All Questions
1,159 questions
2
votes
1
answer
260
views
Relative affine schemes
I was reading these notes by D. Gaitsgory, and I don't understand a claim he makes about relative affine schemes. Namely, on page 3 he says that if $f: Y \rightarrow X$ is an affine scheme over $X$, ...
- 1,325
9
votes
1
answer
634
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{...
- 455
10
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. ...
- 723
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
9
votes
1
answer
546
views
Showing subgroups with equal Lie algebras are equal
Let $k$ be a field. It might as well be algebraically closed, but I do not want to assume that it has characteristic $0$. I will write "group" for "affine group scheme over $k$", ...
- 12.9k
3
votes
1
answer
278
views
"Covering-flat" part in definition of morphism of sites
Let $(\mathcal{C},\mathcal{I})$ and $(\mathcal{D},\mathcal{J})$ be sites where $\mathcal{C}, \mathcal{D}$ are categories and $\mathcal{I}$ and $\mathcal{J}$ are Grothendieck topologies on $\mathcal{C}...
- 6,023
4
votes
0
answers
262
views
de Rham Bloch-Ogus theory in positive characteristic
In their famous paper Gersten's conjecture and the homology of schemes, one of the results that Bloch and Ogus prove is that the second page of the coniveau spectral sequence for $X$ smooth over a ...
- 2,054
2
votes
0
answers
56
views
Conditions for long exact sequence for line bundles on curve to degenerate?
Let $\varphi:X\to Y$ be a morphism of schemes of relative dimension 1, and $\mathcal{L}' \xrightarrow{g} \mathcal{L}$ an injection of line bundles on $X$.
The sequence
$$0\to \mathcal{L}' \xrightarrow{...
- 1,338
17
votes
2
answers
1k
views
A short proof for simple connectedness of the projective line
The Riemann-Hurwitz formula implies that the projective line $\mathbb{P}^1_K$ over any algebraically closed field $K$ is simply connected (i.e., $\pi_1^{et}(\mathbb{P}^1_K) = 1$; equivalently, if $\...
- 3,362
4
votes
1
answer
279
views
Is quotient of projective scheme over arbitrary base by a finite group also projective
This question probably follows from standard geometric invariant theory. If true I'd to know a reference for it.
Given a projective scheme $X\rightarrow S$ over the base $S$. Let's assume a finite ...
- 5,901
3
votes
1
answer
903
views
Cohomological Brauer group vs classical
Let $X$ be a smooth scheme over $\mathbb{C}$.
A $O_X$-algebra $A$ is called Azumaya algebra on $X$
if locally it's ismorphic to matrix algebra: ie for
every $p \in X$ there exist open $U \subset X$ ...
- 6,016
-1
votes
1
answer
895
views
Restriction of a Cartier divisor
Let $X$ be a surface (so $2$-dimensional proper $k$-scheme)
$D \subset X$ an effective Cartier divisor of $X$ which corresponding to an invertible sheaf $\mathcal{L}=O_X(D)$ and
$C \subset X$ a closed ...
- 6,016
8
votes
0
answers
325
views
Moduli stacks and representability of diagonal by schemes
The answer to my question might very well be standard, but I have had trouble finding the right keywords to search for it, so I apologize if this is something well-known to the experts.
I am learning ...
- 28.2k
8
votes
1
answer
774
views
A variant on characteristic $p$ de Rham cohomology
I was thinking about de Rham cohomology in characteristic $p$, and in particular the recent question about Poincare residues, and I came up with the following construction.
Let $k$ be a perfect field ...
- 156k
1
vote
1
answer
2k
views
Pullback map on global sections surjective
Let $X,Y$ be two irreducible, projective $k$-schemes. $k$ is assumed to be algeraically closed. Consider a dominant morphism $f: X \to Y$ between them which is not an isomorphism!
Let $\mathcal{L}$ ...
- 6,016
10
votes
1
answer
1k
views
Gelfand's trick (Gelfand's lemma) in positive characteristic?
I came across this preprint that claims in Lemma 1.1 that Gelfand's trick (also known as Gelfand's lemma) only works in characteristic zero:
Let $H < G$ be finite groups. Suppose we have an anti-...
