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Questions tagged [zariski-topology]

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

The Zariski topology allows using tools of topology for the study of algebraic varieties, even when the underlying field is not a topological field.

This is one of the basic ideas of scheme theory, which allows one to build general algebraic varieties by gluing together affine varieties in a way similar to that in manifold theory, where manifolds are built by gluing together charts, which are open subsets of real affine spaces.

The Zariski topology of an algebraic variety is the topology whose closed sets are the algebraic subsets of the variety. In the case of an algebraic variety over the complex numbers, the Zariski topology is thus coarser than the usual topology, as every algebraic set is closed for the usual topology.

See also: Zariski topology at Wikipedia.

379 questions
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Closure of a point : Zariski Topology.

Let $R$ a ring and $Spec(R)$ the set of prime ideal of $R$. Let $x\in Spec(R)$ and $\mathfrak p_x$ the corresponding ideal. The closure of $x$ is $$\bar x=\{\mathfrak p\in Spec(R)\mid \mathfrak p\supset \mathfrak p_x\}.$$ More generally, if…
user352653
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Mapping of semicubical parabola

Consider the semicubical parabola $ C \subset \mathbb{C} $ given by, $ C:= \{ (x,y) \in \mathbb{A}^2_{\mathbb{C}} \ | \ y^2-x^3=0 \} $. Show that the map $ \varphi: \mathbb{A}^1_{\mathbb{C}} \to \mathbb{C} \\ t \mapsto(t^2,t^3) $ is a…
pseudogeek
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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 and here, but I hadn't found one that explained it…
mi.f.zh
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Proving polynomial function is continuous for Zariski topology on $\mathbb{R}$

I am looking to prove that a polynomial function $f: \mathbb{R} \rightarrow \mathbb{R} $ is continuous for the Zariski topology on $\mathbb{R} $(both domain and range) How do I do this?
Mary
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The poset vs Zariski topology on Spec(R).

Given a commutative ring $R$, it seems to me that there are two obvious topologies one can put on the set $\operatorname{Spec}R$ of its prime ideals. First is the Zariski topology, with closed sets like \begin{equation} V(I) = \{ \mathfrak{p} \in…
jacob
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Characteristics of Zariski dense sets?

What does it mean for a set to be Zariski dense? I mean what are the special that a Zariski dense set has?
urpi
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How can it be shown that D (Zariski closed subset) is irreducible?

Given $A^6_{C}$ with coordinates $(u,v,w,x,y,z)$. Let $D ⊆ A^6$ be the Zariski closed subset given by the vanishing of the 2 × 2 minors of the matrix \begin{bmatrix} u & v & w \\ x & y & z \\ \end{bmatrix} (so $D = V (uy − vx, uz − wx, vz −…
Kalle Blomkvist
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Toward creating the neighborhoods of a point in the Zariski Topology.

In the instance of a line $L$ intersecting with a parabola I need to find out if the intersection is a dense subset. Here's my work / question so far: For another point $P$ on the parabola not in the intersection of $L$, is the entire complement of…
user719023
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Zariski closure in an irreducible component of a set

Let $K$ be a field. We work with $K^n$ with Zariski topology. Let $A\subset K^n$ and let $V_1,\cdots,V_k$ be the irreducible components of the Zariski closure of $A$. Then $A\cap V_i$ is Zariski dense for all $i$. Why?
ABmmm
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Zariski closure of a singleton is irreducible

Closure of singleton is irreducible in Zariski topology Let $R$ be a commutative unitary ring. For any $E \subset R$ define $V(E) = \lbrace P \in \text{spec}(A) \mid E \subset P \rbrace $. Equip $\text{spec(R)}$ with the Zariski topology, that is,…
James Well
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Irreducibility of Zariski closed subset

A Zariski Closed subset $W \subseteq A^n$ is called reducible if $W = > W_{1} \cup W_{2}$ where $W_{i} \nsubseteq W$ and $W_{i}$ closed. Let $W \subseteq A^n$ be closed. Then $W$ is irreducible only if $I(W)$ is a prime ideal. Sufficiency: Suppose…
Ultra
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