# Tag Info

Accepted

### Building algebraic geometry without prime ideals

Actually, you have rediscovered a nice motivation of using prime ideals as points. Indeed, your collection of points are triples $(R, k_x, \mathrm{ev}_x)$ where , $\mathrm{ev}_x \colon R \to k_x$ is a ...
• 9,079
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• 34.5k
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• 37.5k

### Building algebraic geometry without prime ideals

This nice approach to points on schemes in fact becomes crucial once one leaves the world of schemes and travels to the galaxy of stacks. For an algebraic stack $X$, one defines a point of $X$ to be a ...
• 20.8k

### Connеcted components of irreducible algebraic varieties

For a (projective) smooth real plane curve $C \subset \mathbb{RP}^2$ the answer is known. Such a curve is a compact smooth one-dimensional manifold without a boundary, so its connected components are ...
• 65.4k
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### A book on elliptic curves using scheme theory?

Not exactly a book, but there are course notes on abelian varieties from a course that Brian Conrad taught a few years ago. It is definitely from the perspective of scheme theory/functor of points.
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### Smallness of the category of schemes of finite type

You can prove essential smallness of the category of finite type $S$-schemes (without further assumptions), where $S$ is any scheme, as follows: If $S$ is affine and we only consider affine finite ...
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### Can $\mathcal O_X$ be recognized abstract-nonsensically?

I think you are a little confused about what your characterization of $R$ does, and this causes problems as you generalize to sheaves. There is no characterization of $R$ as an element of the ...
• 138k

### Is there an analogue of projective spaces for proper schemes?

I am just posting my comment as one answer. So long as you are only asking about schemes (rather than complex analytic spaces), you can avoid the hard analysis from the previous MathOverflow answer. ...

### Beauville-Laszlo for schemes

Completing the discussion under Will Sawin's answer. The question has been answered completely and affirmatively by Ben-Bassat and Temkin in their paper "Berkovich spaces and tubular descent" (Adv. ...
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### Smooth morphism (algebraic geometry) vs. Submersion (differential geo) & Ehresman's Lemma

One of the many equivalent definitions of smoothness of a morphism $f\colon X\to Y$ of varieties over a field $k$ is that $f$ is smooth if and only if it is formally smooth. The latter means the ...
• 1,790
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### Making the étale topos construction a fully faithful 2-functor from schemes to Grothendieck topoi

Below is a proof that the (pseudo)functor that sends a scheme to its petit étale topos is not fully faithful, for any category of schemes over an algebraically closed base field $k$, assuming that ...
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Etale topology, required to define etale cohomology, is not a topology in the usual sense. It is Grothendieck topology only. In the category of topological manifolds, an etale cover of $X$ is a ...