All Questions
1,159 questions
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
9
votes
1
answer
676
views
What are the rational functions on a noetherian affine scheme?
Let $A$ be a noetherian ring and $X=\operatorname {Spec}A$ the corresponding affine scheme.
There are three rings which might reasonably be called the ring of rational functions on $X$.
a) The total ...
- 47.5k
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 ...
- 5,996
9
votes
1
answer
1k
views
deformation theory in positive characteristic
The idea "Formal deformation theory in characteristic zero is controlled by a differential graded Lie algebra (dgla)" goes back to Goldman-Millson, Deligne, Drinfeld among others; see Lurie's ICM talk....
- 528
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
9
votes
1
answer
315
views
Tube of a mod p point on a smooth Z_(p)-scheme
Let $R$ be a smooth, integral, finite-type $\mathbb{Z}_{(p)}$-algebra of relative dimension $n$ and $\overline{f} \colon R \to \mathbb{F}_p$. Then Hensel's lemma tells us that this lifts to a map $R \...
- 15.4k
9
votes
2
answers
868
views
Nakano vanishing in positive characteristic
Let $X$ be a smooth projective variety defined over a field $k$.
In characteristic zero, the following is a special case of the (Kodaira-Akizuki-)Nakano vanishing theorem:
$(\ast) \quad$ $\mathrm H^...
- 1,072
9
votes
1
answer
833
views
Endomorphism ring of simple ordinary abelian variety
Is there an example of an ordinary and simple abelian variety $A$ over an algebraically closed field $K$ (of characteristic $p>0$) such that ${\rm End}(A)$ is not commutative? Note that the answer ...
- 3,076
9
votes
1
answer
448
views
Torsors trivializing over a fixed finite etale cover
Let $S$ be an integral regular scheme and let $T\to S$ be a finite etale morphism. Let $G$ be a smooth affine finite type group scheme over $S$.
Is the set of $S$-isomorphism classes of $G$-torsors ...
- 151
9
votes
1
answer
943
views
on the local structure of schemes
Let $X$ be an integral finite type scheme over $\mathbb{C}$. Let $x\in X$, such that there exists a neighborhood $U$ of $x$, such that the sheaf of differentials $\Omega^{1}_{U}$ decomposes into:
$\...
- 3,472
9
votes
1
answer
566
views
algorithm for calculating the Chow groups of a variety over a finite field
Is there an algorithm for calculating the Chow groups of a variety over a finite field?
It is know that $H^{2i,i}_\mathrm{mot}(X,\mathbf{Z}) = CH^i(X)$. In how many cases does this help us?
user19475
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
9
votes
1
answer
777
views
Geometric (or intuitive) interpretation of additional derivatives in characteristic p > 0
In characteristic $p > 0$ there are "extra" differential operators, i.e., ones that are outside the algebra generated by first-order derivations.
Is there any interpretation of these operators in ...
- 139
9
votes
1
answer
381
views
Lifting of families of curves to characteristic 0
Let $k$ be a finite field, $X_0$ be a smooth affine variety over $k$ and $C\rightarrow X$ a smooth projective family of curves of genus $\geq 2$.
By a result of Elkik we can always lift $X_0$ to a ...
- 728
9
votes
1
answer
1k
views
Coderivations of $S(V)$ correspond to linear maps $S(V) \to V.$ Only over characteristic $0$?
Definition. Let $k$ be a commutative ring. Let $V$ be a $k$-module. We turn the symmetric algebra $\mathrm{S}\left(V\right)$ of $V$ into a graded Hopf algebra by defining the comultiplication
\begin{...
- 33.8k
9
votes
1
answer
430
views
Existence of certain endomorphism of supersingular elliptic curve
Let $E$ be a supersingular elliptic curve over an algebraically closed field $K$ of characteristic $p$. Let $R = \operatorname{End}(E)$ be its ring of endomorphisms. Then, it is known $R \otimes_{\...
9
votes
1
answer
531
views
Can one check formal smoothness using only one-variable Artin rings?
Let $f:X\rightarrow Y$ be a morphism of schemes over a field $k$. Can one check that $f$ is formally smooth using only Artin rings of the form $k^{\prime}\left[t\right]/t^{n}$, where $k^{\prime}$ is ...
