All Questions
1,159 questions
3
votes
1
answer
933
views
Schemes associated to vector spaces
Let $k$ be a field. Let $F$ be a covariant functor on the category of $k$-algebras to the category of sets. Assume that the opposite functor $F^{op}$ on the category of affine $k$-schemes is a sheaf. (...
- 1,213
3
votes
2
answers
1k
views
Components of an exceptional divisor
Let $X$ be a projective variety and let $\tilde{X}$ be the blow-up of $X$ at a subscheme $Z$. Let $F$ be the exceptional divisor of $\tilde{X}$. I wonder:
What is the number of irreducible ...
- 83
3
votes
2
answers
733
views
If the morphism of root data induced by an isogeny of a reductive group is a Frobenius, is then the isogeny itself a Frobenius?
Let $G$ be a reductive (or just semisimple) algebraic group over an algebraically closed field $k$ of characteristic $p > 0$, let $T$ be a maximal Torus and let $f:G \rightarrow G$ be an isogeny. ...
- 5,243
3
votes
1
answer
191
views
Irreducibility under etale ring map
Let $A\rightarrow B$ be a etale ring map between finite type algebra over algebraically closed field $k$.
If $A$ is one dimensional integral domain, is $B$ direct product of finite type integral ...
- 328
3
votes
2
answers
670
views
Finite etale cover of projective line
If we have a map $f : \mathbb P^1_R \to \mathbb P^1_R$ over $\operatorname{Spec}(R)$, with $R$ a commutative ring, which we assume to be etale, then is it possible to characterize $f$? Must it be an ...
3
votes
1
answer
325
views
vector bundles over projective line over an affine line
Let $k$ be a field and $E$ be a vector bundle over $\mathbb{P}_{k}^{1}\times\mathbb{A}_{k}^{1}$, does it extend to
$\mathbb{P}_{k}^{1}\times\mathbb{P}_{k}^{1}$?
- 3,472
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
3
votes
1
answer
600
views
Why is it useful for the (relative) Picard functor to be representable?
I have been studying Chapter 8 of Neron models by Bosch et al. The first part deals with the relative Picard functor. A lot of work is done to make it representable. My question would be why this work ...
- 393
3
votes
1
answer
260
views
Chevalley devissage
Let $G$ be an algebraic group over a perfect field $k$. Then it is know that it can be written as an extension of an affine algebraic group and a proper algebraic group.
Is there a similar result for ...
- 3,472
3
votes
1
answer
524
views
Flatness of Normalization of regular schemes
I have a followup to the following question: Flatness of normalization.
Suppose that $X$ is a regular scheme (of finite type over a $\mathbb{C}$ if one wants) and $X'$ is the normalization of $X$ in ...
- 2,834
3
votes
1
answer
283
views
When is the gluing of two finite type affine Z-schemes affine?
Let $A$ and $B$ be finitely generated $\mathbf{Z}$-algebra. Suppose that there exists two coprime integers $m$ and $n$ and an isomorphism of $\mathbf{Z}$-algebra
$\phi:A\otimes_{\mathbf{Z}}\mathbf{Z}[...
- 7,609
3
votes
1
answer
459
views
Frobenius functor and length of local cohomology
Let $(R,\mathfrak{m})$ be a Noetherian local ring of positive prime characteristic $p$ and let $F$ be the Frobenius functor. Write $d$ for dimension of $R$. Assume that for some $0\leq i< d $ the ...
- 3,101
3
votes
1
answer
326
views
Classification of fat projective lines?
In section III.3.4 of Eisenbud & Harris's "The Geometry of Schemes," we/they construct an infinite family of double structures on $\mathbb{P}^1 \subset \mathbb{P}^3$ that are distinguished from ...
- 2,218
3
votes
2
answers
374
views
Find a non-quasi-compact scheme s.t. all finitely generated + globally generated quasi-coherent modules are finitely globally generated
Closely related to this question in MSE, but the difference is that we will set $X$ to be scheme and $\mathcal{F}$ to be quasi-coherent.
Let $X$ be a locally ringed space. We say an $\mathcal{O}_X$-...
- 452
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
$$
H^0(X,\mathcal{O})\otimes ...
user159150
3
votes
1
answer
251
views
Do Neron-Severi groups of smooth projective unirational varieties contain $\ell$-torsion?
Let $X$ be a smooth projective unirational variety over an algebraically closed field of characteristic $p>0$, and $\ell\neq p$ a prime. My question: can the Neron-Severi group of $X$ contain (non-...
- 16.9k
3
votes
1
answer
252
views
Is this unipotent group, over characteristic 2, connected?