- 121
3
votes
0
answers
292
views
Generalizations of Artin–Verdier duality?
Constructible étale abelian sheafs on $Spec\ O_\mathbb K$, for number fields $\mathbb K$, satisfy Artin-Verdier duality. Are there known any algebraic schemes or algebraic stacks, other than $Spec\ O_\...
- 2,390
0
votes
1
answer
188
views
Support of a coherent sheaf over a fiber product scheme
I'm trying to prove the following fact which I don't know if it is true since I am not able to find a counterexample:
Let $X,S$ be two $K$-scheme of finite type with $K$ an algebraically closed field....
- 395
9
votes
1
answer
837
views
Representable diagonal map $\Delta: \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ for DM-Stacks/algebraic spaces
Following the standard definitions of a algebraic space or Deligne–Mumford stack one imposed condition is that the diagonal morphism $\Delta: \mathcal{X} \to \mathcal{X} \times \mathcal{X}$
has to be ...
- 6,016
9
votes
1
answer
1k
views
Picard group and reduced schemes
$\DeclareMathOperator\Pic{Pic}$If $A$ is a ring, then we know that $\Pic(A)=\Pic(A_\text{red})$, but for a scheme $X$ it is false in general.
On the other hand, we have that $\Pic(X)=H^{1}_{et}(X,\...
- 3,472
4
votes
0
answers
189
views
If the Frobenius endomorphism of a characteristic $p$ ring is epimorphic, is it surjective?
MO question 19282 is about integral epimorphisms of commutative rings, and a counterexample is given to surjectivity. What about the case of the Frobenius endomorphism of a commutative, characteristic ...
- 4,492
6
votes
0
answers
113
views
$S_n$-invariant polynomials on the dual of reflection representation
Let $W=S_n$ (the symmetric group) acting on $V=k^n$ via permutation of the indices, where $k$ is an algebraically closed field. Closely related to this are the reflection (sub-)representation $V_0=\{ (...
- 1,336
3
votes
2
answers
415
views
Global regular functions and restriction to the fiber
Let $S$ be a local scheme (spectrum of a local ring) with closed point $s$ and $f:X \to S$ a morphism of schemes.
Under which conditions on $f$ and $S$ is the natural map
$$
H^0(X,\mathcal{O})\otimes ...
user159150
2
votes
0
answers
311
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 ...
- 595
8
votes
0
answers
300
views
Which algebra is $\mathbb{F}_p \otimes_{HH^{\cdot} (\mathbb{F}_p)} \mathbb{F}_p$?
Let $HH^{\cdot}(\mathbb{F}_p)$ be the Hochschild cohomology of $\mathbb{F}_p$ over $\mathbb{Z}$, which as a $E_1$ ring is simply $\mathbb{F}_p[x]$ with $x$ in cohomological degree $2$. Then it's clear ...
- 2,356
3
votes
1
answer
348
views
When do generizations ("generalizations") lift uniquely?
If $f : X \to Y$ is proper, then specializations lift along $f$, and uniquely.
(This means, if $R$ is a discrete valuation ring with fraction field $K$ and I choose a factorization $\text{Spec}K \to ...
- 1,104
8
votes
0
answers
351
views
What is the category of coherent sheaves on a logarithmic scheme?
I try to learn basic things on logarithmic geometry, and in particular I don't find much on the category of coherent sheaves on a logarithmic scheme: is it a notion that makes sense or differ from ...
- 333
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
4
votes
1
answer
700
views
The ring of global sections of a regular scheme
Let $X$ be a Noetherian regular scheme. Is $\mathcal{O}_X(X)$ a regular ring? For affine schemes this is true, see 02IU on the Stacks project.
user141316
2
votes
2
answers
645
views
Supersingular elliptic curves and their automorphisms
If $E$ is a supersingular elliptic curve over a finite field of characteristic $p$, what is known about its automorphism group (i.e the stabilizer of a point in algebraic curves terminology).
Do all ...
7
votes
1
answer
498
views
Weyl algebra as an Azumaya algebra over its centre
Assume that $k$ is an algebraically closed field of positive characteristic $p$. On page 3 (page 6 of the PDF file) of Bezrukavnikov, Mirković, and Rumynin - Localisation of Modules for a semisimple ...
