The schemes tag has no wiki summary.
2
votes
1answer
379 views
Can I conclude that a morphism of vector bundles is zero if it is so fiberwise?
Let $f: X \rightarrow Y$ be a flat morphism of locally noetherian schemes and $\varphi: \mathcal U \rightarrow \mathcal V$ a map of vector bundles (locally free sheaves of finite rank) on $X$.
I want ...
0
votes
0answers
81 views
Reducibility of fibers over closed points implies reducibility of the generic geometric fiber?
Suppose that $f\colon X\to Y$ is a proper (or even projective) morphism of (reduced) algebraic varieties over an algebraically closed field $k$. If fibers of $f$ over all closed points of $Y$ are ...
1
vote
0answers
53 views
On the intersection complex
Let $j:U\subset X$ an open immersion between $k$ schemes integral of finite type.
Let $K\in D_{c}^{b}(X,\bar{\mathbb{Q}}_{l})$ a complex of $l$-adic sheaves, such that we have that $IC_{U}$ is a ...
2
votes
0answers
90 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 ...
1
vote
0answers
80 views
smooth morphism from a finite type source
Let $f: X\rightarrow Y$ a smooth morphism over a field $k$. We assume that $X$ is locally of finite type, does it imply that $Y$ is also locally of finite type?
0
votes
0answers
56 views
ind scheme and Jacobson property
Let $G$ a semisimple group over $k$ and $k$ algebraically closed.
Let $G(k((t)))$ the corresponding ind-scheme, does it satisfies the Jacobson property, say closed points are dense in it?
0
votes
0answers
71 views
on the fibers over closed points
Let $X$ and $S$ $k$-schemes of finite type . ($k$ a field) and $U$ an open subset of $X$
Let $f:X\rightarrow S$ a $k$-morphism of finite type.
We assume that for any closed point $s\in S(\bar{k})$, ...
2
votes
0answers
99 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 ...
2
votes
1answer
157 views
flat and finite type morphisms
Let $f:X\rightarrow Y$ a faithfully flat morphism between $k$-schemes. We assume that the fibers are locally of finite type, do we have that $f$ is locally of finite type?
4
votes
0answers
73 views
formal smooth morphism with a formal smooth source
Let $f:X\rightarrow Y$ a morpism between $k$-schemes ( $k$ a field).
We suppose that X is formally smooth and f is formally smooth and surjective.
Do we have that $Y$ is formally smooth?
Or if it's ...
2
votes
0answers
92 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?
1
vote
0answers
47 views
closed subscheme of ind scheme
Let $X$ a ind-scheme of ind-finite type and ind-affine. (e.g, take a k- smooth, affine scheme of finte type $T$, $C$ a smooth projective curve over $k$ and $x$ a closed point, then $X=T(C-x)$ verifies ...
0
votes
2answers
162 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?
0
votes
0answers
151 views
on flat morphisms
Let $j:U\rightarrow X$ an open immersion between k-schemes of finite type and $f:X\rightarrow S$ a surjective k-morphism of finite type.
We suppose that $f\circ j:U\rightarrow S$ is faithfully flat, ...
0
votes
0answers
91 views
on rational singularities
Let a cartesian diagram
Let $X'\rightarrow X$ be a rational resolution of singularities of $k$-schemes of finite type and $Y$ a closed subscheme.
Let $Y'\rightarrow Y$ be the base change to $Y$, we ...
0
votes
1answer
117 views
When is the determinant of the push-forward of an ample line bundle ample
Let $f:X\to S$ be a "nice" morphism of "nice" schemes. Let $L$ be an ample line bundle on $X$.
When is $\det f_\ast L$ also ample?
A "nice" morphism could be anything from "finite type separated" to ...
7
votes
0answers
119 views
Is there a direct proof that affine schemes are fppf quasi-compact?
Let $A$ be a (commutative) ring. A family $(B_i)_{i\in I}$ of $A$-algebras is said to be an fppf cover if it satisfies three properties: (1) each $B_i$ is flat as an $A$-module, (2) each $B_i$ is ...
5
votes
1answer
288 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 ...
2
votes
0answers
104 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 ...
4
votes
1answer
184 views
Which schemes can be presented as limits of smooth varieties?
I can prove a certain statement for any scheme that can be presented as the limit of an essentially affine (filtering) projective system of smooth varieties over a perfect field such the connecting ...
2
votes
0answers
94 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. ...
0
votes
0answers
121 views
How would you call a subscheme of a smooth $S$-scheme?
In my preprint I propose to call $X/S$ quasi-smooth if $X$ can be embedded into a smooth $X'/S$. Does this sound fine?
Upd. So, smoothly embeddable is better? Is it ok to call a morphism smoothly ...
4
votes
0answers
117 views
If $X,Y$ are regular and of finite type over $S$, can $X\times _S Y$ be embedded into a regular $S$-scheme?
It seems to be well-known (see Is there an example of a variety over the complex numbers with no embedding into a smooth variety?) that a general finite type $S$-scheme does not embedd into a regular ...
0
votes
0answers
76 views
Why do I get a morphism $f_P: Spec \mathcal{O}_K \to \mathcal{X}$ for every point $P\in X(K)$? How does this morphism look like?
Hello,
my question probably isn't too hard, but I can't find the answer.
Let $K$ be a number field, $\mathcal{X} /\mathcal{O}_K$ be an arithmetic surface, which is a regular model for a projective, ...
0
votes
0answers
123 views
Quicker way to show that the restriction to a open subvariety is again proper?
Dear all,
Let $f: X \rightarrow Y$ be a morphism of projective varieties over $\mathbb{C}$. Also let $V \subset Y$ be a nontrivial open subvariety and set $U:= f^{-1}(V)$.
