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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5
votes
0answers
292 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
429 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 ...
3
votes
1answer
325 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 ...
29
votes
1answer
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
3answers
915 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
270 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
728 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
534 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
255 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
230 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 ...
2
votes
1answer
336 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
406 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
526 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
290 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
456 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}) ...
10
votes
1answer
771 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
342 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 ...
6
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
711 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
231 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
457 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
268 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
414 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
907 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
314 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
365 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
331 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
808 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
759 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): ...
2
votes
1answer
545 views

k rational points and base change

This could be a tricky question but could help me to better understand these very interesting things. Let $X$ be an algebraic variety over a field $k$ (in the sense of a k-scheme like in Qing Liu), ...
7
votes
1answer
686 views

Closed points of valuation scheme

In the excellent book "Algebraic Geometry 1" of Görtz & Wedhorn, in exercise 3.14, one is asked to show that in the spectrum of a valuation ring with infinitely many primes, the complement of the ...
2
votes
0answers
328 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 ...
7
votes
1answer
423 views

Complex analytic space with no (positive dim.) subscheme ?

Is there an example of a complex analytic space $X$ that doesn't have any (not necessarily open or closed) positive dimensional subspace $Y$ which is analytically isomorphic to (the complex ...
16
votes
2answers
663 views

What is an explicit example of a variety X which is finite over Spec F_p but which does not lift to a scheme Y which is finite and flat over Spec Z_p?

What is an explicit example of a variety X which is finite over Spec F_p but which does not lift to a scheme Y which is finite and flat over Spec Z_p?
1
vote
1answer
452 views

Is every regular (excellent) scheme separated?

Sorry for one more stupid AG question. I need schemes that are regular, excellent and separated. Are these three conditions independent?
4
votes
0answers
296 views

Vector bundles of schemes and their topological realizations

Hi, there is a realization functor $R_\mathbb{R}$ from schemes of finite type over $\mathbb{R}$ to topological spaces and there is also a functor $R_\mathbb{C}$. Does $R_\mathbb{R}$ send an ...
5
votes
1answer
296 views

Topological space associated to a real or complex scheme

Hi, consider a scheme $X$ of finite type over $\mathbb{R}$ (or $\mathbb{C}$). In Hartshorne's appendix B on 'transcendental methods' it is shortly mentioned how to assign a reasonable topological ...
2
votes
0answers
233 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 ...
129
votes
14answers
8k views

What elementary problems can you solve with schemes?

I'm a graduate student who's been learning about schemes this year from the usual sources (e.g. Hartshorne, Eisenbud-Harris, Ravi Vakil's notes). I'm looking for some examples of elementary ...
2
votes
3answers
401 views

Sections of morphisms of schemes up to a finite morphism

Let $f:X\longrightarrow S$ be a flat projective morphism of regular integral noetherian schemes such that that the generic fibre $X_\eta\longrightarrow K(S)$ is a smooth projective connected curve ...
2
votes
1answer
371 views

More on universal homeomorphisms

I would like to understand this notion better; where could I find some examples? In particular, I am interested in the following questions (and references for the answers). Is a universal ...
4
votes
2answers
238 views

Is the pre-image of a regular subscheme with respect to a universal homeomorphism of regular schemes regular?

Let $f:X\to Y$ be a universal homeomorphism of regular (excellent finite-dimensional) schemes, $Z\subset Y$ be a regular subscheme. Is $f^{-1}(Z)$ necessarily regular?
1
vote
0answers
140 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 ...
3
votes
1answer
275 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 ...
5
votes
0answers
511 views

Do all the main properties of constructible and perverse sheaves (in an 'arithmetic' situation) follow from results of Gabber?

This question is a continuation of Bad behaviour of perverse sheaves over 'general' bases? Let $S$ (for example) be a finite type separated scheme over $\mathbb{Z}$. I would like: (1) to ...
1
vote
2answers
660 views

Coproducts of schemes (“gluing construction”) ?

In this MO question it was raised the topic of "gluing constructions" in the category of schemes. I understand the phrase "gluing two schemes along maps to them" as "there exists a coproduct of the ...
1
vote
1answer
244 views

Group scheme of infinite dimensional linear groups ?

Hi there, I know there are fairly straightforward ways to write down the schemes of infinite dimensional projective spaces (not restricting myself to only countable dimensions), but what happens with ...
7
votes
2answers
541 views

Scheme-theoretic account of why every variety embeds in a complete variety

The standard reference for the statement that "any abstract variety is an open subscheme of a complete variety" is Nagata's 1962 paper Imbedding of an abstract variety in a complete variety. ...