34

Let's say $k=\mathbb{C}$ (although something like this should work over any algebraically closed field). Let $V_1=\mathbb{P}^1-\{0,1,\infty,\pi\}$ and $V_2 = \mathbb{P}^1-\{0,1,\infty,e\}$. One can see that $V_1$ and $V_2$ are not isomorphic as varieties/schemes over $\mathbb{C}$: such an isomorphism would extend to a map $\mathbb{P}^1\to\mathbb{P}^1$ taking ...

24

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 $R = k[x]$, and for each $\alpha \in k$ let $R_\alpha = R[y_\alpha]/((x-\alpha)y_\alpha,y_\alpha^2)$. Then $R_\alpha$ is an affine line with an embedded prime $\...

answered Dec 9 '18 at 5:05

R. van Dobben de Bruyn

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22

$\def\dr{d_{\rightarrow}}\def\du{d_{\uparrow}}$ 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 have a perfect pairing $E^{pq}_r \times E^{(n-p)(n-q)}_r \to k$. In particular, $E^{pq}_{\infty}$ and $E^{(n-p)(n-q)}_{\infty}$ are dual and, since $H^k_{DR}(X)...

answered Nov 2 '15 at 4:03

David E Speyer

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21

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 can even get this for subanalytic sets, by a result of Hironaka, in Triangulation
of algebraic sets, Proc. Amer. Math. Soc. Inst. Algebra Geom. Arcata(1974)); ...

21

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 open $U\subseteq X$ are the functions $U\to\mathbb F_2$ (not necessarily continuous). This is a flabby sheaf. I claim that the infinite direct sum $\mathcal F^{\...

20

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 are great theorems which just fall out of the general theory. People quickly figured out more applications, like the Hodge-Deligne polynomial, which is itself a ...

18

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 Separation and Replacement to $\Sigma_n$ formulas. By the reflection principle,$^1$ for each $n$ the theory ZFC proves that there is an ordinal $\alpha_n$ such that $...

answered May 13 '19 at 17:26

Noah Schweber

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17

### 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$_3$] A. Grothendieck, Éléments de géométrie algébrique. IV: Étude locale des schémas et des morphismes de schémas. (Troisième partie). Publ. Math., Inst. Hautes ...

answered Oct 5 '18 at 15:50

R. van Dobben de Bruyn

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17

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 nodal curve with doubled singular point. Then $Y$ is not affine because it is not separated. However, there is a homeomorphism $\mathbb{A}^1\to Y$, which is ...

17

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 of points of $X$. This is of course especially convenient if $X$ is defined by this functor, classical examples being Picard, Quot or various moduli functors. ...

answered May 1 '19 at 7:48

Laurent Moret-Bailly

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16

If $C$ is a sufficiently general rational curve in $\mathbb P^3$ of degree $d$, then the vector space of degree $k$ homogeneous polynomials vanishing on $C$ has dimension precisely equal to
$$\max\left\{0,\binom{k+3}{3} - dk - 1\right\}\;.$$
More geometrically, there exists a degree $k$ surface containing $C$ if and only if $\binom{k+3}{3} - dk - 1 > 0$.
...

16

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 affine open subsets $Y_1$ and $Y_2$, each of them is isomorphic to the affine plane, glued along the complementary subset of their origin. In particular, $Y_1$ ...

15

I think the easiest condition is the fact that the natural morphism
$$
(X, \mathcal O_X) \to \operatorname{Spec}(\mathcal O_X(X))
$$
is an isomorphism.

15

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 $k=\mathbb{C}$, and have similar properties when $k$ is an arbitrary field of characteristic zero. From this point of view, as Grothendieck observed in the 1960'...

15

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 topological groups, group schemes or pro-systems is largely a matter of choice.
How is the étale homotopy group defined? There are many ways, but possibly the ...

14

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-coherent $\mathcal{O}_S$-algebras.
The anti-equivalence is realized by the pushforward of the structure sheaf and the relative spectrum (see Exercise 5.17 in ...

