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Dear Ravi, maybe the simplest example is one by Serre: the holomorphic Stein surface $\mathbb C^\ast\times \mathbb C^\ast $ underlies two non-isomorphic smooth complex algebraic varieties. 1) $\math …

[In front of a blackboard, in an office at Real College] Skeptic: And why should I care about holomorphic functions? Holomorphic enthusiast:$\;$ Can you compute $\quad$ $\sum_{n={-\infty}}^{\infty …

the field of rational functions $\mathcal K_X$ on an integral scheme $X$ ( for example an algebraic variety) is flasque and so is the sheaf of its invertible elements $\mathcal K^\ast_X$. This has a …

Examples : the reals, the complexes, the real quaternions and the octonions of Graves-Cayley. Any such division algebra must necessarily have dimension 1,2,4 or 8 (as in the examples). …

In the theory of holomorphic functions of several variables, Hartogs's theorem that any holomorphic function on a punctured open set of $\mathbb C^n$ ($n\geqslant 2$) can holomorphically be continued …

Dear Timothy, here is a theorem which, according to your wish, "could be understood, and seen to be interesting, by someone who had not already studied the material in that course": Brouwer's celebrat …

Dear Nicojo, since you now have many counter-examples, let me give you a situation where $B$ is finitely generated, in line with your question 2). …

If Pete or someone else is still interested despite the fine answers already given, here is an analysis of what might be the simplest situation. Let $k$ be a field of characteristic $\neq 2$ and defin …

Dear Michal, Munkres presents a regular space that is not completely regular as a very detailed exercise (more than half a page!) to §33 in his book "Topology, Second Edition, Prentice Hall,2000" (pag …

Here are two examples. 1) For $n$ even the sphere $S^n$ and real projective space $\mathbb P^n(\mathbb R)$ are not diffeomorphic since $H^n(S^n) \simeq \mathbb R$ while $H^n(\mathbb P^n(\mathbb R))=0 …

Dear Olivier, in line with the more advanced nature of this site, let me give an example of a less elementary nature. Consider a compact Riemann surface $X$ of genus 2 and on it stable vector bundles …

A strange and difficult question is whether there exists a scheme without any closed point. It is very tempting to think that since an affine scheme does have closed points ( they correspond to maxima …

These seem to be examples not on your list, but I'll let you be the judge of their exotism.... …