All Questions
1,159 questions
234
votes
16
answers
57k
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-...
182
votes
33
answers
32k
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 (...
79
votes
12
answers
13k
views
Is there a high-concept explanation for why characteristic 2 is special?
The structure of the multiplicative groups of $\mathbb{Z}/p\mathbb{Z}$ or of $\mathbb{Z}_p$ is the same for odd primes, but not for $2.$ Quadratic reciprocity has a uniform statement for odd primes, ...
- 118k
60
votes
1
answer
7k
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 ...
54
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 ...
- 63.1k
52
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 $\mathrm{Spec}(B)\to\mathrm{Spec}(A)$. ...
- 5,407
48
votes
5
answers
15k
views
Algebraically closed fields of positive characteristic
I'm taking introductory algebraic geometry this term, so a lot of the theorems we see in class start with "Let k be an algebraically closed field." One of the things that's annoyed me is that as far ...
- 12.6k
47
votes
2
answers
9k
views
current status of crystalline cohomology?
The great references given on Ilya's question make me wonder about the current status of the many conjectures and open questions in Illusie's survey from 1994 on crystalline cohomology. Obviously (...
47
votes
1
answer
1k
views
Summing infinitely many infinitesimally small variables makes sense in algebra
There is an identity $e^x=\lim_{n\to \infty} (1+x/n)^n$, and I always thought it is a purely analytic statement. But then I discovered its curious interpretation in pure algebra:
Consider the ring of ...
- 3,772
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 ...
- 6,229
45
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 ...
- 47.5k
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 ...
- 3,857
39
votes
6
answers
9k
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 \...
- 7,338
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 ...
user138661
37
votes
4
answers
12k
views
Finite extension of fields with no primitive element
What is an example of a finite field extension which is not generated by a single element?
Background: A finite field extension E of F is generated by a primitive element if and only if there are a ...
- 24.1k
37
votes
3
answers
5k
views
Is there a nice proof of the fact that there are (p-1)/24 supersingular elliptic curves in characteristic p?
If $k$ is a characteristic $p$ field containing a subfield with $p^2$ elements (e.g., an algebraic closure of $\mathbb{F}_p$), then the number of isomorphism classes of supersingular elliptic curves ...
- 45.7k
37
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 ...
- 3,772
35
votes
6
answers
8k
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 ...
- 479
34
votes
2
answers
3k
views
The work of E. Artin and F. K. Schmidt on (what are now called) the Weil conjectures.
I was reading Dieudonne's "On the history of the Weil conjectures" and found two things that surprised me. Dieudonne makes some assertions about the work of Artin and Schmidt which are no doubt ...
- 41.4k
32
votes
10
answers
3k
views
Which 'well-known' algebraic geometric results do not hold in characteristic 2?
A smooth curve $X$ in $\mathbb{P}^n$ is strange if there is a point $p$ which lies on all the tangent lines of $X$.
Examples are $\mathbb{P}^1$ is strange and so is $y=x^2$ in characteristic $2$. ...
- 2,215
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 ...
- 15.1k
31
votes
7
answers
4k
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" ...
- 23.5k
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
4k
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 ...
29
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 "...
28
votes
4
answers
7k
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 $...
- 9,497
28
votes
3
answers
2k
views
Intuitive pictures in characteristic p
This is a tough one, but does anyone know of any images that recall characteristic p geometry (over algebraically closed fields) in some sense? It is not enough if it is some picture that can be also ...
- 2,215
27
votes
7
answers
6k
views
Etale covers of the affine line
In characteristic p there are nontrivial etale covers of the affine line, such as those obtained by adjoining solutions to x^2 + x + f(t) = 0 for f(t) in k[t]. Using an etale cohomology computation ...
- 52.7k
27
votes
2
answers
3k
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 ...
- 28.8k
27
votes
4
answers
3k
views
Have people successfully worked with the full ring of differential operators in characteristic p?
This question is inspired by an earlier one about the possibility of using the full ring of differential operators on a flag variety to develop a theory of localization in characteristic $p$. (Here ...
- 57.6k
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 ...
- 11.8k
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^+$ ...
- 363
26
votes
2
answers
3k
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 ...
- 4,605
26
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 ...
- 1,141
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 ...
- 63.1k
25
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. ...
- 5,039
25
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,...
- 1,565
25
votes
3
answers
2k
views
product of all F_p, p prime
Let $R$ be the ring $$R = \prod_{p\ \text{prime}} \mathbb{F}_p$$ where $\mathbb{F}_p$ is the field having $p$ elements.
Is it true that $R$ has a quotient by a maximal ideal which is a field of ...
- 5,163
25
votes
5
answers
7k
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 $p \in X$...
- 1,128
25
votes
0
answers
1k
views
Status of the Euler characteristic in characteristic p
In the introduction to the Asterisque 82-83 volume on `Caractérisque d'Euler-Poincaré, Verdier writes:
Enfin signalons que la situation en caractéristique positive est loin
d'être aussi ...
- 8,723
24
votes
5
answers
6k
views
Wild Ramification
The question is, loosely put, what is known about wild ramification?
Is there a semi-well-established theory of wild ramification that can be furthered in various specific situations? Or maybe there ...
24
votes
2
answers
2k
views
Have we ever proved any non-solvable case of reciprocity without the Langlands program ?
The reciprocity of the title is the following not completely well-posed problem:
Fix $P(X)$ a monic irreducible polynomial of degree $n$, with coefficients in $\mathbb Z$. "Describe"
(in some sense) ...
- 26.1k
24
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\...
- 24.1k
23
votes
1
answer
2k
views
Wanted: Quadratic Space in Characteristic 2 as a Counterexample to a Theorem of Arf
Hi. Peter Roquette sent me an email asking for an example of a quadratic space in characteristic 2 having certain features. I have no idea on this, but maybe someone reading this does.
He would ...
- 50.6k
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 ...
- 1,329
22
votes
3
answers
2k
views
One dimensional (phi,Gamma)-modules in char p
I would like to better understand the simplest case of the correspondence between Galois representations and (phi,Gamma)-modules. Namely, consider 1-dimensional Galois representations of $G_{Q_p}$ ...
- 1,680
21
votes
2
answers
5k
views
State of resolution in positive characteristic?
Heisuke Hironaka's coming talk makes me wonder how the state of the work on that theme is. So far, I noticed (but didn't read) these papers:
Kawanoue, Hiraku, Toward resolution of singularities over ...
- 10.8k
21
votes
5
answers
5k
views
Mirror symmetry mod p?! ... Physics mod p?!
In his answer to this question, Scott Carnahan mentions "mirror symmetry mod p". What is that?
(Some kind of) Gromov-Witten invariants can be defined for varieties over fields other than $\mathbb{C}$...
- 21k