# Questions tagged [zariski-topology]

The zariski-topology tag has no usage guidance.

**4**

votes

**0**answers

85 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**

votes

**0**answers

146 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**

votes

**0**answers

247 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**

votes

**1**answer

276 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

90 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 $...

**1**

vote

**0**answers

123 views

### A family of prime ideals and a family of maximal ideals

Let $R$ be a commutative ring with $1$ and let $\{\frak{p}_\alpha\}_{\alpha\in A}$ and $\{\frak{m}_\beta\}_{\beta\in B}$ be a family of prime ideals and a family of maximal ideals of $R$, respectively,...

**8**

votes

**3**answers

304 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 ...