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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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-...
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 (...
60 votes
1 answer
6k views

Why "open immersion" rather than "open embedding"?

When topologists speak of an "immersion", they are quite deliberately describing something that is not necessarily an "embedding." But I cannot think of any use of the word "embedding" in algebraic ...
53 votes
2 answers
4k views

Connections between various generalized algebraic geometries (Toen-Vaquié, Durov, Diers, Lurie)?

As far as I know, there are four possible ways to generalize algebraic geometry by 'simply' replacing the basic category of rings with something similar but more general: $\bullet$ In the approach by ...
Martin Brandenburg's user avatar
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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46 votes
0 answers
2k views

Mikhalkin's tropical schemes versus Durov's tropical schemes

In Mikhalkin's unfinished draft book on tropical geometry, (available here) (page 26) he defines a notion of tropical schemes. It seems to me that this definition is not just a wholesale adaptation of ...
Jeffrey Giansiracusa's user avatar
44 votes
1 answer
2k views

Useful, non-trivial general theorems about morphisms of schemes

I apologize in advance as this is not a research level question but rather one which could benefit from expert attention but is potentially useful mainly to novice mathematicians. I'm trying to ...
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
40 votes
1 answer
4k views

A closed subscheme of an open subscheme that is not an open subscheme of a closed subscheme?

A morphism $f: V \rightarrow X$ of schemes is a locally closed immersion if it can be factored into a closed immersion followed by an open immersion. It is not hard to show that if $f$ is an open ...
Ravi Vakil's user avatar
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39 votes
6 answers
8k views

What is the inverse image sheaf necessary for in algebraic geometry?

Given a continuous map $f \colon X \to Y$ of topological spaces, and a sheaf $\mathcal{F}$ on $Y$, the inverse image sheaf $f^{-1}\mathcal{F}$ on $X$ is the sheafification of the presheaf $$U \mapsto \...
Charles Staats's user avatar
38 votes
2 answers
3k views

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 ...
user avatar
35 votes
2 answers
3k views

Building algebraic geometry without prime ideals

$\DeclareMathOperator\Spec{Spec}\DeclareMathOperator\ev{ev}$Teaching algebraic geometry, in particular schemes, I am struggling to provide intuitive proofs. In particular, I find it counter-intuitive ...
Anton Mellit's user avatar
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33 votes
6 answers
7k views

How much of scheme theory can you visualize?

I am just starting to learn about schemes and algebraic geometry in general, but I am finding it very hard to visualize things. For example, affine schemes that look like varieties are easily ...
user3860's user avatar
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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
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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31 votes
4 answers
5k views

The Frobenius morphism

I found the following list on the "Frobenius Page" by David Ben-Zvi, described by the author as "an outdated collection of intuitive ways to think about raising to the p-th power". Generates a ...
29 votes
3 answers
3k views

A book on locally ringed spaces?

Are there enough interesting results that hold for general locally ringed spaces for a book to have been written? If there are, do you know of a book? If you do, pelase post it, one per answer and a ...
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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26 votes
1 answer
4k views

Affine scheme on spec(A) of a ring A as the sheafification of a pre-sheave on spec(A)?

It is obvious that there is a parallel between the definition of structure sheaf of $\operatorname{Spec}(A)$ versus the sheafification of a pre-sheaf. The definition of the sheaf $\mathscr F^+$ ...
urelement's user avatar
  • 363
26 votes
1 answer
5k views

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
25 votes
2 answers
2k views

Hodge theory (after Deligne)

In an interview with Deligne on the Simons Foundation website, I heard Robert MacPherson say that at the time Deligne's papers on Hodge theory were being published, the results seemed absolutely ...
THC'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
5 answers
6k views

Reduced scheme and closed points

In The Geometry of Schemes by Eisenbud and Harris, Exercise I-32 asks one to show that a scheme $X$ is reduced if and only if every local ring $\mathcal{O}_{X,p}$ is reduced for closed points $p \in X$...
brunoh's user avatar
  • 1,136
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
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
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
  • 4,919
24 votes
6 answers
4k views

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

By applied algebraic geometry, I don't mean applications of algebraic geometry to pure mathematics or super-pure theoretical physics. Not number theory, representation theory, algebraic topology,...
No One's user avatar
  • 1,545
23 votes
2 answers
2k views

Does Zariski's Main Theorem come with a canonical factorization?

