Questions tagged [schemes]

The first purpose of schemes theory is the geometrical study of solutions of algebraic systems of equations, not only over the real/complex numbers, but also over integer numbers (and more generally over any commutative ring with 1). It was finalized by Alexandre Grothendieck, during the 1950s and the 1960s.

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38 votes
2 answers
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Do Grothendieck universes matter for an algebraic geometer?

I recently learned that some parts of SGA require axioms beyond ZFC. I am just a simple algebraic geometer so I am trying to understand how can this fact impact my life (you may have engaged in a ...
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16 votes
2 answers
3k views

Is there an example of a formally smooth morphism which is not smooth?

A morphism of schemes is formally smooth and locally of finite presentation iff it is smooth. What happens if we drop the finitely presented hypothesis? Of course, locally of finite presentation is ...
David Zureick-Brown's user avatar
232 votes
16 answers
55k views

What elementary problems can you solve with schemes?

I'm a graduate student who's been learning about schemes this year from the usual sources (e.g. Hartshorne, Eisenbud-Harris, Ravi Vakil's notes). I'm looking for some examples of elementary self-...
51 votes
2 answers
7k views

Ring-theoretic characterization of open affines?

Background Recall that, given two commutative rings $A$ and $B$, the set of morphisms of rings $A\to B$ is in bijection with the set of morphisms of schemes $\mathrm{Spec}(B)\to\mathrm{Spec}(A)$. ...
Manny Reyes's user avatar
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26 votes
1 answer
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What are the epimorphisms in the category of schemes?

Is there a known characterization of epimorphisms in the category of schemes? It is easy to see that a morphism $f : X \to Y$ such that the underlying map $\lvert f\rvert$ is surjective and the ...
Martin Brandenburg's user avatar
24 votes
4 answers
6k views

Extending vector bundles on a given open subscheme

Let $U$ be a dense open subscheme of an integral noetherian scheme $X$ and let $E$ be a vector bundle on $U$. Suppose that the complement $Y$ of $U$ has codimension $\textrm{codim}(Y,X) \geq 2$. Let $...
Ariyan Javanpeykar's user avatar
22 votes
1 answer
7k views

What are the monomorphisms in the category of schemes?

Someone recently asked what the epimorphisms in the category of schemes are; the other day I had been wondering about the similar question: what are the monomorphisms in the category of schemes? I am ...
Phillip Williams's user avatar
11 votes
3 answers
2k views

What is the difference between Grothendieck groups K_0(X) vs K^0(X) on schemes?

More specifically, I was wondering if there are well-known conditions to put on $X$ in order to make $K_0(X)\simeq K^0(X)$. Wikipedia says they are the same if $X$ is smooth. It seems to me that you ...
Matt's user avatar
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3 votes
1 answer
881 views

Diagonal is representable then any morphism is representable

Ariyan Javanpeykar said here in comments that, If the diagonal is representable, then isn't any morphism $S\rightarrow \mathcal{X}$ with $S$ a scheme representable? I could not find the statement (...
Praphulla Koushik's user avatar
181 votes
33 answers
31k views

What should be learned in a first serious schemes course?

I've just finished teaching a year-long "foundations of algebraic geometry" class. It was my third time teaching it, and my notes are gradually converging. I've enjoyed it for a number of reasons (...
32 votes
4 answers
3k views

Spectrum of the Grothendieck ring of varieties

Here's a problem that may ultimately require just simple algebraic-geometry skills to be solved, or perhaps it's very deep and will never be solved at all. From the comments, some literature and my ...
Ilya Nikokoshev's user avatar
24 votes
4 answers
6k views

When is an irreducible scheme quasi-compact?

The standard examples of schemes that are not quasi-compact are either non-noetherian or have an infinite number of irreducible components. It is also easy to find non-separated irreducible examples. ...
David Rydh's user avatar
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18 votes
2 answers
2k views

Homotopy types of schemes

Let $X$ be a scheme over $\mathbb{C}$. When does the topological space $X\left(\mathbb{C}\right)$ of $\mathbb{C}$-points have the homotopy type of a finite CW-complex? When does the topological ...
David Carchedi's user avatar
11 votes
1 answer
869 views

About the relation between the categories $\text{Sch}$, $\text{LRS}$ and $\text{RS}$

I've asked this question https://math.stackexchange.com/questions/1407451/about-the-relation-between-the-categories-textsch-textlrs-and-text on math.stackexchange , however I don't think I will ...
user40276's user avatar
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9 votes
1 answer
1k views

Are higher etale homotopy groups topological groups in a natural way?

Since etale fundamental group of a scheme $X$ is the group of natural automorphisms of the fibre functor of the category of finite etale covers of $X$, it comes with structure of a topological group. ...
geometer's user avatar
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3 votes
2 answers
2k views

Smooth morphism (algebraic geometry) vs. Submersion (differential geo) & Ehresman's Lemma

I have a general question about the motivation behind to definition the smooth morphisms as we know it from algebraic geometry. The most common definition of a smooth morphism $: X \to Y$ between two ...
user267839's user avatar
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2 votes
0 answers
280 views

Looking for the exact and the precise statement of Ogus conjecture

I have been looking for several weeks for the exact and the precise statement of Ogus conjecture, but, I cannot find it. The only book which made me discover the statement of this conjecture is that ...
Bradley04's user avatar
  • 487
2 votes
1 answer
550 views

Noetherian stalks imply locally Noetherian

Is there an example of a non-Noetherian integral affine scheme with Noetherian space and Noetherian stalks? What if we replace "integral" with "reduced"?
user avatar
42 votes
2 answers
3k views

Commutative rings to algebraic spaces in one jump?