14

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
$$
H^0(X,\mathcal{O}_X) = H^0(Y,\mathcal{O}_Y) = \Bbbk[x,y,z]/(xz-y^2)
$$
is not.

14

Consider the simplest example:
$$
X = \mathrm{Spec}(\mathbb{C}[\epsilon]/\epsilon^2), \qquad
Y = \mathrm{Spec}(\mathbb{C}[\epsilon]/\epsilon^3).
$$
Definitely, $X_{\mathrm{red}} \cong Y_{\mathrm{red}}$. Also, a simple computation shows that
$$
\Omega_{X/\mathbb{C}} \cong \mathbb{C},
\qquad
\Omega_{Y/\mathbb{C}} \cong \mathbb{C}[\epsilon]/\epsilon^2,
$$
...

13

Let $U \cong \mathbb{A}^n$ be the image of $j$ and let $\Delta = \mathbb{A}^n \setminus U$. Since $U$ is affine, $\Delta$ is purely codimension $1$, so $\Delta$ is the zero locus of some $f \in k[x_1, \ldots, x_n]$. But then $f$ is a unit on $U$, contradicting that $U \cong \mathbb{A}^n$.

answered May 25 '13 at 13:01

David E Speyer

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13

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.
The motivation comes from conjectures on special values of $L$-function. Before the general conjectures were formulated by S.Bloch and K.Kato, there were two ...

13

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 each $U_i \to Y$ factors as
$$U_i \stackrel \pi \to \mathbb A^d_Y \to Y,$$
with $\pi$ étale. Mnemonic: smooth morphisms have étale coordinates.
See ...

13

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

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
traditionally mysterious and problematic. In critical special cases,
i.e., curves, it made sense, i.e., the set of curves of genus greater
than one had a natural ...

answered Oct 12 '18 at 16:29

Carlo Beenakker

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12

Consider a morphism $f:\mathbb{P}^1 \to \mathbb{P}^1$ of degree $e> 2$. Using Grothendieck's splitting lemma, it is straightforward to compute that $f_*\mathcal{O}(1)$ is isomorphic to $\mathcal{O}^{\oplus 2} \oplus \mathcal{O}(-1)^{\oplus (e-2)}$. Thus the determinant is $\mathcal{O}(-(e-2))$, which is anti-ample. Thus, I don't see any reason at all ...

12

Suppose the structure morphism $g: X\to \operatorname{Spec}(A)$ is separated and of finite type, and $f: \operatorname{Spec}(B)\to \operatorname{Spec}(A)$ is faithfully flat; furthermore, assume $A, B, X$ are all Noetherian. Denote $X\times_A B$ as $X_B$ and let $f': X_B\to X$ be the base change of $f$ along $g$, and $g': X_B\to \operatorname{Spec}(B)$ the ...

12

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 type $S$-schemes, these correspond to finite type $\Gamma(S)$-algebras. These are isomorphic to algebras of the form $\Gamma(S)[x_1,\dotsc,x_n]/I$ for some ...

12

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. The argument below sketches this. The main additional detail beyond Hilbert scheme techniques is a strong variant of Chow's Lemma. Let $k$ be an ...

12

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. Math. 2013), but one has to be careful when globalizing $\hat A$ and (especially) $\hat A[1/f]$.
(1) Beauville and Laszlo treat the non-affine case in the ...

12

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 relative schemes over something else than a scheme.
In Grothendieck's philosophy, a relative scheme $f\colon X\to S$ functions as a family of schemes $X_s$ (for $...

11

If I understand the question correctly, the category of schemes is regularly wellpowered.
A locally closed immersion factors as $Y\xrightarrow{i}U\xrightarrow{j}X$ where $j$ (resp. $i$) is an open (resp. closed) immersion. Now open immersions correspond bijectively to open subspaces of the underlying space of $X$ (this is clearly small), while closed ...

answered Mar 18 '14 at 8:20

Laurent Moret-Bailly

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