Zariski's Main Theorem (EGA IV, Thm 8.12.6): Suppose $Y$ is a quasi-compact and quasi-separated scheme, and $f:X\to Y$ is quasi-finite, separated, and finitely presented. Then $f$ factors as $X\...
Anton Geraschenko'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
21 votes
1 answer
2k views

Are all formal schemes *really* Ind-schemes?

$\newcommand\LRS{\mathsf{LRS}}\newcommand\FormalSch{\mathsf{FormalSch}}\DeclareMathOperator\Spf{Spf}\newcommand\IndSch{\mathsf{IndSch}}\newcommand\ALRS{\mathsf{ALRS}}\newcommand\FSch{\mathsf{FSch}}$I'...
Saal Hardali's user avatar
  • 7,549
20 votes
3 answers
6k views

Closed vs Rational Points on Schemes

Background: When Ueno builds the fully faithful functor from Var/k to Sch/k he mentions that the variety $V$ can be identified with the rational points of $t(V)$ over $k$. I know how to prove this on ...
Matt's user avatar
  • 970
20 votes
1 answer
1k views

Functorial characterization of open subschemes?

Given a morphism of schemes f: U → X, can one determine when f is an isomorphism of U onto an open subscheme of X in terms of some induced functors between the categories of quasicoherent modules ...
Manny Reyes's user avatar
  • 5,102
20 votes
1 answer
819 views

Why would one "attempt" to define points of a motive as $\operatorname{Ext}^1(\mathbb{Q}(0),M)$?

I'm a novice when it comes to motives. (I've read multiple introductory texts.) I'm attempting to read Galois Theory and Diophantine geometry by Minhyong Kim. In it, he says that "One might attempt, ...
Quinlan Aktaş's user avatar
20 votes
1 answer
1k views

Does every relative curve have a Picard scheme?

More precisely: Let $X \to S$ be a smooth proper morphism of schemes such that the geometric fibers are integral curves of genus $g$. Must the fppf relative Picard functor $\operatorname{\bf ...
Bjorn Poonen's user avatar
  • 23.6k
18 votes
4 answers
2k views

What are the Benefits of Using Algebraic Spaces over Schemes?

I have heard that algebraic spaces have better formal properties than schemes. What are these benefits? Also, is there a natural way to go straight from affine schemes to algebraic spaces bypassing ...
Dinakar Muthiah's user avatar
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
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
3 answers
2k views

Can $\mathcal O_X$ be recognized abstract-nonsensically?

This question has been asked by Teimuraz Pirashvili many years ago. I forgot about it after a while and remembered only now by accident. He probably knows the answer by now, but I still don't. In the ...
მამუკა ჯიბლაძე's user avatar
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
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
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
  • 22.7k
18 votes
0 answers
2k views

History of the functor of points

Until now, I thought the functor of points approach was introduced by Grothendieck at the 1973 Buffalo seminar. However, in this note by Lawvere the author writes: "I myself had learned the ...
Arrow's user avatar
  • 10.3k
17 votes
1 answer
936 views

Is a direct sum of flabby sheaves flabby?

Consider a family of flabby (= flasque) sheaves $(\mathcal F_i)_{i\in I}$ of abelian groups on the topological space $X$. My question : is their direct sum sheaf $\mathcal F=\oplus _{i\in I} \mathcal ...
Georges Elencwajg'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
17 votes
1 answer
3k views

A book on elliptic curves using scheme theory?

I'm interested in learning some stuff about elliptic curves. I've been learning scheme theory, and I'm interested in seeing these tools "in action". It seems that the standard introduction to elliptic ...
Alex Mathers's user avatar
17 votes
2 answers
1k views

Construction of the petit Zariski topos out of the gros topos of a scheme

Let S be a scheme. Let (Sch/S) be a small category of schemes over S (including essentially all finitely presented schemes affine over S). Let E = (Sch/S)zar denote the gros Zariski topos with its ...
Marc Nieper-Wißkirchen's user avatar
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
16 votes
1 answer
1k views

Classical algebraic varieties VS $k$-schemes VS schemes

We know that there is an equivalence of categories between the two following categories: $1)$ Classical varieties over $k$, where $k$ is an algebraically closed field. (Informally I mean locally ...
Dubious's user avatar
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