Let $E_{ij}(x)\in \mathrm{Mat}_{7\times7}(\overline{\mathbb{F}}_2)$ be the matrix with zeros everywhere, except for the value $x$ at $ij$. Set $$a(x)=1+E_{12}(x)+E_{34}(x)+E_{56}(x),\quad b(y)=1+E_{23}...
- 4,593
3
votes
1
answer
1k
views
Diagonal is representable then any morphism is representable
Ariyan Javanpeykar said here in comments that,
If the diagonal is representable, then isn't any morphism $S\rightarrow \mathcal{X}$ with $S$ a scheme representable?
I could not find the statement (...
- 6,023
3
votes
1
answer
270
views
Restriction of separable map
If $f: X\to Y$ is a separable map between varieties that is a bijection on closed points, is it true that $f$ remains separable when restricted to an integral subscheme $Z\subset X$?
If we drop the ...
- 1,537
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
3
votes
1
answer
419
views
Affine hulls and base change
Let $S$ be a scheme. We consider the functor, called affine hull, from the category of quasicompact and quasiseparated $S$-schemes to the category of affine $S$-schemes, defined as a left adjoint to ...
- 6,700
3
votes
1
answer
424
views
Are Isom-schemes geometrically connected
This question is about properties of Isom-schemes that are well-known over algebraically closed fields.
Let $K$ be a field of characteristic zero, let $C$ be a smooth projective geometrically ...
- 123
3
votes
1
answer
390
views
Torsion of elliptic curves is finite
Let $S$ be an integral 1-dimensional scheme with function field $K$.
Let $E$ be an elliptic curve over $K$. The torsion of $E$ over $K$ is not necessarily finite. As an example consider an elliptic ...
- 295
3
votes
1
answer
339
views
Galois invariant line bundle and base change
Let $K$ be a number field and consider a finite Galois extension $L|K$. Moreover let $X$ be a projective, regular, integral variety over $K$. After a base change we obtain a morphism of varieties $f:...
- 321
3
votes
1
answer
245
views
Can non-geometrically reduced reduced subschemes happen for reductive groups?
The title is meant to be punchy, but also a tongue-in-cheek acknowledgement of the prevalence of ‘reduce’-derived words in this area. (Unfortunately, I overlooked the fact that the question in the ...
- 13k
3
votes
1
answer
188
views
Maximal closed subscheme stable under the action of a finite connected group scheme
Let $k$ be a field of characteristic $p>0$, $X$ a smooth projective $k$-variety and $Y\subseteq X$ a closed irreducible subvariety. Let $G$ be a connected finite $k$-group scheme acting on $X$.
...
- 728
3
votes
3
answers
423
views
Are there characteristic-dependent Betti numbers in characteristic not equal to two?
Let $\Delta$ be a finite simplicial complex on $n$ vertices. Let $S = \mathbb{k}[\mathbf{x}]$ be a polynomial ring over a field $\mathbb{k}$ in $n$ variables and $I$ be the Stanley-Reisner ideal of $\...
- 984
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
3
votes
1
answer
282
views
Frobenius splitting for an excellent, non $F$-finite, $F$-pure hypersurface
Let $p$ be an odd prime. Let $k$ be a field of characteristic $p$ such that $[k:k^p]=\infty$ (i.e. $k$ is not $F$-finite ) .
Also assume that $-1$ is not a square in $k$ . Consider the homogeneous ...
- 642
3
votes
1
answer
444
views
Projection formula for flat morphisms
Let $f\colon X\to Y$ be a flat morphism between two smooth projective varieties. Let $L$ be a locally free sheaf on $X$ and $\mathcal{F}$ a coherent sheaf on $Y$. How to prove $f_*(L\otimes f^*\...
- 91
3
votes
1
answer
480
views
Arc space & formal loops in motivic integration
One of the most essential ingredients in the theory of motivic integration are the space of arcs of a given $k$-variety
$X$. This is a scheme, whose $k$-rational points are the $k[[t]]$-valued points ...
- 6,016
3
votes
1
answer
317
views
What is known about lower etale cohomology of unirational varieties?
Which information is currently known about $H^1_{et}(X,\mathbb{Z}_l)$ and $H^2_{et}(X,\mathbb{Z}_l)$, where $X$ is a smooth unirational variety over an algebraically closed field of finite ...
- 16.9k
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
3
votes
1
answer
1k
views
Extension of morphism of quasiprojective varieties
I am trying to understand when a morphism defined in an open dense subset of a variety can be extended to the whole variety.
For curves, it is known that if $f:C \to C’$ is a rational morphism from ...
