# Questions tagged [zariski-topology]

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

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### How exactly does Zariski-Topology relate to Standard topology on $\Bbb{C}$?

I'm currently self-studying algebraic geometry and I tried to construct some proofs / counterexamples for relations between open / closed sets in Zariski and Standard Topology on $\mathbb{C}$ but I'm ...
1 vote
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### Let $G$ be a linear algebraic group and $C_G(x)$ be the centralizer of $x$. Show that $C_G(x)$ is a closed subgroup for all $x\in G$.

Let $G$ be a linear algebraic group and $C_G(x)$ be the centralizer of $x$. Show that $C_G(x)$ is a closed subgroup for all $x\in G$. I want to enstablish a homomorphism from $G$ to $G$, and the ...
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### $X\times Y$ and the product topology [duplicate]

I'm studing Andreas Gathmann's notes on algebraic geometry (pdf here: https://agag-gathmann.math.rptu.de/de/alggeom.php). In chapter 4 (about Morphisms) he was using the universal property of products ...
1 vote
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### Can $\operatorname{Spec}(R)$ be not homeomorphic to $\operatorname{Spec}(S)$ but be isomorphic as a poset?

Looking at the definition of the spectrum of a ring made me wonder when/whether is its topology determined by the poset of prime ideals of the ring. Let $R$ and $S$ be commutative unital rings. Can ...
56 views

### If the product of closed sets is closed in the product topology shouldn't {(0,0)} be closed in $A^1 \times A^1$ zariski topology.

I am not sure what is incorrect about the statement. If the product of closed sets is closed in the product topology shouldn't {(0,0)} be closed in $A^1 \times A^1$ zariski topology, i.e singletons ...
1 vote
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### Zariski open set in $\mathbb{\bar Q}^2$ and a question on shifted-sparse interpolation

In [1 (linked PDF here)], there is mention of a Zariski open set in Corollary 3 and they mention that "generic" means $(b_1, b_2)$ belongs to a Zariski open set in $\mathscr{\bar F}^2$. ...
1 vote
68 views

### How is a morphism between projective curves the union of morphisms of affine curves?

I am reading a paper, and it uses the following definition: A morphism ${f : C \to C'}$ of affine curves is the restriction to C of a map of the following form, where ${f_1}$ and ${f_2}$ are elements ...
1 vote
146 views

### Atiyah MacDonald Exercise 1.21 iv) Proof of continuity

Let $\Phi: A \rightarrow B$ be a surjective ring homomorphism. Let $X = \text{Spec}(A)$ and $Y = \text{Spec}(B)$, then $\Phi^*=\Phi^{-1}: Y \rightarrow V(Ker(\Phi))$. I already showed that $\Phi^*$ ...
53 views

### Proving a set is algebraic.

I want to investigate about if the set of points of the form $(x,1)$ with $x\neq 0$ (horizontal line without a point) is closed or not in $\mathbb{A}_k^n$ with the zariski topology. My intuition say ...
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### Dimension of affine varieties preserves for open subsets

My problem: Let $X$ be an affine variety and $U$ an open subset of $X$. Then $dim U = dim X$. My attempt: If I take a chain of irreducible and closed sets $Z_1\subsetneq... \subsetneq Z_n$ in $U$, ...
1 vote
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### Does the Zariski-closure of an irreducible subgroup $G$, also act irreducible?

Let $G \subset GL(V)$ be a subgroup for a complex vectorspace $V$. Let $H=\overline{G}$ be its Zariski-closure. Then I want to check if the following statement (1) implies statement (2): $G$ acts ...
1 vote
52 views

### Is a closed subgroup of a linear algebraic group, again a linear algebraic group?

Let $G$ be a linear algebraic group over an algebraically closed field $K$. For a subgroup $H \subset G$, we denote with $\overline{H}$ the closure of $H$ in $G$ with respect to the Zariski topology. ...
That wikipedia page has a proof of Zariski's lemma, which states that if $A$ is a Jacobson ring (in particular when $A$ is a field) and $B$ is a finitely-generated algebra of $A$, which is a field, ...