- 111
12
votes
3
answers
4k
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 ...
- 945
2
votes
1
answer
680
views
Why care about Grothendieck topology? [closed]
Noah Schweber said here the following:
Why would you want a notion of sheaf theory for objects more general
than topological spaces? Well, the original motivation (to my
understanding) was to ...
- 6,023
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
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
3
votes
1
answer
349
views
Field extensions in Grothendieck rings
Let $k$ be a field, and consider the Grothendieck ring of $k$-varieties, $K_0(V_k)$. Let $K/k$ and $K'/k$ be field extensions of $k$. We view $\mathrm{Spec}(K)$ and $\mathrm{Spec}(K')$ as $k$-schemes ...
- 4,605
13
votes
0
answers
749
views
Rings whose Frobenius is flat
Let $R$ be a ring of characteristic $p>0$. The (absolute) Frobenius is the map of rings $F_R:R\rightarrow R$ defined by $x\mapsto x^p$.
I am interested in rings for which $F_R$ is flat (hence ...
anon1432
5
votes
0
answers
257
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 ...
- 5,901
1
vote
1
answer
591
views
Galois action on $\overline{k}$-valued points extends to action on $k$-scheme $X$
Let $X$ be $k$ variety or more genrally a $k$-scheme. Denote the algebraic closure of $k$ by $\overline{k}$. it's a fact that $X(\overline{k}):=Hom(\operatorname{Spec} \ \overline{k}, X)$ as set is ...
- 6,016
8
votes
1
answer
2k
views
Why is, for a group scheme of finite type, "smooth" (resp. irreducible) equivalent to "geometrically reduced" (resp. geometrically irreducible)?
I have some questions about two statements from Bosch's "Algebraic Geometry and Commutative Algebra" about algebraic varieties (page 479). Since I still don't have the permission to add images I quote ...
- 6,016
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
19
votes
1
answer
977
views
Lang's Jacobian identity: slicker, elementary proof?
In Jeffrey Lang, A Jacobian identity in positive characteristic, J. Commut. Algebra, Volume 7, Number 3 (2015), pp. 393--409, the following result is proven:
Theorem 1. Let $p$ be a prime. Let $\...
- 33.8k
2
votes
1
answer
587
views
Fixed point scheme definition
I'm sorry if this is a trivial question, but it seems I can't find a clear answer.
I have a finitely generated Poisson algebra $A$, the Poisson scheme $X=Spec(A)$ and an automorphism $g$.
What is ...
- 175
3
votes
1
answer
307
views
How to show analytification functor commutes with forgetful functor?
Let $k$ be a field complete with respect to a non-trivial non-archimedean
absolute value (so that rigid $k$-space makes sense). Denote $K$ a finite field extension of $k$.
Denote $X\rightsquigarrow X^{...
- 452
13
votes
1
answer
542
views
Is there an analogue of projective spaces for proper schemes?
Does there exist a countable set of connected proper smooth $\mathbb{C}$-schemes such that any connected proper smooth $\mathbb{C}$-scheme admits a $\mathbb{C}$-immersion into one of them?
user137671
4
votes
2
answers
460
views
Are Chow groups invariant under universal homeomorphisms?
Let $f:X\to Y$ be a universal homeomorphism of schemes, $R$ a coefficient ring. Which assumptions on $f$ and $R$ are suffient to ensure that the pullback map $f^*$ of $R$-linear Chow groups is ...
- 16.9k
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
1
vote
0
answers
156
views
integral subschems of geometrically integeral schemes
Let $X$ be a geometrically integral surface over a field $k$ and let $C$ be an integral curve on $X$. I want to know whether $C$ is geometrically irreducible over $k$ or not.
- 19
1
vote
0
answers
125
views
Help to use Statistics and algebra books for community [closed]
My father has 2000 statistics and higher algebra books (schaum series etc). Need to use these for community since he passed away (India) kindly guide me
I just need to know if we can donate these ...
- 21
1
vote
0
answers
124
views
Combinatorics of projective planes over commutative rings
An axiomatic projective plane is a point-line incidence structure with the following axioms:
any two distinct points are collinear (via a unique line);
any two distinct lines meet in a unique point;
...
- 4,605