All Questions
1,159 questions
2
votes
0
answers
345
views
Examples of semi-stable models of curves
Let $R$ be a discrete valuation ring with fraction field $K$ of characteristic zero and residue field $k$ of characteristic $p>0$. Assume $k$ is algebraically closed. I want to produce examples of ...
- 2,323
2
votes
0
answers
148
views
Purely inseparable $k$-rational dominant maps between an absolutely irreducible $k$-surface and $\mathbb{P}^2$
Let $X$ be an absolutely irreducible surface over an algebraically closed or finite field $k$ of characteristic $p$. Is it true that the following conditions are equivalent?
There is a purely ...
- 1,833
2
votes
0
answers
117
views
representability of some mapping stack
Let $S$ be an Artin stack of finite type.
We assume that it contains a point as an open dense.
Is it always true that the mapping stack:
$Hom^{0}(\mathbb{P}^{1},S)$
which consists of sections ...
- 3,472
2
votes
0
answers
476
views
Does the numerical equivalence relation coincide with the homological one for 1-cycles (in positive characteristic)?
Is the Grothendieck's Standard Conjecture D (stating that the numerical equivalence relation for algebraic cycles with rational coefficients coincides with the homological one) known to be true for ...
- 16.9k
2
votes
0
answers
118
views
extending local systems on a neighbourhood
Let $Y$ an affine finite type scheme over an algebraically closed field $k$.
Let $S$ be a closed subscheme of $Y$ and $Y'$ the henselization of $Y$ along $S$.
If we have a $\mathbb{Z}_{\ell}$ local ...
- 3,472
2
votes
0
answers
212
views
Description of the equalizer of $\prod _j R/I_j \rightrightarrows \prod _{i,j}R/(I_i+I_j)$
This is a crosspost of this MSE question.
I have asked several questions in an attmept to get a general version of the Chinese remainder theorem without conditions on the ideals which will trivially ...
- 10.5k
2
votes
0
answers
394
views
Blow up along a section of a smooth morphism
Let $C$ be a ground locally notherian and quasiprojective scheme. Let $\pi:S\to B$ be a $C$-morphism of finite type $C$-schemes. We call $\pi$ a $C$-smooth morphism if the morphisms $S\to C$ and $B\to ...
- 43
2
votes
0
answers
416
views
In how many ways can one extend the zero section of the affine line with a double origin
Let $X$ be the affine line with a double origin over $\mathrm{Spec}\,\mathbb Z$. Let $X_\eta$ be its generic fibre, the affine line with a double origin over $\mathrm{Spec}\,\mathbb Q$.
Let $0$ be ...
- 21
2
votes
0
answers
127
views
Locally free sheaves of algebras vs. algebra bundles
It is well known that there is a bijective correspondence between locally free sheaves of modules and vector bundles (cf. https://rigtriv.wordpress.com/2008/04/09/locally-free-sheaves-and-vector-...
- 357
2
votes
0
answers
186
views
completion of non-finitely generated ideal
Let consider $A=k[x_{1},x_{2}...]$, the polynomial ring with countably many indeterminates.
Then we can consider the completion $\hat{A}=\varprojlim_{r,l}k[x_{1},x_{2},..]/(x_{1}^{r},..x_{l}^{r},x_{l+...
- 3,472
2
votes
0
answers
160
views
Covering a finite set of points of height 1 by an affine open
Let $R$ be a Noetherian ring and let $X$ be a finite type, separated $R$-scheme that is normal and integral. Let $x_1, \dotsc, x_n \in X$ be points of height $1$. Does there exist an open affine $U \...
- 1,103
2
votes
0
answers
285
views
infinite dimensional germs of schemes and tangent spaces
(The question of the type "how to define?")
Let $(R,\mathfrak{m})$ be a local ring over a field $k$ of zero characteristic. Consider the matrices over this ring, $Mat(m,R)$. I think of $Mat(m,R)$ as ...
- 2,232
2
votes
0
answers
244
views
Descent theory of line bundles on abelian varieties under isogenies (in char p>0)
I have a couple of questions regarding the descent theory of line bundles on abelian varieties under isogenies in positive characteristic.
Let $X$ be an abelian variety and $L\in Pic(X)$ a line ...
- 614
2
votes
0
answers
102
views
semicontinuity of the conductor defined by Temkin
We say a principal pair $(X,\mathcal{I})$ where $X=Spec(A)$ is affine scheme and $\mathcal{I}=\tilde{I}$ where $I\subset A$ is a principal ideal generated by $\pi$ wich is a non zero divisor.
For a ...
- 3,472
2
votes
0
answers
141
views
scheme of sections over complete local ring
Let $f:X\rightarrow S= Spec(k[[\pi]])$ a finite type faithfully flat morphism.
Let $U\subset X$ be an open subset such that $U$ is smooth and surjective on $S$.
We consider the $k$-scheme $X(k[[\pi]]...