I would like to show that ...
4
votes
0answers
195 views
Is the pushout of smooth varieties along a smooth closed subvariety again a variety?
The following question is motivated by a desire to find a rough analog in algebraic geometry of the usual notion of gluing of smooth bordisms.
Suppose k is an algebraically closed field of ...
6
votes
1answer
299 views
Which local ringed spaces are schemes?
(This was originally asked on math.stackexchange, but didn't get any responses. I figured it might be worthwhile to move it here and try again.)
This paper gives a proof that the underlying ...
3
votes
1answer
203 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 ...
14
votes
2answers
501 views
Minimal number of generators for $A^n$
Let $A$ be a commutative ring and $n \in \mathbb{N}$. What is the minimal number $e_A(n)$ of generators of the $A$-algebra $A^n$? Here is what I already know (I can add proofs if necessary) from a ...
4
votes
1answer
241 views
Finitely-affine morphisms; cohomological dimension of schemes
Let $f\colon X\to U$ be a morphism of Noetherian schemes such that the scheme $U$ is affine and the scheme $X$ is separated and, e.g., quasi-projective over affine. Let $U=\bigcup_\alpha U_\alpha$ be ...
3
votes
1answer
419 views
(Mixed) Tate motives
Hi there,
in recent times I was reading texts about motives, and I want to ask
something about Tate motives which is not clear to me (as I came across
different definitions in different texts).
Let ...
2
votes
0answers
150 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).
...
5
votes
0answers
63 views
Category of the smooth formal p-groups over a local ring
Fontaine showed in Asterisque 47-48 that the category of finite dimensional smooth formal $p$-groups over the ring $A=W(k)$ of the Witt vectors over a finite field $k$ is anti-equivalent to the ...
4
votes
0answers
464 views
Two definitions of smoothness?
This is confusing, there appear to be possibly two definitions of smoothness in algebraic geometry for a morphism $f: X \rightarrow Y$ of schemes of finite type over an arbitrary field $k$.
...
0
votes
1answer
181 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 ...
0
votes
0answers
227 views
G-torsor whose ring of regular functions is a field.
I already asked this question on stackexchange but didn't get any answer. Maybe it is better suited for mathoverflow.
Let $G$ be an affine group scheme (not necessarily of finite type) over ...
0
votes
0answers
114 views
Projective spaces with nonconstant regular functions
I can construct a scheme by patching that represents a projective space over an arbitrary ring. I can also prove that, if the ring is a Jacobson domain, the only regular functions on it are constants.
...
5
votes
3answers
309 views
Irreducible “family” of relative effective divisors of a smooth morphism
Let $\pi:X\rightarrow Y$ be a smooth proper (assume projective if needed) morphism of schemes with $Y$ locally noetherian, and let $Z\subset X$ be an irreducible integral closed subscheme containing ...
2
votes
2answers
144 views
Induced maps of an automorphism of a curve on the tangent ot its jacobian and on its differential forms
Let $C$ be a smooth projective curve of genus $g$ over a field $k$ and $J$ be its jacobian (defined over $k$). Let $\sigma: C \rightarrow C$ be a $k$-automorphsm of $C$. This automorphism $\sigma$ ...
2
votes
2answers
285 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 ...
0
votes
0answers
249 views
When are points of a scheme a variety?
I think of an affine scheme over a field $k$ as an extension of the concept of a (possibly reducible) variety over $k$, by extending the affine $n$-space $k^n$ to $A^n$, where $A$ is a commutative ...
2
votes
1answer
214 views
Proper morphisms: Lie groups vs. group schemes
A Lie group can (often) be recovered as the $\mathbb{R}$-points of a group scheme. I am wondering if this parallelism carries over to proper actions.
In particular, let $G$ be a Lie group acting on a ...
1
vote
1answer
325 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
vote
1answer
151 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 $f:V\to \mathbb{P}^1_k$, ...
4
votes
1answer
365 views
technical question about a paper on Belyi's theorem
I'm trying to understand Belyi's theorem, as presented here: http://eprints.soton.ac.uk/29785/1/b45h1koe.pdf
He defines a curve $X$ over a field $C$ to be a smooth projective geometrically connected ...
5
votes
0answers
244 views
What is known about “singularity types” in the Murphy's Law sense?
In his "Murphy's Law" paper, Vakil gives a definition equivalent to the following:
The singularity type of a pointed scheme $(X,p)$ its equivalence class, under the following equivalence relation: ...
4
votes
2answers
283 views
trying to understand the support of the sheaf of relative differentials
So I'm trying to understand a proof of Belyi's theorem from http://eprints.soton.ac.uk/29785/1/b45h1koe.pdf
specifically lemma 3.4.
The setup is as follows: Let $X/\mathbb{C}$ be a curve, and let $t ...
2
votes
1answer
238 views
Spreading out of flat morphisms of schemes
In EGA IV, Chapter 8, projective systems of schemes (and morphisms between them) are considered. Let $(S_{\lambda})_{\lambda \in L}$ be a projective system of schemes and let $S$ be the projective ...
25
votes
1answer
1k views
Connections between various generalized algebraic geometries (Toen-Vaquié, Durov, Diers, Lurie)?
As far as I know, there are four possible ways to generalize algebraic geometry by 'simply' replacing the basic category of rings with something similar but more general:
$\bullet$ In the approach by ...
4
votes
3answers
620 views
Do Disjoint Unions and Fiber Products Commute?
Do disjoint unions and fiber products commute?
In other words, is the following statement true?
Statement: Let $C$ be a category with (infinite) coproducts and fiber products. Let {$U_{i}$} be a ...
