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All I know is that when I had to prove this result a long time ago, I came up with this proof a few months after I had started learning affine schemes and I was exhilarated at the thought that I could …

Since in 2007-2008 you evoked [ Class 24, §1.8, The problem with locally free sheaves] the equivalence between locally free sheaves and vector bundles on a scheme, the following point, potentially co …

in the category of affine schemes, but actually in the category of schemes, thanks to the string of equalities (where $X$ is a not necessarily affine scheme) $$ Hom_{Schemes} (X, Spec(A))= Hom_{Rings … a reflexive subcategory of the category of schemes. …

$P_1$ and $P_2$ the two special closed points, $A_1$and $A_2$ their respective open complements and $A_{12}=A_1\cap A_2$, so that $A_i\simeq \mathbb A^1_k$ and $A_{12}\simeq\mathbb G_m$, all affine schemes … [In categorical language: $\Gamma$ is an anti-equivalence from the category of affine schemes to that of rings] Proof 6 Every global function $P(T)\in \Gamma(X,\mathcal O_X)=k[T]$ (see Proof 5) takes …

I shall assume that $X,Y$ are integral, locally noetherian schemes and that $f$ is dominant. …

I remember it took me a long time to realize this and when I did I lost some of my fear of schemes. …

I have only seen the fact that direct sums of flabby sheaves are flabby (correctly) used on noetherian spaces, actually schemes, so that my question originates just from idle curiosity... …

As to the suggestion "it was discovered that there were far more propositions about preschemes than about schemes, and people decided that this was ridiculous": considering the God-like status of Grothendieck …

If you decide to teach a more arithmetically flavoured algebraic geometry, students should be made aware that schemes over a ring $A$ are stranger than they might think. …

Let $A$ be a noetherian ring and $X=\operatorname {Spec}A$ the corresponding affine scheme. There are three rings which might reasonably be called the ring of rational functions on $X$. a) The total r …

The terminology is disastrous if one considers schemes over $\mathbb C$, since these schemes have a holomorphic structure for which "meromorphic" has a completely different meaning. …

Be warned that this is an advanced monograph and that general knowledge of schemes is insufficient : techniques like positivstellensatz, real spectra, Nash functions,...are de rigueur here. …

Dear Giovanni, here is a point of view orthogonal to the examples you mention (although I am not sure they will satisfy your wish that they be "stranger"... ). Given a morphism $X \to S$, inste …

Here then is the statement you need : Let $f:X\to Y$ be a closed morphism between schemes of finite type over a field $k$. …