The concept of scheme mimics the concept of manifold obtained by glueing pieces isomorphic to open balls, but with different "basic" glueing pieces. Properly, a scheme is a locally ringed space locally isomorphic (in the category of locally ringed spaces) to an affine scheme, spectrum of a ...

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7
votes
1answer
67 views

“Lifting” fibres of morphism of arithmetic schemes to get rid of “nongeometric” ramification

This is a soft question and really a request for pointers towards a certain rigorous formulation of geometric intuition I've had for some "arithmetic schemes". I'm looking for ideas and key references ...
1
vote
0answers
52 views

Does affine open set equal to distinguished open subset in an affine scheme?

For $A$ a commutative ring, does it always hold that all affine open subschemes of $\text{Spec }A$ lie over a distinguished open subset of $\text{Spec} A$?
1
vote
1answer
76 views

Is surjectivity preserved in open neighborhoods?

Let $X,S$ be schemes of finite type over a field and let $f:X\times S\to S$ be the projection. Suppose we have a morphism of coherent sheaves $\phi:\mathscr E\to \mathscr F$ on $X\times S$. Is it ...
1
vote
1answer
61 views

Differential forms on a scheme: unclear equation

Disclaimer: In this question I assume that the reader is familiar with the construction of the module of differentials $\Omega^1_{B|A}$ where $B$ is an $A$-algebra. (If you need more details about ...
3
votes
0answers
113 views

Is the function field of a variety a function field?

Let $X$ be an integral Noetherian scheme of dimension $n$ over a field $k$ (arbitrary field). The function field of $X$ is defined as $K(X):=\mathcal O_{X,\eta}$ where $\eta$ is the generic point of ...
0
votes
1answer
31 views

Function field on a regular scheme of dimension $1$

Let $(X,\mathcal O_X)$ be a locally noetherian scheme of dimension $1$ and suppose that $X$ is regular, that is: $\mathcal O_{X,x}$ is a regular local ring. We have no other hypothesis on $X$. What ...
0
votes
1answer
31 views

Is the set of affine morphisms the smallest set in this specific sense?

One motivation for affine morphisms I have seen is that: $\operatorname{Spec} A \to \operatorname{Spec} \Bbb Z$ should be an affine morphism for any ring $A$. The set of affine morphisms should be ...
4
votes
1answer
54 views

What is the relative version of a reduced scheme?

I have often heard it said that it is important to think of properties of a scheme $X$ as really a special case of a property of morphisms applied to the morphism $X\to \operatorname{Spec} \Bbb Z$. ...
4
votes
0answers
52 views

Is the scheme-theoretic image stable under taking products?

Let $f:X\to Y$ be a morphism of schemes and let $Z\subset Y$ be its scheme-theoretic image. If $T$ is any other scheme, consider the induced morphism $g=f\times 1_T:X\times T\to Y\times T$. ...
0
votes
1answer
45 views

Stalks of the ideal sheaf of an irreducible subscheme

Suppose that $X$ is a noetherian scheme such that $Z\subseteq X$ is a closed subscheme. Clearly $Z$ define an ideal sheaf $\mathscr I\subset\mathscr O_X$. Now let $z\in Z$ be a point such that it is ...
6
votes
2answers
142 views

Vakil's definition of smoothness — what happens at non-closed points?

The following is definition 12.2.6 in Vakil's notes. A $k$-scheme is $k$-smooth of dimension $d$, or smooth of dimension $d$ over $k$, if it is pure dimension $d$, and there exists a cover by ...
1
vote
0answers
29 views

Calculating sheaf of differential operators for smooth scheme

I have heard that if $X$ is a smooth scheme over $k$, then we can calculate the sheaf of differential operators $\mathcal{D}_X$ by considering étale morphisms from an affine open set to ...
1
vote
1answer
64 views

Hartshorne Exercise II.2.18(d)

The Exercise: Let $\phi: A \rightarrow B$ be a ring homomorphism and let $X = \operatorname{Spec} A, Y = \operatorname{Spec} B$. Let $f: Y \rightarrow X$ be the morphism of schemes induced by $\phi$. ...
3
votes
1answer
49 views

Local properties of morphisms of schemes

In Hartshorne Proposition II.5.8, he shows, given a morphism $f \colon X \to Y$ where X and Y are schemes and $\mathcal{G}$ a quasi-coherent sheaf of $\mathcal{O}_{Y}$- modules, that ...
2
votes
1answer
38 views

Quasicoherent sheaf on the functor of points is the same as on the scheme itself

I've seen the definition of a quasicoherent sheaf $\mathcal{F}$ on an arbitrary functor $$ X : CRing \to Sets $$ as a specification of an $R$-module $\mathcal{F}(x)$ for each $R$-point $x \in X(R)$ ...
1
vote
1answer
22 views

Infinitely many non-isomorphic degree 8, dimension zero schemes in the plane

In Geometry of Schemes by Eisenbud and Harris, it is claimed in Exercise II-19 that: There are infinitely many isomorphism types of degree 7 subschemes supported at the origin in 3-space, and ...
7
votes
1answer
83 views

How does Hartshorne's definition of group schemes encode the law for the neutral element?