Typically, in the functor of points approach, one constructs the category of algebraic spaces by first constructing the category of locally representable sheaves for the global Zariski topology (...
42 votes
6 answers
6k views

Arbitrary products of schemes don't exist, do they?

Thinking of arbitrary tensor products of rings, $A=\otimes_i A_i$ ($i\in I$, an arbitrary index set), I have recently realized that $Spec(A)$ should be the product of the schemes $Spec(A_i)$, a ...
Georges Elencwajg's user avatar
31 votes
7 answers
3k views

Categorical construction of the category of schemes?

The answer to the following question is probably well known or the question itself is well known not to have a reasonable answer. In the latter case could you please let me know what the "right" ...
algori's user avatar
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28 votes
1 answer
2k views

Why and how did preschemes become schemes?

Originally (e.g., in the first edition of EGA and in Mumford's Red Book), what are now called "schemes" were referred to as "preschemes." The word "scheme" was reserved for what are now called "...
27 votes
1 answer
1k views

Motivation for relative schemes: why should one work with schemes over a ringed topos?

Recently I've been trying to learn more about relative schemes. These were developed in M. Hakim's thesis Topos annelés et schémas relatifs under Grothendieck's guidance and appear in many of later ...
Emily's user avatar
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25 votes
2 answers
2k views

Reference for de Rham cohomology in positive characteristic

It is known in characteristic $0$ that (algebraic) de Rham cohomology is a Weil cohomology theory. However, in characteristic $p > 0$ it isn't, if only because it has mod $p$ coefficients, whereas ...
R. van Dobben de Bruyn's user avatar
25 votes
2 answers
8k views

Intuition behind generic points in a scheme

In a scheme, each point is a generic point of its closure. In particular each closed point is a generic point of itself (the set containing it only), but that's perhaps of little interest. A point ...
ssquidd's user avatar
  • 1,101
18 votes
1 answer
849 views

"Real algebraic varieties" vs finite type separated reduced $\mathbb{R}$-schemes with dense $\mathbb{R}$-points

This question is partly motivated by a few comments here. Let me denote by $R$ the (real-closed) field of real numbers $\mathbb{R}$; everything is probably the same over an arbitrary real-closed field....
Qfwfq's user avatar
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18 votes
1 answer
2k views

Geometric generic fibre

This is a pretty elementary question about schemes, but it came up in the course of research, so let's try it here rather than MSE. Question 1: Are the fibres of a family of complex varieties ...
Lazzaro Campeotti's user avatar
18 votes
3 answers
2k views

Is there an example of a variety over the complex numbers with no embedding into a smooth variety?

Is there an example of a variety over the complex numbers with no embedding into a smooth variety?
David Zureick-Brown's user avatar
18 votes
2 answers
2k views

Images and monomorphisms of schemes

If $X$ is an object in an arbitrary category, there is a natural definition of a subobject of $X$ as a monomorphism into $X$ (or really an equivalence class of monomorphisms). If $X$ is a scheme, ...
Harold Williams's user avatar
17 votes
2 answers
967 views

What is an explicit example of a variety X which is finite over Spec F_p but which does not lift to a scheme Y which is finite and flat over Spec Z_p?

What is an explicit example of a variety X which is finite over Spec F_p but which does not lift to a scheme Y which is finite and flat over Spec Z_p?
David Zureick-Brown's user avatar
12 votes
1 answer
3k views

Is the degree of a finite morphism stable by base change

Let $f:X\longrightarrow Y$ be a finite morphism of schemes of degree $n$. Let $S\to Y$ be a morphism of schemes. Is the degree of the finite morphism $X\times_Y S \longrightarrow S$ equal to $n$? If ...
Taicho's user avatar
  • 225
12 votes
3 answers
3k views

Do disjoint unions and fiber products commute?

Do disjoint unions and fiber products commute? In other words, is the following statement true? Statement: Let $C$ be a category with (infinite) coproducts and fiber products. Let {$U_{i}$} be a ...
Hiro's user avatar
  • 945
11 votes
4 answers
2k views

The category of finite locally-free commutative group schemes

I'm trying to understand the properties of the category $\mathcal{FL}/S$ of finite locally-free commutative group schemes over an arbitrary base-scheme $S$. I know it is not in general an abelian ...
fls's user avatar
  • 111
11 votes
5 answers
8k views

When is the push-forward of the structure sheaf locally free

Let $f:X\longrightarrow Y$ be a morphism of noetherian schemes. Under what conditions is $f_\ast \mathcal{O}_X$ a locally free $\mathcal{O}_Y$-module? Example 1. Suppose that $f$ is affine. Then $f_\...
Ariyan Javanpeykar's user avatar
11 votes
2 answers
2k views

Is a scheme Noetherian if its topological space and its stalks are?

