All Questions
1,159 questions
3
votes
0
answers
218
views
Is there a proper smooth variety in characteristic $p$ whose Hodge-to-de Rham spectral sequence does not degenerate at $E_1$?
By Deligne-Illusie, such a variety has no lifting to $W_2(k)$. In their paper they state that they do not know if such an example exists. Has this question been answered since then?
- 4,164
11
votes
2
answers
1k
views
Representations of $\mathrm{SL}(2)$ in characteristic 2
$\DeclareMathOperator\SL{SL}$In characteristic zero one can use the Clebsch-Gordan rule to decompose tensor products of $\SL(2)$-modules. In characteristic $p$, things are more complicated.
I am ...
- 605
11
votes
2
answers
1k
views
Cotangent complex of perfect algebra over a perfect field
Let $A$ be a perfect $\kappa$-algebra over a perfect field $\kappa$ of positive characteristic $p$. Then the algebraic (= classical) cotangent complex $L_{A/\kappa}^{\operatorname{alg}}$ is known to ...
- 2,011
3
votes
1
answer
251
views
Do Neron-Severi groups of smooth projective unirational varieties contain $\ell$-torsion?
Let $X$ be a smooth projective unirational variety over an algebraically closed field of characteristic $p>0$, and $\ell\neq p$ a prime. My question: can the Neron-Severi group of $X$ contain (non-...
- 16.9k
4
votes
0
answers
233
views
Gluing two affine schemes along a different intersection
Given a quasi-projective $X$ variety that is the union of two affines $\text{Spec(A)}$, $\text{Spec(B)}$ with intersection $\text{Spec}(C)$. Let $f\in C$. Then $\text{Spec}(C_f)$ is an open in both of ...
- 5,901
3
votes
1
answer
405
views
Krull dimension of the ring of global sections
Let $X$ be an irreducible scheme. Can the Krull dimension of $\mathcal{O}_X(X)$ exceed that of $X$?
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 ...
2
votes
1
answer
350
views
Base change of normalization map and scheme-theoretic surjectivity
Let $C$ be an affine, integral curve and $f: \widetilde{C} \to C$ be its normalization. Let $g:D \to C$ be a finite, affine, surjective morphism (note $D$ need not be reduced, but can assume ...
- 2,032
2
votes
0
answers
207
views
Rank of the top Chow group
Let $X$ be a regular integal scheme of finite type over $\mathbb Z$ and assume that $X$ has dimension $d$. In general it is not known if the Chow groups $CH^q(X)$ ($q$ is the codimension) are finitely ...
- 1,237
7
votes
0
answers
660
views
What is the left adjoint to base change of schemes?
Restriction of Scalars and Functoriality of Presheaves.
Let $\phi\colon R\longrightarrow S$ be a morphism of rings. There is associated to $\phi$ a natural functor from $\mathrm{Alg}_S$ to $\mathrm{...
- 11.8k
0
votes
0
answers
405
views
hypersurface of degree d Hilbert polynomial
I am in trouble to solve part (2) Exercise 1.13 from "Moduli of Curves"
by Harris and Morrison on page 9:
Exercise (1.13)
2) Fix a subscheme $X \subset \mathbb{P}^r$. Show, by taking ...
- 6,016
2
votes
1
answer
330
views
Upper semi-continuity of intersection numbers
Consider a smooth projective morphism of schemes $X \rightarrow S$ with relative dimension $n$ (the application I have in mind is with $S$ = an open subset of $\text{Spec } \mathbb{Z}$) and assume ...
- 100
9
votes
3
answers
2k
views
Characterisation for separable extension of a field
Can someone verify this for me.. or tell me what reference shows me this... is this true:
Let $k$ be a field. Then a field extension $K$ of $k$ is separable over $k$ iff for any field extension $L \...
- 2,275
2
votes
0
answers
658
views
Preserved invariants by a flat family
Let $X, C$ be schemes and $f: X \to C$ be a "flat family". That is $f$ is flat morphism. For sake of simplicity we can say that $f$ is surjective and $C$ is an irreducible curve that "parametrizes" ...
- 6,016
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'...
- 7,799
4
votes
2
answers
669
views
A proper flat family with geometrically reduced fibers
Let $R$ be a Noetherian commutative unital ring. Let $f:X\rightarrow Y=\mathrm{Spec}\,R$ be a proper flat morphism of schemes with geometrically reduced fibers. We want to prove that the induced map $...