- 558
3
votes
1
answer
410
views
Derivations of central extensions of simple Lie algebras
Let $L$ be a finite-dimensional Lie algebra over a field of characteristic zero. It is not difficult to see (and also follows from Theorem 4.4 of [G. Hochschild: Semi-simple algebras and generalized ...
- 5,878
3
votes
1
answer
392
views
Embedded resolution of curves on smooth varieties
As far as I understand, embedded resolution of singularities means the following: given a variety $X$ over an algebraically closed field, and a closed subvariety $Y$, there exists a birational map $f:...
- 16.2k
3
votes
1
answer
335
views
resolution property and perfect stacks
Recall that for a scheme $X$, it has the resolution property if every coherent sheaf $E$ on $X$, is the quotient of a finite locally free $\mathcal{O}_X$-module.
On the other hand, Ben-Zvi-Nadler-...
- 3,472
3
votes
1
answer
412
views
When Hom scheme has projective components?
The Hom scheme of two projective varieties over some field is constructed as an open subfunctor of the Hilbert scheme of the product of the two schemes by Grothendieck. So it is a countable union of ...
- 5,901
3
votes
1
answer
212
views
Algebraic spaces in the étale topology (proof from Stacks project)
I have a question about the proof of Lemma 78.12.1 from Stacks Project. The aim of the last paragraph of the proof is to verify that the map of sheaves in the étale topology $F \to U/R$ is an ...
- 6,016
3
votes
1
answer
633
views
Representability of Grassmannian functor by a scheme
I am having some trouble following a proof that the Grassmannian functor is representable by a scheme. I am following the proof in EGA 9.7.4. It is only a small step that I am stuck on. For reference, ...
- 453
3
votes
2
answers
279
views
Schemes with no finite morphisms onto themselves
Let $X$ be an integral Noetherian scheme that has no automorphisms other than the identity. Is it possible to characterize (in a not completely tautological way) $X$ such that
there is no finite ...
user138661
3
votes
1
answer
275
views
Sections of morphisms up to fppf covering
Let $f:X\to S$ be a finite type affine morphism of schemes where $S$ is an integral noetherian affine regular scheme whose function field is of characteristic zero.
Assume that all geometric fibers ...
- 33
3
votes
1
answer
416
views
Can the Grothendieck ring of varities over a field $k$ be defined for non separated schemes?
The Grothendieck ring of varieties over a field $k$ is the abelian group generated by isomorphim classes $[X]$ of separated, reduced $k$-schemes $X$ of finite type with the relation
$[X]=[Y] + [X\...
3
votes
1
answer
276
views
Mumford-Ramanujam examples in characteristic p [and in Arakelov geometry]
For a compact Riemann surface $B$ of genus $\geq 2$, it is a consequence of the Narasimhan-Seshadri theorem that there exist rank-$2$ vector bundles $E \to B$ of degree zero, all of whose symmetric ...
- 13.8k
3
votes
2
answers
500
views
Varieties that do not extend to flat families
Is it easy to give an example of a function field $K$ and a smooth proper variety $X$ over $K$ that does not extend to a flat scheme over $B$, where $B$ is a smooth proper variety with function field $...
- 31
3
votes
1
answer
288
views
Do $Q_l$-etale Euler characteristics of Chow motives coincide for all $l$?
I am interested in (Chow) motives over (algebraically closed) characteristic $p>0$ fields. For $H$ being $\mathbb{Q}_l$-adic cohomology, one can consider $Ch_l(M)=\sum (-1)^i\dim_{\mathbb{Q}_l} H^i(...
- 16.9k
3
votes
1
answer
1k
views
Quasi-coherent sheaves of O_X-algebras
Let $X$ be a quasi-compact scheme and let $\mathcal{A}$ be a quasi-coherent sheaf of $\mathcal{O}_X$-algebras on $X$. $X$ being quasi-compact, we can write $X = U_1 \cup \dots \cup U_n$ with each $...
- 2,799
3
votes
1
answer
283
views
Is the theta characteristic attached to an etale double cover of a plane quintic arising from a cubic threefold even in characteristic two?
We work over an algebraically closed field $k$ of characteristic $2$. Let $X$ be a cubic threefold realized as a conic bundle via $f: X \to \mathbb{P}^2$ (after blowing up some line). Let $C \to \...
- 679
3
votes
1
answer
129
views
Schur multiplier of finite-dimensional simple Lie algebras in positive characteristic
The Schur multipliers of finite simple groups are known and easily accessible:
https://en.wikipedia.org/wiki/List_of_finite_simple_groups
Moreover, as a consequence of the second Whitehead's Lemma, if ...
- 630
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