29
votes
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 ...
- 8,406
25
votes
Accepted
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[...
- 23.2k
23
votes
Reference for de Rham cohomology in positive characteristic
$\def\dr{d_{\rightarrow}}\def\du{d_{\uparrow}}$ This is true. I don't know a reference, but here is a proof. I will show, more strongly, that, at every stage in the Hodge-de Rham spectral sequence, we ...
- 144k
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 ...
- 19k
22
votes
Accepted
Homotopy types of schemes
Any scheme which is separated of finite type, has at least a triangulation, hence is, in particular, a CW-complex. In fact, by a theorem of Lojasiewicz, this is true for any semi-algebraic set (one ...
- 12.6k
22
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 ...
- 38.2k
21
votes
Accepted
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 ...
- 1,670
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 ...
- 19.1k
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 ...
- 12.6k
17
votes
Accepted
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$...
- 23.2k
17
votes
Accepted
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 ...
- 16k
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 ...
- 8,871
16
votes
Accepted
An apparent equivalence of the category of affine schemes over $S$ and the category of quasi-coherent $\mathcal{O}_S$-algebras
No, it's not true in general (EGA 2, (1.2.3)).
The following example is taken from EGA 2, (1.3.3). Over a field $K$, let $S$ be the affine plane with a doubled origin. Then $S$ is the union of two ...
- 12.6k
16
votes
Accepted
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 $...
- 32.6k
15
votes
An apparent equivalence of the category of affine schemes over $S$ and the category of quasi-coherent $\mathcal{O}_S$-algebras
What is true is that there is an antiequivalence between the category of schemes affine over $S$ (that is $S$-schemes for which the preimage of an open affine of $S$ is an open affine) and quasi-...
- 16k
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 ...
- 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
...
- 161k
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 ...
- 151
15
votes
Accepted
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 ...
- 12.6k
14
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 ...
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
$$
H^0(X,\mathcal{O}_X) = H^0(Y,...
- 34k
14
votes
Accepted
schemes having same reduced underlying space and same cotangent sheaf are isomorphic?
Consider the simplest example:
$$
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}...
- 34k
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 ...
- 19.3k
13
votes
On Q-Cartier Divisors
Maybe I can say something useful here. The main confusion seems to be how to find the sheaves/ideals/modules associated to multiples of divisors. As Martin Bright points out, symbolic power of a ...
- 19.7k
13
votes
Why would one "attempt" to define points of a motive as $\operatorname{Ext}^1(\mathbb{Q}(0),M)$?
In the spirit of You Could Have Invented Spectral Sequences by T.Chow, I claim you could have invented $\operatorname{Ext}^{1}(\mathbb Q(0),M)$ as group of "rational points" of a motive. Here is how.
...
- 9,680
13
votes
Accepted
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.
13
votes
Accepted
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 ...
- 8,406
12
votes
Accepted
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 ...
- 5,334
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 ...
- 123k
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. ...
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