Hartshorne's Algebraic Geometry says A scheme $X$ with a morphism to another scheme $S$ is a group scheme over $S$ if there is a section $e\colon\;S\to X$ (the identity) and a morphism ...
8
votes
1answer
231 views

Geometric intuition for the Stein factorization theorem?

What is the intuition behind the Stein Factorization Theorem? I understand that it was originally a theorem in several complex variables, so I was wondering if there's some geometric explanation that ...
1
vote
0answers
50 views

Geometric intuition for normalization as intersection of valuation rings?

Why should the normalization of a ring correspond to the intersection of valuation rings containing it? I am looking for a geometric explanation, if possible. I understand that normalization at a ...
0
votes
0answers
29 views

Completion along locally closed subscheme

If $X$ is any scheme over $k$ then we know that the image of the diagonal $\Delta(X)$ is locally closed in $X \times_k X$, so that there is an open set $W$ of $X \times_k X$ with $\Delta(X)$ closed in ...
2
votes
1answer
38 views

Proving a set is open in a locally ringed space $(X,\mathscr O_X)$

Let $(X, \mathscr O_X)$ be a locally ringed space and let $A = \Gamma(X,\mathscr O_X)$ be the global sections. For $f\in A$, define the "distinguished open base" as: $$D(f) = \{x\in X : \pi_x(f) ...
3
votes
2answers
55 views

Open covering of a scheme and global sections

Let $X$ be a scheme. For a global section $f\in\Gamma(X,\mathcal O_X)$, let $X_f=\{x\in X\mid f_x\not\in\mathfrak m_x\}$. For $f_1,...,f_n\in\Gamma(X,\mathcal O_X)$, I wish to know if the following ...
1
vote
0answers
38 views

Compatibility of isomorphisms between distinguished opens

Let $f\colon X\to Y$ be a morphism of schemes. Let $\operatorname{Spec}A,\operatorname{Spec}C$ be affine open subschemes of $Y$ such that $\operatorname{Spec}A_g=\operatorname{Spec}C_f$ for some $g ...
2
votes
0answers
24 views

A category of closed immersions

Fix a scheme $Z$, and consider a category whose objects are schemes $X$ equipped with a closed immersion $Z\to X$. Obviously, a morphism $f:X\to Y$ should commute with the respective closed ...
3
votes
2answers
94 views

Residue fields of schemes of finite type (over $\mathbb{Z}$)

Suppose $X$ a scheme of finite type over $\mathbb Z$. I want to prove that: (1) The residue fields of closed points of $X$ are finite; (2) For a given $q=p^n$ with $p$ prime, there is only a finite ...
0
votes
1answer
46 views

Scheme morphism properties that aren't stable under taking triangles?

Let $\mathcal{P}$ be the collection of properties of morphisms of schemes that satisfy the following conditions: Stability under arbitrary pullbacks Stability under composition There's a nice list ...
2
votes
1answer
61 views

Composition of morphisms of locally ringed spaces

I have a specific question about defining the composition in (locally) ringed spaces. The definition I had formulated myself while reading Hartshorne, since he conveniently neglected to suggest any ...
2
votes
0answers
51 views

Construction of Tate curve and formal schemes

In the notes websites.math.leidenuniv.nl/geom/tate.ps (and probably in other places), there is a construction of the Tate curve, where the steps are summarized below. 1) Take ...
10
votes
1answer
167 views

What is the geometric meaning of representability?

Representable functors play a large role in algebraic geometry when developed through the 'functor of points' approach. One finds schemes represent Zariski sheaves and this gives access to the great ...
0
votes
1answer
54 views

Closed subscheme defined by kernel of diagonal homomorphism

Let $f: X \to Y$ be a morphism of schemes. Let $\Delta : X \to X \times_Y X$ denote the diagonal morphism. Take $U = \textrm{Spec } A$ and $V = \textrm{Spec } B$ be affine open subsets of $Y$ and $X$ ...
6
votes
1answer
82 views

Is representability of Zariski sheaves local on the base?

Let $F: \mathsf{Sch_{/S}}^{op} \to \mathsf{Set}$ be a Zariski sheaf on the category of $S$-schemes. $F$ being a sheaf means it satisfies the following property: Sheaf condition: For every ...
4
votes
1answer
93 views

Geometric intuition behind $V(M)=\operatorname{Spec}(\operatorname{Sym}(M))$?

