Articles of zariski topology

Show that affine varieties are quasi-compact.

A topological space $X$ is called quasi-compact if whenever ${(U_i)}_{i∈S}$ are a family of open subsets such that $∪_{i∈S}(U_i) = X$ then there are a finite number of $(U_i)$’s which actually cover X. I’m thinking first we show that if $(U_{fi})$, $i ∈ S$ (where ${f_i}$, $i ∈ S$ a collection of elements of $R[X]$) […]

Closure of image by polynomial of irreducible algebraic variety is also irreducible algebraic variety

How to show that closure of image by polynomial of irreducible algebraic variety is also irreducible algebraic variety. $A$ -irreducible algebraic variety, $F$ – polynomial map ($F:A \to B$), proof that $\bar B$ – is irreducible algebraic variety. Closure of some set is always algebraic variety as definition of closure. But how to show, that […]

Prove that this condition is true on an Zariski open set

Why is there an Zariski open set of $P\in GL(n,\mathbb{R})$ such that $P\,\text{diag}(1,\dots,1,-1)P^{-1}$ can be conjugated by a diagonal matrix $D$ to get an orthogonal matrix? Note that $M=P\,\text{diag}(1,\dots,1,-1)P^{-1}$ is involutory ($MM=I$) and $\det(M)=-1$. Possible starting point: Consider $$\{P\in GL_n(\mathbb R),\ D \text{ invertible and diagonal s.t. } DP\text{diag}(1,\dots,1,-1)P^{-1}D^{-1} \text{ is orthogonal}\}$$ In general, this […]

Why do we take the closure of the support?

In topology and analysis we define the support of a continuous real function $f:X\rightarrow \mathbb R$ to be $ \left\{ x\in X:f(x)\neq 0\right\}$. This is the complement of the fiber $f^{-1} \left\{0 \right\}$. So it looks like the support is always an open set. Why then do we take its closure? In algebraic geometry, if […]

A question about the Zariski topology

Let $(\mathfrak a_i)$ be an infinite family of ideals in commutative ring $R$. Is $\bigcap\limits_{i=1}^\infty \mathfrak a_i$ not defined? I am trying to understand Zariski topology. Here, $V(\bigcap_i \mathfrak a_i)= \bigcup\limits_{i} V(\mathfrak a_i)$. If $\bigcap\limits_{i=1}^\infty \mathfrak a_i$ is defined, which I think it should be, then we would have an infinite union of sets of […]

Ring with spectrum homeomorphic to a given topological space

I would like to ask whether given a topological space $X$, we can find a commutative ring with unity $R$ such that $\operatorname{Spec} R$ (together with the Zariski topology) is homeomorphic to $X$. Since the spectrum is a compact space, this is obviously only possible if $X$ is compact. Furthermore, from this answer we obtain […]

Why Zariski topology?

Why in algebraic geometry we usually consider the Zariski topology on $\mathbb A^n_k$? Ultimately it seems a not very interesting topology, infact the open sets are very large and it doesn’t satisfy the Hausdorff separation axiom. Ok the basis is very simple, but what are the advantages?