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 […]

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 […]

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 […]

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 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?

Intereting Posts

How to prove that ab $\le \int_{0}^{a} \phi(x) dx + \int_{0}^{b} \phi^{-1}(y) dy$
Is there an analogue of the jordan normal form of an nilpotent linear transform of a polynomial ring?
Does $\int_0^{2 \pi} \sqrt{1-(a+b \sin\phi)^2} d\phi $ have a closed form in terms of elliptic integrals?
Index notation for tensors: is the spacing important?
Modern book on Gödel's incompleteness theorems in all technical details
How to show $f,g$ are equal up to n'th order at $a$.
Use implicit function theorem to show $O(n)$ is a manifold
Showing every knot has a regular projection using differential topology
Compactness of Algebraic Curves over $\mathbb C^2$
How to solve this Poisson's equation
An alternative approach to constructing the free group.
Finding a pair of functions with properties
concrete examples of tangent bundles of smooth manifolds for standard spaces
Change of limits in definite integration
Is the pointwise maximum of two Riemann integrable functions Riemann integrable?