The first purpose of schemes theory is the geometrical study of solutions of algebraic systems of equations, not only over the real/complex numbers, but also over integer numbers (and more generally over any commutative ring with 1).It was finalized by Alexandre GROTHENDIECK, during the 1950's and ...

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8
votes
1answer
375 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
229 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
2answers
543 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
319 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
1answer
823 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
191 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
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
0answers
576 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
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 ...
1
vote
0answers
275 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
229 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
407 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
2answers
235 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
2answers
472 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
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
1answer
232 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
384 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
163 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
428 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
301 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
475 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
1answer
346 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 ...
30
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 ...
6
votes
3answers
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
2answers
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
0answers
274 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
2answers
793 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
0answers
581 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
2answers
261 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
1answer
239 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
1answer
357 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
1answer
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
1answer
542 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
1answer
296 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
3answers
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
1answer
860 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
1answer
377 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
2answers
1k 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
1answer
780 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 ...
7
votes
5answers
1k views

pushforward of locally free sheaf is locally free?

Hi, Is there an example of a proper smooth map of schemes $f:X\to Y$ and a vector bundle $E$ on $X$ such that $f_*E$ is not locally free on $Y$? Thanks
1
vote
1answer
241 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 ...
3
votes
1answer
494 views

Vector space structure on the tangent bundle of a scheme and relation to the tangent sheaf

First a word of warning: I am not a trained algebraic geometer, so it is possible (likely) that these questions are inappropriate for MO, if so: my apologies. Said this: As far as I understand the ...
3
votes
1answer
278 views

The restriction of a global section which is not a zero divisor is still an non-zero divisor?

Let X be a scheme. U is an open subscheme of X. Assume f is a global section on X which is not a zero divisor, then the restriction of f to U is still an non-zero divisor? If X is affine, the answer ...
2
votes
2answers
446 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 ...
22
votes
0answers
963 views

Mikhalkin's tropical schemes versus Durov's tropical schemes

In Mikhalkin's unfinished draft book on tropical geometry, (available here) (page 26) he defines a notion of tropical schemes. It seems to me that this definition is not just a wholesale adaptation of ...
0
votes
1answer
317 views

Is the following morphism etale

Let $Y$ be a reduced noetherian $1$-dimensional scheme such that the normalization morphism $f:X \longrightarrow Y$ is finite. Let $g:Y\longrightarrow Z$ be a finite flat morphism, where $Z$ is a ...
5
votes
1answer
372 views

Limits of reduced schemes question from Eisenbud and Harris

My question pertains to exercise II-16 in Eisenbud and Harris' "The geometry of Schemes". For an algebraically closed field $K$ the question is as follows: Consider zero-dimensional subschemes ...
4
votes
1answer
339 views

When does Zariski closure commute with base change?

This should be an elementary question for anyone who knows SGA by heart (alas, not for me). It smells a lot like a descent problem. All schemes are supposed to be noetherian, and all morphisms to be ...
3
votes
2answers
842 views

Hartshorne's associated scheme for a variety

This question comes from Proposition 2.6 in Chapter 2 of Hartshorne's Algebraic Geometry. In my edition, that's on page 78. For a variety $V$, Hartshorne defines the topological space $t(V)$ to ...
4
votes
2answers
799 views

morphisms of affine schemes question

So, in chapter 2, section 2 of Hartshorne, (prop 2.3), he describes how if $\varphi : A\rightarrow B$ is a homomorphism of rings, then you get a morphism of (affine schemes): ...