All Questions
1,159 questions
2
votes
1
answer
234
views
Glueing modules over $\{x\}\times \operatorname{Spec} R$
Let $k$ be a field and $(C,\mathcal{O}_C)$ be a smooth geometrically irreducible projective curve over $k$ of function field $k(C)$ and let $x$ be a closed point on it. From Laszlo-Beauville's lemma, ...
- 1,479
7
votes
2
answers
1k
views
Hochschild Kostant Rosenberg theorem for varieties in positive characteristic?
Is there is a known version of the HKR theorem as proved in say Swan's paper "Hochschild Cohomology of Quasiprojective Varieties" in positive characteristic? I assume something is known about this as ...
- 3,758
3
votes
0
answers
151
views
Other interesting notions when we change topology on $\text{Sch}/S$
Let $\text{Sch}$ be the category of schemes. Let $S$ be an object of $\text{Sch}$. Consider the category $\text{Sch}/S$.
Some interesting topologies on $\text{Sch}/S$ are Zariski, fpqc, étale, fppf.....
- 6,023
4
votes
1
answer
334
views
Affine open with irreducible complement
Let $X$ be an integral Noetherian separated scheme. Under what conditions can we find a non-empty affine open in $X$ whose complement is irreducible?
user139859
1
vote
0
answers
890
views
Cohomology of the pullback of a coherent sheaf (considered as an abelian sheaf)
Let $X$ be an affine Noetherian scheme, $Y$ a separated Noetherian scheme, $f:X\rightarrow Y$ a morphism of schemes inducing a homeomorphism on the underlying topological spaces.
Let $F$ be a ...
user137767
2
votes
1
answer
451
views
Tie-Breaking Trick for Log Canonical Pairs and F-pure pairs in Positive Characteristic
Let $X$ be a projective 3-fold in characteristic $p>0$. Let $(X, D)$ be a klt pair, and $D'$ a $\mathbb{R}$-Cartier divisor such that $D'=A'+B'$, where $A\geq 0$ is an ample $\mathbb{Q}$-divisor ...
- 313
1
vote
0
answers
246
views
Frobenius twist of a field
Let $k$ be a field of characteristic $p>0$ (not necessarily perfect). Consider the Frobenius endomorphism $F : k \to k$, $x \mapsto x^p$. I am curious about what happens when we take $k$ as a $k$-...
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
10
votes
3
answers
2k
views
Pullback along Frobenius morphism
Let $X$ be a scheme over a finite field $\mathbb{F}_q$ and let $F : X \to X$ be the absolute Frobenius morphism. If $\mathcal{L}$ is an invertible $\mathcal{O}_X$-module, then there is a natural ...
- 63.1k
3
votes
1
answer
484
views
Harish-Chandra isomorphism for characteristic $p$
I am trying to understand the proof of Theorem 1 from this paper V. Kac and B. Weisfeiler (Indag. Math. 1976, DOI link).
Theorem 1. Let either $p\neq 2$ or $\varrho\in X(\mathscr{T})$. Then $\gamma(...
- 385
2
votes
1
answer
495
views
Linear systems over non-algebraically closed field
Let $X$ be a regular, irreducible projective scheme, of finite type over an arbitrary field $k$. Both Weil divisors and Cartier divisors are defined on $X$ and naturally correspond. If $k$ is ...
- 3,555
7
votes
0
answers
387
views
Algebraic geometry "over the function field" of the base
This is vaguely similar to, but quite different from, this question.
In the above linked question the focus is on fields of large cardinality per se. Here we fix a base field $k$ (say algebraically ...
- 23.4k
0
votes
0
answers
221
views
Fiber product of singular varieties
Let $f\colon X\to Y$ be a morphism between irreducible $\mathbb{C}$-varieties, such that $f$ is 1-1 and surjective, but $df$ fails to be injective along a subvariety $Z$. (This forces $Y$ to be ...
- 287
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
1
vote
0
answers
189
views
Relate Codimensions of Integral Schemes and their Generic Fibers
I have a question about an argument in the proof of Thm 8.2.5 (Dimension formula) from Liu's "Algebraic Geometry" (page 334):
Firstly the problem is reduced to the affine case with $Y:=Spec \text{ }\...
- 6,016
6
votes
1
answer
193
views
Restricted Lie algebras with no nonzero proper restricted subalgebras
Let $L\neq 0$ be a restricted Lie algebra over a field $F$ of characteristic $p>0$. If $F$ is algebraically closed, then it is known that $L$ has no nontrivial restricted subalgebras if and only ...
- 630
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_{\...
