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29 votes
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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 ...
Leo Alonso's user avatar
  • 9,079
25 votes
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Is a scheme Noetherian if its topological space and its stalks are?

This is false. The easiest counterexample I could come up with is the following "affine line with embedded points at every closed [rational] point": Example. Let $k$ be an infinite field, let $R = k[...
R. van Dobben de Bruyn's user avatar
23 votes

Hodge theory (after Deligne)

A brief answer. First of all, the results are miraculous. Deligne's Hodge II and Hodge III give just a few example applications of the kind of results you can prove using mixed Hodge theory; these ...
Dan Petersen's user avatar
  • 39.4k
23 votes

Do Grothendieck universes matter for an algebraic geometer?

It's inherently difficult to give a negative answer to a question like this, but here's a technical fact that pushes in that direction: Let ZFC$_n$ be the subtheory of ZFC gotten by restricting ...
Noah Schweber's user avatar
21 votes
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Is a direct sum of flabby sheaves flabby?

No, a direct sum of flabby sheaves need not be flabby. Take $X=\{1,1/2,1/3,1/4,\dots\}\cup\{0\}$ with the subspace topology from $\mathbb R$, and let $\mathcal F$ be the sheaf whose sections over an ...
Alexander Betts's user avatar
20 votes

When are valuative criteria useful?

One very important point, implicit in the comments by @zzy and @abx, is that valuative criteria allow to check a property (e.g. properness) of (say) an $S$-scheme $X$ directly in terms of the functor ...
Laurent Moret-Bailly's user avatar
18 votes

Making the étale topos construction a fully faithful 2-functor from schemes to Grothendieck topoi

There is no way it is true in general, but there are results in this direction nevertheless: Theorem 3.1 in this paper of Voevodsky establishes (a kind of fully) faithfulness for normal schemes of ...
D.-C. Cisinski's user avatar
17 votes
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Why Use Hypercohomology When Defining the de Rham Cohomology of a Smooth Scheme over $k$?

This is pretty much explained in the comments, but let me put it into an answer. One wants algebraic de Rham cohomology to be isomorphic to the usual de Rham cohomology (using $C^\infty$ forms) when $...
Donu Arapura's user avatar
  • 34.5k
17 votes
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Defining abstract varieties and their morphisms over a finitely generated subfield of the base field

This is treated (in much greater generality) in EGA IV$_3$, Théorème 8.8.2. The existence of $X_0$ and $Y_0$ follows from part (ii), whereas the existence of $f_0$ is part (i). References. [EGA IV$...
R. van Dobben de Bruyn's user avatar
17 votes
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Are higher etale homotopy groups topological groups in a natural way?

TL;DR The higher étale homotopy groups are the homotopy groups of the profinite completion of the shape of the étale topos. As such they are profinite groups. If you choose to see profinite groups as ...
Denis Nardin's user avatar
  • 16.3k
17 votes

The underlying space of a scheme remembers its affineness?

Here is a counterexample. Fix a field $k$, and let $Y$ be built from two copies of the affine nodal curve $y^2=x^3+x^2$, glued together on the complement of the singular point. In other words $Y$ is a ...
Julian Rosen's user avatar
  • 8,951
15 votes

Useful, non-trivial general theorems about morphisms of schemes

Here's a theorem I find useful: Theorem. Let $\phi \colon X \to Y$ be a smooth morphism of schemes of relative dimension $d$. Then there exists an open cover $X = \bigcup U_i$ of $X$ such that ...
15 votes

Research in applied algebraic geometry that essentially needs a background of modern algebraic geometry at Hartshorne's level

I will ignore the issue of what is "applicable" and what is only "potentially applicable", and the issue of whether something could be translated into classical language, and simply offer an example ...
Pop's user avatar
  • 867
15 votes

Hodge theory (after Deligne)

The laudatio for the Wolf prize explains it like this: Central to modern algebraic geometry is the theory of moduli, i.e., variation of algebraic or analytic structure. This theory was ...
Carlo Beenakker's user avatar
15 votes

Do Grothendieck universes matter for an algebraic geometer?

Tim Chow drew my attention to this thread, and asked if I cared to comment. Actually I wrote up a detailed version of my thoughts several years ago, On doing category theory within set theoretic ...
Nath Rao's user avatar
  • 151
15 votes
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Motivation for relative schemes: why should one work with schemes over a ringed topos?

The comments to your question discuss the variation of relative schemes over a topos, vs relative schemes over a site. But it seems your question stood at the more basic level of the relevance of ...
ACL's user avatar
  • 12.8k
15 votes
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The Serre duality theorem intuition

First of all, dualizing sheaves are unfortunately not treated in EGA. The treatment in Hartshorne has some limitations. Perhaps some of them are related to your questions. For pointers to more recent ...
Leo Alonso's user avatar
  • 9,079
14 votes

The ring of global sections of a regular scheme

The answer is no. For instance, take the quadratic cone $$ Y = \mathrm{Spec}(\Bbbk[x,y,z]/(xz-y^2)) $$ and let $X$ be its blowup at the vertex. Then $X$ is regular, but $$ H^0(X,\mathcal{O}_X) = H^0(Y,...
Sasha's user avatar
  • 37.5k
14 votes
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schemes having same reduced underlying space and same cotangent sheaf are isomorphic?

Consider the simplest example: $$ X = \mathrm{Spec}(\mathbb{C}[\epsilon]/\epsilon^2), \qquad Y = \mathrm{Spec}(\mathbb{C}[\epsilon]/\epsilon^3). $$ Definitely, $X_{\mathrm{red}} \cong Y_{\mathrm{red}...
Sasha's user avatar
  • 37.5k
14 votes

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 ...
Daniel Loughran's user avatar
14 votes

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 ...
Francesco Polizzi's user avatar
13 votes
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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.
12 votes
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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 ...
HeinrichD's user avatar
  • 5,402
12 votes

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 ...
Will Sawin's user avatar
  • 138k
12 votes

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. ...
12 votes

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. ...
Piotr Achinger's user avatar
12 votes
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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 ...
Alexander Betts's user avatar
12 votes
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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 ...
Jens Hemelaer's user avatar
11 votes
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Why care about Grothendieck topology?

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 ...
Bugs Bunny's user avatar
  • 12.1k
11 votes

Applications of schemes to mathematical physics

The Hilbert scheme of points on a K3 surface plays an important rôle in providing a strong coupling test of S-duality by Vafa and Witten. This is the original paper on what is known as Vafa-Witten ...
José Figueroa-O'Farrill's user avatar

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