

Suppose $K$ is a finite field extension of the p-adic numbers $\mathbb{Q}_p$. Let $T$ be the algebraic torus over $\mathbb{Q}_p$ obtained by the Weil Restriction of scalars from $K$ to $\mathbb{Q}_p$ ...
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Is $\mathbb{R}$ the smallest uncountable Zariski closed set in $\mathbb{C}^n$ and countably infinite zeroes of a complex function? [closed]

I have few questions which are related to a problem I am solving. Let $v = (v_1,\dots, v_n)$ be a fixed vector in $\mathbb{C}^n$ and $n>1$. [1] Is $\mathbb{R}$ an algebraic set in $\mathbb{C}^n$ ...
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Zariski closure of a set in $k^3$

Let $k$ be any field, and we give $k^3$ the Zariski topology. Then the question is how to compute the Zariski closure of the set $S=\{(x,y,z)\in k^3|xz=y, x+1=z^2, x\neq0\}$ in $k^3$. The ...
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Zariski-closure of finite linear groups

I have come across the following Proposition from Milne: Let $G$ be an affine algebraic group over a field $k$ and $S$ a closed subgroup of $G(k)$. There is a unique algebraic subgroup $H$ of $G$ ...
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47 views

If $R_P$ is reduced, is there a Zariski basic open set $D(f)$ containing $P$ such that $R_f$ is also reduced?

Let $R$ be a commutative ring with unity. If $P\in \operatorname{Spec} R\:$ is such that $R_P$ is reduced, then is it necessarily true that $\exists f\in R\setminus P$, i.e., $f\in R$ such that $P\in ...
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33 views

Seeing if an arbitrary subring or set is dense in the Zariski topology

In another question, it was suggested that I check if a certain subring of a ring was dense in the Zariski topology. The subring, however, was not an ideal, so I wasn't sure how to represent it in $...
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Going from varieties to schemes

I've been studying the Nullstellensatz and the Zariski topology. I understand the basic gist is that in $\Bbb C[x_1, x_2, ..., x_n]$, varieties are in one-to-one correspondence with radical ideals. ...
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47 views

Is every point in the spectrum of a ring $R$ closed?

I am just getting started on spectrums of rings. I see how it is natural to augment the set of prime ideals with the Zariski topology, but from my poor intuition on the topic I don't see how any of ...
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31 views

Question about topology on the product of affine spaces (question related to variety being complete)

Let $k$ be an algebraically closed field and denote $\mathbb{A}^n$ as the $n$-tuple of elements in $k$ where we put the Zariski topology. Say we have $\mathbb{A}^n \times \mathbb{A}^m$ where we put ...
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190 views

Does the zero set of a real-analytic function in several variables form a subvariety?

This is probably a very naive question: I am trying to understand if the zero-set of a real-analytic function in several variables can be "wilder" than the zero-set of a polynomial in several ...
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41 views

Zariski topology: closed set of $\mathbb{A}_F^1$ that contains $\mathbb{Z}$ equals $\mathbb{A}_F^1$

I have a question about a closed set in Zariski topology. I might be overlooking something easy, but I’m stuck. Let $A$ be a closed set in $\mathbb{A}_F^1$ where $F$ is a field of characteristic zero ...
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47 views

Prove that $\overline {B^*} = B.$

Let $R$ be a commutative ring with identity. Let $\operatorname {Max} (R)$ denote the set of all maximal ideals of $R$ and let $\operatorname {J} (R)$ denote the set of all prime ideals of $R$ which ...

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