0
votes
0
answers
195
views
Hyperelliptic Curve (Liu's Book)
Let $Y:=\mathbb{P}^1_k$ and $X$ a hyperelliptic curve. We working with the definition 4.7 from Liu's "Algebraic Geometry and Arithmetic Curves" (page 288)
Namely there exist finite separable map $\pi:...
- 6,016
1
vote
0
answers
149
views
Formula for fibre square (from Fulton's Intersection Theory)
I have a question about argumentation used in the proof of Proposition 1.7 in Fulton's book on Intersection Theory on page 18:
Proposition 1.7 Let
$\require{AMScd}$
\begin{CD}
X' @>{g'}>> ...
- 6,016
4
votes
1
answer
260
views
Embedding a finite morphism into a finite morphism of smooth varieties
Let $f\colon X\to Y$ be a finite morphism of quasiprojective varieties (I’m most interested in the case that $f$ is normalization and the varieties are over $\Bbb C$). Is the following statement true (...
- 3,079
0
votes
0
answers
325
views
Grothendieck topology on a scheme equivalent to the circle
Suppose a class of morphisms of schemes is reasonable enough so that we can associate a small site to any scheme (just like small étale site). Is there a simple class of morphisms such that there is a ...
user138661
3
votes
0
answers
159
views
$G_0(X) \cong G_0(X_{red})$ where X is a noetherian scheme
Let $\textbf {X}$ be a noetherian scheme,
$\textbf {M(X)}$ be the categroy of coherent sheaf over the scheme $\textbf {X}$.
We denote $ \textbf {K$_0$(M(X))}$ to be $ \textbf {G$_0$(X)}$.
Now I ...
user139827
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_\...
- 9,497
1
vote
0
answers
157
views
Application of Stein factorisation: rigidity lemma
Let $X,Y$ Noetherian schemes and $f:X \to Y$ proper map. The Stein factorisation factorizes $f$ as $X \xrightarrow{g} Spec \text{ } f_* \mathcal{O}_X \xrightarrow{h} Y$ with $h$ finite and $g$ has ...
- 6,016
4
votes
1
answer
180
views
When can a scheme be recovered from its descent groupoid?
Suppose that $ Y $ is a scheme and $ f\colon X\to Y $ a covering of $ Y $ in some Grothendieck topology on the category of schemes (i.e. if $\{ U_i\to Y\}$ is a covering in the topological sense, then ...
- 295
2
votes
0
answers
139
views
Formally smooth maps of schemes and factorization systems
I am thinking about how formally smooth maps of schemes relate to factorization systems.
Let $C$ be the category of schemes. Let $E$ be the class of morphisms of schemes consisting of closed ...
user30211
1
vote
0
answers
160
views
Riemann–Hurwitz Formula for Normal Projective Curves
My question refers to the proof of Theorem 7.4.16 from Liu's "Algebraic Geometry" at page 290 :
QUESTION: Why does $e'_x= length_{\hat{B}}(W_{\hat{B}/\hat{A}}/\hat{B})$ hold?
During the proof we ...
- 6,016
1
vote
2
answers
636
views
Is finite union of locally closed subscheme, a scheme
Let $X$ be a projective, noetherian $k$-scheme for an algebraically closed field $k$ of characteristic zero. Let $Y_1,...,Y_r$ be locally closed subschemes (open subschemes of closed subschemes) of $X$...
- 1,981
3
votes
0
answers
240
views
Smallest non-vanishing index of Local-Cohomology and Ext functor, as in the definition of depth, for not necessarily affine schemes
Let $(Z,\mathcal O_Z)$ be a closed subscheme of a Noetherian scheme $(X,\mathcal O_X)$. Then there is an ideal sheaf $\mathcal J$ on $X$ such that $i_*(\mathcal O_Z) \cong \mathcal O_X/\mathcal J$ , ...
- 642
3
votes
0
answers
143
views
What kind of equivalences exist between categories of characteristic $0$ and characteristic $p$?
The tilting equivalence for perfectoid algebras gives an equivalence of categories $$K\text{-perf} \cong K^\flat\text{-perf}$$
where the left-hand-side are algebras in characteristic zero and the ...
- 4,164
21
votes
4
answers
2k
views
Simplest example of jumping of cohomology of structure sheaf in smooth families?
Using Hodge theory (and the ill-defined Lefschetz principle), one can show that in characteristic 0, given a proper smooth family $X \rightarrow B$, the cohomology groups of the structure sheaf of the ...
