# Questions tagged [zariski-topology]

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13
questions

**1**

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51 views

### 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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153 views

### 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, ...

**3**

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207 views

### 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}_{...

**0**

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107 views

### 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**answer

214 views

### 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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153 views

### 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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172 views

### 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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94 views

### 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 ...

**0**

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154 views

### 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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259 views

### 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^...

**9**

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**1**answer

380 views

### 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**

vote

**1**answer

101 views

### 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 $...

**12**

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**5**answers

772 views

### 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 ...