**0**

votes

**1**answer

241 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 ...

**9**

votes

**1**answer

411 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 ...

**6**

votes

**1**answer

389 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

**0**answers

129 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

**1**answer

251 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

**0**answers

124 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

**0**answers

147 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 ...

**5**

votes

**0**answers

159 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

**0**answers

92 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

**0**answers

188 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

**0**answers

235 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 ...

**8**

votes

**1**answer

387 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

**1**answer

230 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 ...

**15**

votes

**2**answers

545 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

**1**answer

329 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 ...

**5**

votes

**1**answer

859 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

**0**answers

192 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

**0**answers

68 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

**0**answers

586 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

**1**answer

209 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 ...

**2**

votes

**0**answers

277 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

**0**answers

233 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

**3**answers

414 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 ...

**3**

votes

**2**answers

243 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 $\sigma$ ...

**5**

votes

**2**answers

484 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

**0**answers

278 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

**1**answer

233 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

**1**answer

389 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

**1**answer

165 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

**1**answer

432 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

**0**answers

303 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

**2**answers

481 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 ...

**4**

votes

**1**answer

352 views

### Spreading out 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 ...

**31**

votes

**1**answer

2k 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 ...

**6**

votes

**3**answers

1k 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 ...

**16**

votes

**2**answers

2k views

### Intuition behind generic points in a scheme

In a scheme, each point is a generic point of its closure. In particular each closed point is a generic point of itself (the set containing it only), but that's perhaps of little interest. A point ...

**1**

vote

**0**answers

277 views

### “reduction” of a module

Let $X$ be a scheme over a field $k$. There is a well-known
antiequivalence between locally free sheaves of
$\mathcal{O}_X$-modules and vector bundles over $X$. Given a module
$\mathcal F$ and a ...

**6**

votes

**2**answers

811 views

### explanation on a scheme which is not affine scheme

Hartshorne at the end of page 76 of his Algebraic Geometry book gives an example of a scheme which is not an affine scheme. The scheme is constructed by gluing two affine lines together along their ...

**4**

votes

**0**answers

605 views

### Zariski tangent space of a scheme as the vector space of derivations

A standard lemma says that for a scheme $X$ of finite type over an algebraically closed field $k$ the set of derivations $\mathcal{O}_{X,x} \to \kappa(x)=k$, is isomorphic to the Zariski tangent ...

**0**

votes

**2**answers

266 views

### Automorphism group of a scheme, 2

Hi,
I have the following two questions about automorphism groups of schemes.
First of all, let $S$ be a scheme, and $S^c$ its set of closed points. What
is the connection between $Aut(S)$ and ...

**1**

vote

**1**answer

240 views

### not locally of finite type implies not universally closed?

A proper morphism is defined as separated, of finite type and universally closed. I wonder if the requirement of being of finite type is superfluous, i.e. if being not of finite type implies not ...

**3**

votes

**1**answer

359 views

### Flat cover by a locally Noetherian scheme

Les S be a scheme. Does there exist a faithfully flat morphism T to S with T a locally Noetherian scheme?

**8**

votes

**1**answer

409 views

### Nonnegative additive functions on coherent sheaves

Let $(X,\mathcal{O}_X)$ be a Noetherian integral scheme and let $g$ be a (numerical) additive nonnegative function from coherent $\mathcal{O}_X$-modules to $[0,\infty)$. This question may be well ...

**6**

votes

**1**answer

549 views

### Proper morphism sending coherent to coherent

Hello,
Is there a proof that the push forward by a proper morphism of Noetherian schemes sends coherent sheaves to coherent ones, without passing in the argument through projective morphisms?
Thank ...

**4**

votes

**1**answer

303 views

### Flat family of normal schemes over a normal base

Let $f \colon X \to Y$ be a flat morphism of schemes over $\mathbb{C}$. Suppose that $Y$ is normal and that the fibers over the closed points of $Y$ are all normal.
Can I say something about the ...

**3**

votes

**3**answers

463 views

### Are schemes pushouts of neighbourhoods and formal neighbourhoods?

Hello,
I have two questions, the first less important.
Let $X$ be a scheme, $x \in X$ a schematic point.
What is an elegant way of defining/characterizing the map $\operatorname{Spec}(O_{X,x}) ...

**11**

votes

**1**answer

877 views

### Affine scheme on spec(A) of a ring A as the sheafification of a pre-sheave on spec(A)?

It is obvious that there is a parallel between the definition of structure sheaf of $\operatorname{Spec}(A)$
versus the sheafification of a pre-sheaf.
The definition of the sheaf $\mathscr F^+$ ...

**0**

votes

**1**answer

392 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 ...

**7**

votes

**2**answers

2k views

### Jacobian criterion for smoothness of schemes

An affine scheme $X = Spec(A)$ is said to be smooth if for any closed embedding
$X\subset\mathbf A^n$, of ideal $I$, it is true that, locally on $x\in X$, the ideal $I$
can be generated by a sequence ...

**3**

votes

**1**answer

794 views

### Is the degree of a finite morphism stable by base change

Let $f:X\longrightarrow Y$ be a finite morphism of schemes of degree $n$. Let $S\to Y$ be a morphism of schemes.
Is the degree of the finite morphism $X\times_Y S \longrightarrow S$ equal to $n$?
If ...