- 41
0
votes
0
answers
100
views
Quotient of $\text{Proj}(A)$ by the action of a finite group
Let $X$ be $ \operatorname{Proj}(A)$ for some graded ring A, and let $G$ be a finite group acting on $A$ with morphisms of graded rings; consequently $G$ acts on $X$.
I know I can write $X = \bigcup_{...
- 1
3
votes
1
answer
638
views
Can not tell colimits from limits
Proposition 71 here reads:
Let $X$ be a concentrated scheme and $F$ a quasi-coherent sheaf of modules. The
following are equivalent:
(a) The functor $\mathrm{Hom}(F, −):Qco(X)\rightarrow Ab$ ...
user139957
13
votes
1
answer
554
views
On non-representability of certain hom schemes
Let $k$ be an algebraically closed field. It is well-known that the isom-sheaf Isom$(\mathbb{A}^1_k,\mathbb{A}^1_k)$ is not representable by an algebraic space. (To be clear, the functor Isom$(\mathbb{...
- 131
5
votes
2
answers
1k
views
Map of coherent sheaves inducing isomorphism on the stalks at the generic point
Let $f:X\rightarrow Y$ be a finite morphism between Noetherian integral schemes that is surjective on the underlying topological spaces. Does there exist an integer $n>0$ and a coherent $O_X$-...
user138661
0
votes
0
answers
220
views
Use of flattening stratification (from Nitsure's construction of Hilbert and Quot schemes)
I study Nitin Nitsures paper on the Construction of Hilbert and Quot Schemes and not understand the propetry (F) completely:
In previous chapter (Embedding Quot into Grassmanian) it was
proved that ...
- 6,016
17
votes
2
answers
3k
views
Why is one interested in the mod p reduction of modular curves and Shimura varieties?
Why is one interested in the mod p reduction of modular curves and Shimura varieties?
From an article I learned that this can be used to prove the Eichler-Shimura relation which in turn proves the ...
user19475
7
votes
3
answers
927
views
Lefschetz fixed-point theorem for the Frobenius map
Where can one find a proof of Lefschetz fixed-point theorem for the Frobenius map on elliptic curves over algebraic closures of $F_{p}$ ?
This could immediately follow if their coholomogies (for the ...
- 305
12
votes
1
answer
2k
views
Basic questions about formal schemes
I have some questions related to formal schemes. Essentially I would like to understand how much formal schemes are different from usual schemes. First of all, let me specify the definition I prefere ...
- 445
3
votes
0
answers
348
views
Sections of non-reduced schemes
Let $X$ be an affine, irreducible (complex), generically reduced, scheme containing an embedded point, say at $x \in X$. Suppose further dimension of $X$ is strictly positive (can assume to be one ...
- 2,032
6
votes
2
answers
957
views
An apparent equivalence of the category of affine schemes over $S$ and the category of quasi-coherent $\mathcal{O}_S$-algebras
I had asked something very similar before on math.se (deleted now) but unfortunately it hadn't received a lot of attention. I decided to re-ask here.
Let $S$ be a fixed scheme. Is the following true?
...
- 7,799
11
votes
4
answers
3k
views
What does ramification have to do with separability?
Does ramification have anything to do with inseparability? It feels like an extension of Q in which p ramifies should somehow correspond to an extension of F_p(t). Does totally ramified <--> purely ...
- 15.4k
1
vote
2
answers
531
views
Equivalence between categories of affine schemes over $X$ and representable functors $\operatorname{Points}(x) \to \operatorname{Sets}$
I am reading Strickland - Formal schemes and formal groups. For a functor $X\colon \operatorname{Rings}\to \operatorname{Sets}$, he defines (2.14) the category of $\operatorname{Points}(X)$ in the ...
- 455
5
votes
0
answers
208
views
Use of Flattening Stratification part 2 (Nitsure's construction of Hilbert and Quot schemes)
I study Nitin Nitsure's paper Construction of Hilbert and Quot Schemes (arXiv:math/0504590) and have some problems with the content of imposed universal property (F) in the section "Use of ...
- 6,016
2
votes
0
answers
92
views
A Subfunctor of Quot-functor compatible with pullbacks
Let $X$ be a smooth projective irreducible algebraic curve over field $k$. For $d,r,k,m >0$ the representable Quot scheme $\mathcal {Quot}_X^{r,d}(\mathcal{O}_X(m)^k)$ is given for
any test scheme $...
- 6,016
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 ...