- 3,472
2
votes
0
answers
250
views
fpqc, formal smoothness
Based on Possible formal smoothness mistake in EGA, let $X$ and $Y$ $k$-schemes ($k$ a field),
let $f:X\rightarrow Y$ a fpqc morphism such that $f$ is formally smooth and $X$ formally smooth, do we ...
- 3,472
2
votes
0
answers
236
views
descent for formally smooth maps
Let $f:X\rightarrow Y$ a morphism between schemes and $Y'\rightarrow Y$ a fpqc morphism
such that the base change $f'$ of $f$ to $Y'$ is formally smooth, does it imply that $f$ is formally smooth?
- 3,472
2
votes
0
answers
255
views
Lang isogeny for group stacks
Let $G$ be a commutative algebraic group stack over $\mathbb{F}_q$ (I don't really care about the precise definition: I'm secretly thinking about the Picard stack of a projective curve). To what ...
- 3,605
2
votes
0
answers
606
views
Kawamata-Viehweg Vanishing Theorem for Excellent Surfaces
Is there any version of Kawamata-Viehweg vanishing theorem that holds for excellent surfaces? I will be very glad if there is one such, but if not then is it at least true for a surface over a non ...
- 165
2
votes
0
answers
175
views
Regular subscheme of a projective limit of schemes
Let $S\cong \varprojlim S_i$, where $S$ and all $S_i$ are separated regular excellent of finite Krull dimension. Let $Z$ be a closed regular subscheme of $S$. As Theorem 8.8.2 of EGA4 shows, $Z$ comes ...
- 16.9k
2
votes
0
answers
143
views
Dualizing sheaf in mixed characteristic for regular schemes.
I've been looking many places, but everything I find seems to either talk about (a) varieties or (b) extremely general situations with dualizing complexes. As I am not in the situation of (a) (i.e. ...
- 419
2
votes
0
answers
245
views
Segre class of cones and Base change of projective cones
I'm trying to work out a result in Fulton's intersection theory and I think I need the following basic result about base change of projective cones (whose support may not be the entire base scheme).
...
- 3,555
2
votes
0
answers
279
views
deRham cohomoloy of CM liftings of Jacobians
Let $k$ be field of characteristic $p>0$ and $W=W(k)$ the ring of Witt vectors of $k$. We call a smooth curve over $k$, ordinary, when the Jacobian of $J(X)$ of $X$ is an ordinary abelian variety. ...
- 637
2
votes
0
answers
464
views
understanding Milne's article "Duality in the flat cohomology of a surface"
I have trouble understanding a point of Milne's article "Duality in the flat cohomology of a surface" http://jmilne.org/math/articles/1976a.pdf
see the "Alternatively" on p. 177, paragraph before ...
user19475
2
votes
0
answers
631
views
cohomology of projective limit of schemes
Hello,
Suppose that $X_i$ is a projective system of schemes and $F_i$ is a compatible
projective system of abelian sheaves on the $X_i$ (i.e. if $p_{ij} : X_i \to X_j$ is the
transition map, then we ...
- 2,842
2
votes
0
answers
370
views
Can any radiciel morphism be presented as the composition of a universal homeomorphism with an immersion?
Let $f:X\to Y$ be a radiciel (=universally injective) morphism, where $X$ is a regular connected scheme. Can it be presented as the composition of a universal homeomorphism with an immersion? This ...
- 16.9k
2
votes
0
answers
220
views
When inverse image is conservative; a reference or a generalization?
I am interested in the following question: for $f$ being a morphism of schemes, which conditions ensure that $Rf^*_{et}$ is conservative? This is true if $f$ has a section or if $f$ is an \'etale ...
- 16.9k
2
votes
0
answers
298
views
Induced groupoid schemes
This is a more direct version of this question, which was perhaps a bit obtuse. This is a more elementary formulation.
Recall that for a groupoid scheme (or indeed any internal groupoid) $X = (X_1 \...
- 35.5k
2
votes
0
answers
321
views
Dimension of fibres of moment maps in characteristic $p$
Suppose $G$ is a connected semisimple linear algebraic group with Lie algebra $\mathfrak{g}$ and $X$ is a homogeneous $G$-space with isotropy subgroup $H$ (associated Lie algebra $\mathfrak{h}$) that ...
- 3,545
1
vote
2
answers
492
views
open immersion between affine spaces
Let $j:\mathbb{A}^{n}\rightarrow\mathbb{A}^{n}$ an open immersion over a field $k$. Is it an isomorphism?
- 3,472
1
vote
2
answers
788
views
Defining algebraic manifold without referring to schemes
Let $M$ be a complex manifold admitting an atlas with each chart biholomorphic to $\mathbb{C}^n$ and transition maps being rational functions.
Is it true that there exists a smooth integral ...
