# Questions tagged [zariski-topology]

The zariski-topology tag has no usage guidance.

20
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### Unifying categorical equivalences and dualities for the functors : Gelfand spectrum and Zariski spectrum, Structural sheaf and Continuous sections

I would be very grateful for any references I might be led to, from a categorical point of view for the functors:
$\textsf{Spec}_{\mathscr{Z}\textrm{arisky}}(-)$, related to $\mathcal{O}(-)$, which ...

0
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0
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### Is the set of common-zeros of the systems of polynomials Zariski dense?

Let $K$ be an algebraically closed field of characteristic $0$.
Let $f: K^m \to K^m$ be a map defined by $f(x_1,x_2,\cdots,x_m)=(f_1,f_2, \cdots,f_m)$ where $f_i \in K[x_1,x_2, \cdots, x_m]$. Let $Z$ ...

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2
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### Intrinsic topology on the Zariski spectrum

In a big topos whose objects are a kind of "space", it sometimes happens that when we define some "set" internally to the topos, the "topology" it automatically acquires ...

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### Defining path on the prime spectrum

If $p$ and $q $ are two prime ideals of a commutative ring $R $ such that $p \subseteq q$, then we can easily define a continuous function (a path) $f$ from the unit interval $ [0,1]$ to the prime ...

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### When is a constructible set locally closed?

Let $X$ be a topological space (or more specifically, $\mathbb{C}^N$ endowed with the Zariski topology), and let $S \subseteq X$ be a constructible set, i.e. $S=\cup_{i=1}^n C_i \cap U_i$, where the $...

1
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2
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109
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### Rank of a linear combination of linear operators

I asked this question a few days ago in MathExchange and received no satisfatory answer. I hope it is well suited for MathOverflow.
Suppose I have two linear operators $X,\,Y$ on $\mathbb{C}^n$. Now ...

2
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0
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172
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### When is there a path between two minimal prime ideals?

Recall a path from a point $x$ to a point $y$ in a topological space $X$ is a continuous function $f$ from the unit interval
$[0,1]$ to $X$ with $f(0) = x$ and $f(1) = y$. Now let $R$ be a ...

1
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0
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63
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### When minimal prime ideals are maximal with respect to not containing an element

Let $\{ P_i \}$ be the set of all minimal prime ideals of a commutative ring $R $. Is there any conditions on $R $ under which there exists an element $x\in R $ such that $P_i $ is an ideal of $R $ ...

8
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### Continuous functions on a compact $T_1$ space

Let $X$ be a compact $T_1$ topological space consisting of more than one point, and suppose that $X$ is locally compact (i.e. every point has a local base of compact neighbourhoods), second countable, ...

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### Do these Zariski-dense subgroups of $\operatorname{SO}_{6}(\mathbb C)$ have non-empty intersection with this subset?

Let $G\leq \operatorname{SO}_{6}(\mathbb Z)$ be a finite-index normal subgroup, so it's a Zariski dense subgroup of $\operatorname{SO}_{6}(\mathbb C)$; and let $H$ be the subset of $\operatorname{SO}_{...

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### Symmetric products of varieties and projective bundles

Given a smooth projective geometrically connected curve $C$, a symmetric product of $C$ has the structure of a projective bundle over the Jacobian of $C$ (e.g. see Symmetric powers of a curve = ...

5
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1
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296
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### Locally affine varieties and du Val singularities

Let me start with an apologetic disclaimer: I am very far from an algebraic geometer, so this question might be crudely formulated.
I have a specific question about du Val singularities, but while ...

0
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0
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172
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### The Grothendieck ring of varieties with classical Zariski

Consider $\mathcal{V}_k$, the category of $k$-varieties over the finite field $k = \mathbb{F}_q$ with $q$ elements. We see varieties in the old "classical" sense of the word, foreseen with the old ...

6
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### On the prime spectrum of $R[[X]]$ when the prime spectrum of $R$ is Noetherian

All rings below are commutative with unity.
If $R$ has a.c.c. on radical ideals i.e. if $Spec R$ is Noetherian under Zariski topology, then so is $R[X]$, this is Theorem 2.5 in the following paper ...

4
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### A relation between $Spec((1+I)^{-1}R)$ and $Spec(R/J)$

Let $R$ be a commutative ring with identity and let $I$ and $J$ be two finitely generated ideals of $R$. Clearly $1+I:=\{1+i:i\in I\}$ is a multiplicative closed subset of $R$. We can consider the ...

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### Convergent sequences in projective varieties

It's very well known that if $X$ is an irreducible projective variety (feel free to assume that the base-field is $\mathbb{C}$), then any two points $x,y\in X$ can be connected by the image of a non-...

4
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0
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276
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### Bézout and products in algebraic groups

Let $G$ be a linear algebraic group -- be it a Lie group or a group of Lie type. Let $V$, $W$ be subvarieties of $G$. Of course, $V\cap W$ is also a variety (not necessarily irreducible) and $V\cdot W^...

12
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1
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685
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### Points of the big Zariski site

It's relatively simple to show that the geometric morphisms $ \mathbf{Set} \to \mathrm{Sh}(\mathbf{CRing}^\mathrm{op}_{\mathrm{fp}}, \mathrm{Zar})$ correspond to local rings.
More precisely, since ...

1
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1
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148
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### An ideal and its J-radical

Let $R$ be a commutative ring with $1$ and $I$ be an ideal of $R$. Now let $J=\cap_{I\subseteq m\in Max(R)}m$. Set $A:=\{p\in Spec(R): I\subseteq p\}$ and $B:=\{p\in Spec(R): J\subseteq p\}$, where $...

15
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5
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### Locales as spaces of ideal/imaginary points

I posted this question on MSE a few days ago, but got no response (despite a bounty). I hope it will get more answers here, but I'm afraid it might not be appropriate as I'm not sure it's actually ...