- 15.5k
3
votes
2
answers
582
views
Quasi-compactifying schemes
Let $X$ be a scheme. Does there exist an open immersion $X\rightarrow Y$ with $Y$ quasi-compact?
user138661
2
votes
0
answers
68
views
Foundational question: to nonunitial commutative rings correspond to schemes?
Affine schemes correspond to unitial commutative rings, of course.
Further, let us draw upon the Gelfand correspondence, where commutative $C^*$-algebras are dual to compact hausdorff spaces. If we ...
user30211
5
votes
0
answers
148
views
algebraic de Rham cohomology of toric varieties (reference request)
I haven't been able to find anything workable yet, but I'm looking for a reference on the de Rham cohomology of toric varieties, where as many as possible of the following conditions are handled:
...
- 1,142
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
5
votes
0
answers
195
views
Algebraic Space: Two equivalent constructions
According to Wikipedia
there are two common ways to define algebraic spaces:
they can be defined as either quotients of schemes by étale
equivalence relations,
or as sheaves on a big étale site that ...
- 6,016
-1
votes
1
answer
191
views
Number of distinct scheme structures on a set [closed]
Given a cardinal number $|X|$, how many isomorphism classes of schemes with the cardinality of the set of points equal to $|X|$ are there?
user142033
1
vote
0
answers
411
views
Separable morphism of curves
A proof from Janos Kollar's Lectures on Resolution of Singularities Kollar (p 37) works as follows:
Theorem 1.58 (M. Noether, 1871). Let $k$ be an algebraically closed
field and $C \subset \...
- 6,016
1
vote
0
answers
96
views
Valuation diagram of scheme
Let $f: X\rightarrow Y$ be a morphism of schemes, then if $f$ is quasi-compact, then there exist a valuation ring $A$ and its fraction field $K$ satisfying the following commutative diagram. The proof ...
user169802
-1
votes
1
answer
324
views
property of rational functions on projective curves
I have a couple of question about the proof of Lemma 1.20.5 from Janos Kolloar's Lecture on Resolution of Singularities (page 19):
Lemma 1.20.5 Let $C$ be a reduced, irreducible projective curve (=1D ...
- 6,016
2
votes
0
answers
139
views
Closed map of schemes and Frobenius reciprocity
A map of locales $f : X \rightarrow Y$ is closed if it satisfies the reciprocity relation $f_*(f^*(X) \vee Y) \cong X \vee f_* (Y)$.
How can we express a that a map of schemes $f : X \rightarrow Y$ ...
user30211
4
votes
1
answer
557
views
Is a universally closed monomorphism a closed immersion?
The question is essentially in the title: $f\colon X \rightarrow Y$ is a monomorphism of schemes that is universally closed; does this imply that $f$ is a closed immersion? Any such $f$ is quasi-...
- 2,663
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 ...
3
votes
1
answer
777
views
Proving the representability of a functor that is covered by open subfunctors
I already posted this on math.stackexchange some while ago, but haven't received any answers yet. (https://math.stackexchange.com/questions/3221006/proving-the-representability-of-a-functor-that-is-...
- 215
11
votes
2
answers
2k
views
Idea behind Grothendieck's proof that formally smooth implies flat?
From this answer I learned that Grothendieck proved the following result.
Theorem. Every formally smooth morphism between locally noetherian schemes is flat.
The book Smoothness, Regularity, and ...
- 10.5k
3
votes
0
answers
234
views
Zero section of quasi-coherent bundle
Let $S$ be a scheme and let $\mathcal{E}$ be a quasi-coherent $\mathcal{O}_S$-module. Then we can construct a graded quasi-coherent $\mathcal{O}_S$-algebra $\mathscr{A}:= Sym(\mathcal{E})$ and define ...
- 6,016
19
votes
2
answers
3k
views
Bertini theorems for base-point-free linear systems in positive characteristics
Suppose that $X$ is a smooth algebraic variety over an algebraically closed (uncountable if it helps) field of characteristic $p > 0$. Suppose that $L$ is a line bundle, probably ample or at least ...
- 20.5k
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 (...
1
vote
0
answers
188
views
Affine scheme as algebraic space
We working in the following with Knutson's definition of an algebraic space
(ie via equivalence relation; there is also another equivalent def via
sheaves but let us work here with the following one):
...
- 6,016
5
votes
1
answer
291
views
The Idea of Kroneckerian geometry
Let $X$ be a complex, projective algebraic variety and assume that $X$ has a model $X_0$ over $\mathbb Z$ i.e. $X\cong X_0\times_{\operatorname{Spec }\mathbb Z}\operatorname{Spec }\mathbb C$.
Let's ...