- 125
1
vote
2
answers
3k
views
Pullback of a constant sheaf
Let $\varphi:X\to Y$ be a surjective morphism of schemes which are connected and of finite type.
Let $A$ be an abelian group, $\mathscr{F}$ be the constant sheaf on $X$ with fibers $A$ and $\mathscr{...
- 251
1
vote
1
answer
445
views
Are morphisms of schemes generically affine
Let $f \colon X \to Y$ be a morphism of schemes, where $X$ and $Y$ are separated integral Noetherian schemes. Does there necessarily exist a nonempty open affine $U \subset Y$ such that $f^{-1}(U)$ is ...
- 7,338
1
vote
1
answer
110
views
Torsor of finite presentation and surjectivity of map of $\overline{k}$-valued points
I have a question about the content of remark 2.6.6. (i) (p 18) from M. Brion's notes on structure of algebraic groups.
Let $G$ be a group scheme over certain fixed base field $k$ (as all other ...
- 6,006
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
1
vote
2
answers
619
views
A curve is proper iff the space of global sections is finite-dimensional
Let $k$ be a field, $X\rightarrow \mathrm{Spec}\,k$ be a separated morphism of finite type of relative dimension$\leq 1$ (as defined here). Is it true that $f$ is proper iff $f_* \mathcal{O}_X$ is ...
- 11
1
vote
1
answer
1k
views
Codimension of points in fibered products
This is a question about a proof in Hartshorne, but let me try to formulate it without reference to Hartshorne.
Let $X$ be a noetherian scheme (which is also integral, separated and regular in ...
- 33.8k
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 ...
- 6,006
1
vote
1
answer
343
views
Smoothness and smoothness over formal neighborhood
Let $f:X\rightarrow Y$ a locally finitely presented map. Let $x\in X$ and $y=f(x)$.
We assume that the map on the level of fomal neighborhoods $X_{x}\rightarrow Y_{y}$ is formally smooth, can we find ...
- 3,472
1
vote
1
answer
443
views
What kind of conditions we need to make morphisms of schemes quasi-projective?
What kind of conditions we need to make morphisms of schemes quasi-projective?
I am really interested in the following case:
If $f : X \to Y$ is an etale, of finite type and separated morphism of ...
- 945
1
vote
1
answer
244
views
Are there nonaffine schemes over which every exact sequence of vector bundles is split?
Is there an example of a non-affine scheme $X$ such that every short exact sequence of vector bundles over $X$ splits? If there are such examples then what if we ask it to be true of all (not ...
- 187
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
1
vote
2
answers
508
views
Base change of a finite morphism
Let $X$, $Y$, $Z$ be integral schemes of finite type over a field $K$ (i.e., locally affine opens are finitely-generated algebras over $K$). Suppose we have the following condition$\colon$
$f \colon ...
- 563
1
vote
1
answer
226
views
flatness of restriction of structure sheaf over ring of global sections
Let $X$ be an affine scheme. $U \subseteq X$ open. Then I want to show that $\mathcal{O}_X(U)$ is flat over $\mathcal{O}_X(X)$.
But I want to prove it only by knowing the definition of structure sheaf ...
1
vote
1
answer
694
views
Under what conditions is an open subscheme of an affine scheme affine and what ring corresponds to it?
It is well known that an open subscheme of an affine scheme is not necessarily an affine one. But what are (if possible the most general) sufficient conditions for its affinity? And is it known how, ...
- 6,962
1
vote
1
answer
149
views
When is $R$ a direct summand of Frobenius pushforwards?
Let $(R,\mathfrak m)$ be a reduced Noetherian local ring of prime characteristic $p$. For integer $e>0$, let $F^e_* R$ denote the $R$-module which is $R$ as an abelian group, but the $R$-module ...
- 413
1
vote
1
answer
217
views
Meaning of torsion points in a Roitman's theorem
I am having some problems to understand the meaning of the following theorem due to Roitmann. I found this theorem in Voisin's book: Hodge Theory and Complex Algebraic Geometry, Volume II, page ...
- 519
1
vote
1
answer
2k
views
Finiteness of surjective etale morphisms
Is every surjective etale morphism from a connected separated scheme to $A^n_{\mathbb{C}}$ of finite type? Is it finite? We use Stacks project's definitions.
EDIT: From Jason Starr's answer, we ...
- 723
1
vote
1
answer
280
views
Norms of elements in Artin-Schreier extensions
The following is claimed in the proof of Theorem 7.5 of Auslander, Goldman, "The Brauer group of a commutative ring":
Let $k$ be a nonperfect field of positive characteristic $p$, let $K := k(x)$ ...
- 428
1
vote
1
answer
212
views
Putting two complete varieties in a family over the projective line
Let $X$ and $Y$ be two proper varieties of dimension $n$ over a field $k$. I'm looking for "reasonable" conditions, under which, there exists a proper and dominant morphism $f:V\to \mathbb{P}^1_k$, ...
- 137