In the equivalence between geometric vector bundles and locally free sheaves we assign to a locally free sheaf $M$ the bundle $V(M)=\operatorname{Spec}(\operatorname{Sym}(M))$. I don't doubt the ...
1
vote
0answers
43 views

About normalizer subgroup scheme

Let $S$ be a scheme and let $G$ be a group scheme over $S$. Let $X$ be an $S$-subscheme of $G$; we can define the controvariant functor: $$ \mathbf{N}_G(X):\mathbf{Sch}_{S}\to\mathbf{Group}\\ \forall ...
1
vote
0answers
72 views

Action on algebraic variety and adjoint bundles

Let $X$ be a complex algebraic variety and let $G$ be a complex algebraic group; I mean that $X$ is a reduced, separated scheme of finite type on $Spec\mathbb{C}$, and the underlying set of $G$ is a ...
0
votes
1answer
99 views

Classifying non-reduced points in noetherian schemes

Let $X$ be a noetherian scheme. So in particular $X$ is a finite, locally finite, union of its irreducible components $X = \bigcup^n_i K_i$. Non-reduced points in $X$ fall into 2 categories: Fat ...
2
votes
0answers
41 views

If $X$ is a proper scheme over $k$, is $X/G$ separated?

Let $X$ a proper scheme over a field $k$ and let $G$ a finite group of its automorphism (as $k$-scheme). Let suppose that the quotient $X/G$ exists, is it is separated? How to prove it? If the ...
0
votes
1answer
53 views

Finite type and finite fibers implies quasi-finite

I am trying to understand different finiteness conditions, in particular I am looking at the following exercise from Algebraic Geometry and Arithmetic Curves by Qing Liu: Let $f:X\rightarrow Y$ be a ...
3
votes
0answers
33 views

Motivation of completions of schemes?

Sorry if this is a stupid question, but what are the uses of a completion of a scheme along a closed subscheme? Are there any nice universal properties it satisfies, or do certain morphisms factor ...
4
votes
1answer
139 views

How much algebraic geometry is there in complex geometry (for example, Demailly)?

I wonder how much of modern algebraic geometry (schemes, etc) is there in complex (algebro-analytic) geometry. What I mean is complex algebraic (analytic) geometry in the sense of Demailly, Lasarsfeld ...
1
vote
1answer
57 views

Schemes whose ring of global sections separate points

Let $X$ be a scheme over an algebraically closed field. We say that $\Gamma(X,\mathcal{O}_X)$ separates points iff for every $x,y \in X$ there's an $f \in \Gamma(X,\mathcal{O}_X)$ with $f(x)\ne f(y)$. ...
1
vote
2answers
49 views

Example of a module which is free at an isolated point

I'm looking for the most simple example of a quasicoherent sheaf $\mathcal{F}$ over a scheme $X$ (preferably affine for simplicity) which has a free stalk $\mathcal{F}_x$ at a point $x \in X$ and yet ...
2
votes
1answer
56 views

Hypersurfaces meet everything of dimension at least 1 in projective space

The following exercise is taken from ravi vakil's notes on algebraic geometry. Suppose $X$ is a closed subset of $\mathbb{P}^n_k$ of dimension at least $1$, and $H$ is a nonempty hypersurface in ...
3
votes
0answers
47 views

Can the theorem on semicontinuity of fiber dimension be explained as a semi-continuty of rank of a quasi-coherent sheaf?

Here are two theorems: If $F$ is a finite type quasi-coherent sheaf on a scheme $X$, then $ rank(F)(p) = dim_{k(p)} F_p \otimes_{O_p} k(p)$ is a upper semicontinuous function on $X$. If $\pi : X \to ...
1
vote
1answer
74 views

$\Gamma(X,-)$ for Quasi-Coherent/Coherent Sheaves Maps to R-modules/Finitely Generated R modules

I'm trying to show that the functor $\Gamma(X,-)$ from the category of quasi-coherent sheaves maps a quasi-coherent sheaf to an R-Module, and also that for coherent sheaves the same functor takes the ...
3
votes
1answer
28 views

Valuations coming from Prime Divisors

I'm trying to understand where the valuation defined by a prime divisor on an integral, Noetherian separated scheme regular in codimension 1 comes from. In particular, I'm looking at this example: ...
3
votes
1answer
44 views

What is the “module of twisted global sections”?

Let $X$ be a projective variety. Suppose we have computed the graded modules corresponding to $\Omega_{\mathbb P^n}$ (the cotangent sheaf) and $\mathcal O_X$. One way to get a representation for ...
1
vote
1answer
35 views

Fully faithful functor from Preschemes to $Funct((Rings),(Sets))$

I am trying to fill in the details of the proof of Proposition 2, II. Preschemes, §6, in Mumford's Red Book (page 114, 2nd edition). For any two preschemes $X_1,X_2$, ...
1
vote
0answers
27 views

A characterization of normal schemes (clarification of a statement of proposition)

The following is taken from 4.1 in Liu's book. Definition: A scheme $X$ is normal at $x \in X$ if $O_{X,x}$ is normal. $X$ is normal if it is irreducible and normal at every point. ...
13
votes
1answer
261 views

When can stalks be glued to recover a sheaf?

Let $\mathcal{F}$ be a sheaf over some topological space. The stalks are $\mathcal{F}_x= \underset{{x\in U}}{ \underrightarrow{\lim}} \mathcal{F}(U)$. Is there a special name for a sheaf that ...
1
vote
0answers
30 views

Construction of relative projective space via glueing

I would like to gain further practice on glueing schemes by constructing projective space over a ring. I am considering the following: I wonder how we get that $\mathcal{O}_{X_i} (X_{ij}) = ...