- 3,857
4
votes
0
answers
235
views
Krull dimension of schemes locally of finite type over PID
Let $R$ be a commutative unital ring that is a PID. Assume that $R$ is not a DVR. Let $X$ be an integral scheme locally of finite type over $\mathrm{Spec}\,R$. Can the Krull dimension of $\mathcal{O}...
user138661
4
votes
2
answers
340
views
On families of supersingular abelian surfaces over the projective line
Let $k=\mathbb{F}_q$. I recently learned that there are non-isotrivial families $f:X\to \mathbb{P}^1_k$ of supersingular abelian surfaces. In particular, the Kodaira-Spencer map of this family is non-...
- 43
16
votes
2
answers
1k
views
Is the tangent space functor from PD formal groups to Lie algebras an equivalence?
The previous version of this question was rather badly broken, and I hope this version makes some sense.
There have been a few questions on MathOverflow about how much representation-theoretic ...
- 45.7k
3
votes
0
answers
352
views
Irreducible Smooth Proper one-dimensional Schemes isomorphic to $\mathbb{P}^1$
I have a curious question about an argument/hint given in following thread:
https://math.stackexchange.com/questions/3062986/irreducible-smooth-proper-one-dimensional-mathbbz-schemes
The OP asked if ...
- 6,016
4
votes
1
answer
861
views
Properties of log smooth schemes
Let $k$ be a field and $M$ be sharp monoid (with no invertible element) consider the log point $\eta_M=(\operatorname{Spec}(k), M)$. Let $X$ be a fine saturated scheme over $\eta_M$ such that the ...
- 171
1
vote
1
answer
199
views
Projective subvarieties of blow-ups of affine varieties
Let $X$ be an integral affine scheme of finite type over a field $k$. Let $Y\subset X$ be an integral closed subscheme of codimension $n>0$. We blow up $X$ along $Y$ and get a $k$-scheme $X'$. Is ...
- 231
5
votes
0
answers
241
views
Schemes monomorphing into affine scheme of dimension 1
Let $Y$ be an affine scheme of Krull dimension 1. Let $X\rightarrow Y$ be a monomorphism in the category of schemes. If $X$ is connected, is $X$ necessarily affine? What if we assume that $Y$ is a ...
user137767
7
votes
0
answers
296
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 ...
user137676
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
2
votes
1
answer
169
views
Embedding smooth proper schemes into smooth proper schemes
Do there exist connected proper smooth $\mathbb{C}$-schemes $X_i$ ($\forall i\in \mathbb{Z}_{>0}$) with $\mathrm{dim}_{\mathbb{C}}X_i=i$ such that $X_i$ admits an immersion into $X_{i+1}$ and any ...
user74900
1
vote
1
answer
317
views
Subschemes of the affine line over PID
Let $R$ be a PID with infinitely many prime ideals. Suppose we have two integral locally closed subschemes of $\mathrm{Spec}\,R[x]$ such that
both have non-empty intersection with the affine open $\...
user137767
18
votes
3
answers
3k
views
Lifting varieties to characteristic zero.
If you want to compute crystalline cohomology of a smooth proper variety $X$ over a perfect field $k$ of characteristic $p$, the first thing you might want to try is to lift $X$ to the Witt ring $W_k$ ...
- 4,015
2
votes
0
answers
226
views
Does the structure morphism matter in GAGA?
Let $X$ be an integral Noetherian separated scheme. Can there exist two morphisms of finite type $X\rightarrow \mathrm{Spec}\,\mathbb{C}$ such that the corresponding complex-analytic spaces are not ...
user140820
5
votes
0
answers
537
views
Basic questions about crystals and Grothendieck connections
I have a few basic questions about Grothendieck connections and crystals. I know it's bad practice to ask a bunch of different questions at once, however I feel they naturally come bundled together. ...
- 10.5k
4
votes
0
answers
169
views
Fibered surfaces degenerating to Frobenius
Let $R$ be a DVR with algebraically closed, positive characteristic residue field $k$. Let $X\rightarrow Spec(R)$ and $C\rightarrow Spec(R)$ be smooth projective morphisms of relative dimension 2 and ...
- 599
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 ...
- 970
5
votes
0
answers
217
views
Exact differential forms in characteristic $p>0$
Let $k$ be an algebraically closed field of characteristic $p>0$. Suppose $1< e_i <p$ for $i=1,2, \ldots, n$ are integers ($n \ge 2$). What are the conditions on the $e_i$'s so that the ...
- 245
1
vote
0
answers
223
views
Extend a Morphism of Schemes
I have a question about following argument used in Szamuely's "Galois Groups and Fundamental Groups" in the excerpt below (or look up at page 159):
Let $X,Y$ schemes which are finite and locally free ...
- 6,016
1
vote
0
answers
298
views
Fully faithful functor from schemes to spaces
Is there a fully faithful functor from the category of schemes
to the category of topological spaces and continuous maps (or some other sufficiently topological objects, e.g. smooth manifolds and ...
user137517