Is a scheme being Noetherian equivalent to the underlying topological space being Noetherian and all its stalks being Noetherian?
G.-S. Zhou's user avatar
10 votes
1 answer
824 views

Is the functor of points of a scheme cofinally small?

Background: In functorial algebraic geometry one would like to consider the category of all functors $\mathsf{CRing} \to \mathsf{Set}$ and define/characterize the category of schemes as a full ...
Martin Brandenburg's user avatar
9 votes
2 answers
582 views

Is the category of schemes wellpowered? regularly wellpowered?

Wellpowered means that for every scheme $X$, the subobject lattice of monormophisms $Y \to X$ is essentially small; regularly wellpowered means that for every scheme $X$, the regular subobject lattice ...
Tim Campion's user avatar
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9 votes
1 answer
321 views

When are free modules on sheaves of sets quasicoherent?

This question was previously asked over at math.SE. Let $X$ be a scheme. Let $\mathcal{E}$ be a sheaf of sets on $X$. Then we can define $\mathcal{O}_X\langle\mathcal{E}\rangle$, the free module over ...
Ingo Blechschmidt's user avatar
9 votes
2 answers
2k views

Diagonal map and "infinitesimal points"

Let $f:X\to Y$ be a morphism between schemes. To construct the relative sheaf of differentials on $X$ (relative to $Y$), we first consider the diagonal map $\Delta: X \to X\times_Y X$ and then define $...
Brian's user avatar
  • 1,502
9 votes
1 answer
600 views

Simple examples of colimits of affine schemes (evaluated in the presheaf category) which are not affine schemes

Notation and Setting: let $\operatorname{Aff}$ denote the category of affine schemes whose objects are covariant representable functors $\operatorname{X}:\operatorname{Ring}\rightarrow\operatorname{...
sagirot's user avatar
  • 455
8 votes
1 answer
720 views

What about schemes built up out of graded rings?

Toen-Vaquié construct a category of schemes relative to some complete cocomplete closed symmetric monoidal category $C$. Affine schemes correspond by definition 1:1 to commutative monoid objects in $C$...
Martin Brandenburg's user avatar
8 votes
1 answer
424 views

Spin structures on schemes

This is a very naive question, but I have been wondering about the role of spin geometry and spinor structures in the context of algebraic geometry. I know the definition of spin structures and ...
Bilateral's user avatar
  • 3,064
7 votes
1 answer
876 views

Are relative curves $X \to S$ determined by their fibers?

Consider relative curves $X \to S$, defined to be flat, integral, projective schemes of relative dimension 1 over $S$. When are these objects determined by their fibers? So if $X,Y$ are $S$-schemes ...
PrimeRibeyeDeal's user avatar
7 votes
0 answers
279 views

Weil homotopy theory

In algebraic geometry, we have something called Weil cohomology theories, which formalize the notion of a "good" cohomology theory of smooth projective varieties. I believe that for $l$-adic ...
user avatar
7 votes
1 answer
595 views

Does Grothendieck's algebraization imply existence of colimits of schemes?

I am a little bit confused about two lemmas regarding Grothendieck's algebraization. Assume all algebras are defined over some field. Here is the short version of my question: Does Tag 09ZT ("...
user127776's user avatar
  • 5,831
6 votes
2 answers
309 views

How to understand the "boundary" of subscheme, as defined in "An elementary characterisation of Krull dimension"

In An elementary characterisation of Krull dimension and A short proof for the Krull dimension of a polynomial ring, Coquand, Lombardi, and Roy give an elementary characterization of Krull dimension, ...
Somatic Custard's user avatar
6 votes
2 answers
1k views

The underlying space of a scheme remembers its affineness?

Let $f:X\rightarrow Y$ be a morphism of schemes. We know that if $Y$ is affine and $f$ induces homeomorphism on the underlying spaces then $X$ is affine. Is it true that if $X$ is affine and $f$ ...
user avatar
6 votes
1 answer
301 views

Irreducible of finite Krull dimension implies quasi-compact?

Let $X$ be the underlying space of a scheme. If $X$ is irreducible of finite Krull dimension, is it necessarily quasi-compact? Is it necessarily Noetherian? What if we assume not only that Krull ...
schematic_ftm's user avatar
5 votes
0 answers
250 views

Colimit of nilpotent thickenings in the category of schemes

This question is highly related to this and this one. Given a ring $A$ and an ideal $I$, the direct system of schemes $\text{Spec}(A/I)\rightarrow \text{Spec}(A/I^2)\rightarrow \ldots$ has a colimit ...
user127776's user avatar
  • 5,831
5 votes
0 answers
1k views

Given a morphism of schemes, when does bijective + isomorphic tangent spaces = isomorphism?

Let $f: X \to Y$ be a morphism of schemes over a field $k$ such that $f$ induces (1) a bijection between their closed points, and (2) an isomorphism of their Zariski tangent spaces. Under these ...
user48134's user avatar