- 10.5k
9
votes
1
answer
356
views
Reference request for indecomposable representations of $\mathfrak{sl}(2)$ over an algebraically closed field of characteristic $p > 0$
Is there a good reference on the classification of indecomposable representations of the Lie algebra $\mathfrak{sl}(2)$ over an algebraically closed field of characteristic $p > 0$ (of course, in ...
- 239
9
votes
1
answer
332
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 ...
- 3,312
9
votes
0
answers
388
views
True on stalks, false on affine opens [closed]
In scheme theory, there are some properties that can be specified purely on the stalks of the structure sheaf but they "lift" to the properties of the values of structure sheaf on affine opens, e.g.
...
user142126
9
votes
0
answers
400
views
Weierstrass division theorem for henselian rings
Let $A$ be an henselian local noetherian ring. There is an old result of Lafon ("Anneaux henséliens et théorème de préparation" (1967)), which says that if $A$ is analytically normal and of ...
- 3,472
9
votes
0
answers
560
views
Function fields of characteristic p modular curves, and mod p reductions of the classical modular equation
Let l and p be distinct primes, l>2. There are "characteristic p modular curves" X_0(l) and X(l), defined over an algebraic closure, K, of Z/p, solving moduli problems for elliptic curves with some ...
- 5,422
9
votes
0
answers
1k
views
Ample vector bundles, $H^1=0$ and global generation in characteristic $p$
This is a follow up from Ample vector bundles on curves question, in characteristic $p>0$. First, some background. It is known that in characteristic $0$, a vector bundle $E$ on say a projective ...
- 2,976
8
votes
2
answers
4k
views
Closed subschemes and pulling back the structure sheaf via the inclusion map
I would just like a clarification related to closed subschemes.
If $(X,{\cal O}_X)$ is a locally ringed space and $A\subset X$ is any subset with the subspace topology then $i^{-1}{\cal O}_X$ will be ...
- 2,814
8
votes
2
answers
8k
views
What does "supersingular" mean?
Are supersingular primes and supersingular elliptic curves related?
(this was essentially a subquestion in my earlier question, but still looks sufficiently different to me to deserve a separate post)...
- 15.1k
8
votes
2
answers
1k
views
K-Theory of Schemes: Monoidal vs. Exact
There are several ways for defining the K-Theory of a category depending on which structure it admits. The K-Theory of schemes is commonly defined as the "group completion" of the category of ...
- 333
8
votes
1
answer
779
views
Connеcted components of irreducible algebraic varieties
I am wondering what is the possible (or maximum) number of connected components for an irreducible algebraic variety in $\mathbb R^n$ defined by a degree $d$ polynomial (i.e. hypersurface) in $\mathbb ...
- 185
8
votes
2
answers
1k
views
Lefschetz on étale fundamental group for quasi-projective varieties
If $X$ is a smooth projective variety of dimension at least $3$ over $\mathbb{C}$, Lefschetz's Hyperplane theorem says that for every hyperplane section $H$
$$\pi^1(H)\to\pi^1(X)$$
is an isomorphism, ...
- 483
8
votes
1
answer
3k
views
Orbits of group scheme action
I am interested in orbits of the action of a group scheme on a scheme and I'm particularly interested in the following special case: Let $k$ be an algebraically closed field, let $G$ be an affine ...
- 83
8
votes
2
answers
814
views
Are there varieties with non finitely generated Picard group and vanishing irregularity?
Let $X$ be a smooth projective variety over an algebraically closed field $k$.
Can it happen that $q(X) := \dim H^1(X,\mathcal O_X) =0$ and $\textrm{Pic} \,X$ is not finitely generated?
Certainly, ...
- 83
8
votes
1
answer
1k
views
obstruction to smooth lifting of smooth schemes
According to general theory, for a square zero thickening defined by an ideal I: SpecA -> SpecA', there is an obstruction of lifting a smooth scheme X over A to a smooth scheme over A' living in H^2(X,...
- 5,052
8
votes
1
answer
472
views
Nonnegative additive functions on coherent sheaves
Let $(X,\mathcal{O}_X)$ be a Noetherian integral scheme and let $g$ be a (numerical) additive nonnegative function from coherent $\mathcal{O}_X$-modules to $[0,\infty)$. This question may be well ...
- 3,101
8
votes
2
answers
910
views
Is this finite surjective flat morphism of 2 dimensional schemes a local complete intersection
Let $X$ be a regular integral noetherian scheme of dimension 2 and let $D$ be a simple normal crossings divisor in $X$.
EDIT: Let $U = X-D$.
Consider a finite etale morphism $V\longrightarrow U$ ...
- 9,497
8
votes
1
answer
387
views
Are schemes obtained from reduced schemes by gluing infinitesimal stuff along a reduced closed subscheme?
Let $X$ be a scheme of finite type (say over the complex numbers). The set of points for which the local ring is reduced is then an open subset $U\subseteq X$.
Is it true that there is a closed ...
- 23.4k
8
votes
1
answer
741
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 $C$...
- 63.1k
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 ...
- 5,996
8
votes
1
answer
603
views
absolute Galois group of the function field of a curve over $\mathbb{F}_p^{alg}$
In their 2008 paper "Torelli theorem for curves over finite fields" Bogomolov, Korotiaev and Tschinkel mention in the beginning of Section 9 that absolute Galois groups of curves over $\mathbb{F}_p^{...
- 4,070
8
votes
1
answer
943
views
Automorphisms over finite field that do not lift to an automorphism in characteristic zero
My main question is the following: is there an automorphism of the affine space $\mathbb{A}^n$ (automorphism of an algebraic variety) defined over a finite field which does not lift to an automorphism ...
- 7,722
8
votes
2
answers
500
views
Double points in the Grothendieck ring
Let $K_0(V_k)$ be the Grothendieck ring of $k$-varieties, and consider the scheme $X = \mathrm{Spec}(k[x]/(x^2))$.
I understand that this scheme has one point, but I am missing the fact that in $K_0(...
- 4,605
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
8
votes
1
answer
331
views
If C is a fusion category over a field of nonzero characteristic and dim C = 0, is Z(C) ever fusion?
If $C$ is a fusion category and $\dim(C) \neq 0$ (the latter is automatic in characteristic zero, but not in nonzero characteristic), then the Drinfel'd center $Z(C)$ is fusion. More generally, if $C$...
- 28.1k
8
votes
2
answers
2k
views
Functorial characterization of morphisms of schemes
This question is akin in spirit to this one:
Functorial characterization of open subschemes?
In the above MO question, a "functorial" characterization is given for closed immersions and open ...
- 1,503
8
votes
1
answer
808
views
Automorphisms of curves in positive characteristic
It is well known that over an algebraically closed field of characteristic zero a general curve (for an open subset of $M_g$) of genus $g\geq 3$ is automorphism-free.
Is this result still true over ...
user68440
8
votes
3
answers
570
views
Variations on a theme of O'Bryant, Cooper and Eichhorn concerning power series over $\mathbb Z/2\mathbb Z$
Define 2 power series over the field $\mathbb Z/2\mathbb Z$ by $f=1+x+x^3+x^6+\dots$, the exponents being the triangular numbers, and $g=1+x+x^4+x^9+\dots$, the exponents being the squares. Write $f/g$...
- 5,422
8
votes
1
answer
342
views
The Grothendieck topology of closed immersions on schemes
Let $S$ be a scheme. Let's define a Grothendick topology on $\mathrm{Sch}/S$ where a covering family $\{f_i:Z_i\rightarrow X\}_{i\in I}$ on an $S$-scheme $X$ is a collection of closed immersions of $S$...
- 329
8
votes
1
answer
436
views
Presentation for a Finite Etale Cover of an (Affine) Elliptic Curve
I posted this question on MSE a few days ago, but I did not get much interest in it. So I thought I would try my luck here. If you are interested in answering the question, there is a bounty over on ...
- 181
8
votes
1
answer
483
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 ...
- 2,818
8
votes
1
answer
982
views
Is there a really big ring of differential operators in characteristic p?
$k$ is a field of characteristic $p$.
$k[t]$ has canonical first-order differential operator $\partial$
As an endomorphism of $k[t]$, $\partial^p=0$.
First way to fix it:
Use the divided power ...
8
votes
1
answer
1k
views
Proper morphism sending coherent to coherent
Hello,
Is there a proof that the push forward by a proper morphism of Noetherian schemes sends coherent sheaves to coherent ones, without passing in the argument through projective morphisms?
Thank ...
- 5,562
8
votes
1
answer
536
views
Voisin examples in $p$-adic geometry
Let $K$ be an algebraic closure of p-adic rationals. Does there exist a proper smooth rigid-analytic variety over $K$ whose etale homotopy type is not isomorphic to etale homotopy type of a proper ...
user137712