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
86 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 $\rho\colon\...
8
votes
1answer
247 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
58 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
41 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) \not\...
3
votes
2answers
56 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
26 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
106 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
47 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
62 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
62 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 $\mathbb{P}^{1}_{\mathbb{...
10
votes
1answer
185 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
58 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
84 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 $S$-...
4
votes
1answer
95 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
48 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
77 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
114 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
44 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
60 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
34 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
151 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
59 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
66 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
70 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
75 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
50 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
36 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 $X_1,X_2$, $$\operatorname{hom}(X_1,...
1
vote
0answers
28 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
297 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
32 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}) = \...
1
vote
0answers
24 views

Pulling back the sheaf of ideals under diagonal embedding

Let $X$ be a scheme (over a field $k$) and $\Delta:X\to X\times X$ be a diagonal embedding. Let $I$ be a sheaf of ideals of the diagonal $\Delta(X)$ in $X\times X$. For positive integer $n$ how could ...
1
vote
0answers
12 views

Classification of group schemes of order 2: Why is $sx=x\otimes 1+1\otimes x-cx\otimes x$?

In their paper Group Schemes of Prime Order, Tate and Oort state Suppose $$G=\operatorname{Spec}(A)\text,\quad S=\operatorname{Spec}(R) \text,$$ and suppose the augmentation ideal $I=\...
0
votes
1answer
93 views

An algebraic variety as a scheme

If we take the definition of a variety as a reduced integral scheme of finite type over an algebraically closed field $k$, then a variety is in particular a scheme over $k$, so is a scheme $X$ with a ...
1
vote
0answers
34 views

Milne's definition of $G$-torsors

I am trying to understand the proof of proposition 4.1 of Milne's book Étale Cohomology (p.120) and I am getting really confused with some points of the reverse implication: if I understand correctly ...
2
votes
1answer
88 views

Inverse function theorem for etale morphisms

Looking around stackexchange, it seems there are many related questions, but I'm a beginner and I can't find a proof on the internet (without going through the more general results in stacks project). ...
1
vote
0answers
50 views

Pushforward of a quasicoherent sheaf on a noetherian scheme is quasicoherent

I'm reading through and trying to understand the proof of Proposition 5.8 in Chapter II of Algebraic Geometry by Hartshorne that the pushforward of quasicoherent sheaf $\mathcal{F}$ by a morphism $\...
2
votes
1answer
58 views

Confusion Over Geometric Irreducibility

Let $X$ be a scheme over a field $k$. For an extension field $K$ of $k$, we can change the base to obtain the scheme $X_K$. Supposedly it is possible that $X$ is irreducible while $X_K$ is reducible, ...
4
votes
0answers
121 views

Learning Galois theory geometrically?

Recently I started poking at algebraic geometry and commutative algebra. My background is basic category theory and basic algebraic topology. I don't know a lot of other mathematics. I noticed Galois ...
3
votes
1answer
41 views

All open subsets of the spectrum of a number field are principal

Let $K$ be a number field, $\mathcal{O}_k$ its ring of integers, $\operatorname{Cl}(K)$ its class group and $h_K = \lvert \operatorname{Cl}(K)\rvert$ its class number. Let $X = \operatorname{Spec}(\...
1
vote
0answers
31 views

Relative affine schemes as zero-locuses

Let $f : Y \to X$ be an affine morphism of schemes (say of finite type over a field $k$). I read in some notes that $Y$ can be written as $Y \cong E \times_{E'} X$ where $u : E \to E'$ is a morphism ...
1
vote
1answer
45 views

$f:X\rightarrow Y$ is a closed immersion iff $f:f^{-1}(U_i)\rightarrow U_i$ is a closed immersion.

I came across the following property of closed immersions on Wikipedia - A morphism $f:Z\rightarrow X$ is a closed immersion iff for some (equivalently every) open covering $X=\bigcup U_j$ the ...
0
votes
0answers
32 views

A finite morphism between affine schemes is closed. How can we generalize this to show any finite morphism of schemes is closed?

I am working on Exercise II.3.5 b) in Hartshorne. It asks to show that a finite morphism is closed. I am working on a way to show that this is true in the affine case, but I am having trouble showing ...
0
votes
0answers
100 views

Is this solution to Hartshorne Exercise III.10.2 correct?

Exercise III.10.2 in Hartshone's Algebraic Geometry is discussed in this question. The accepted answer proceeds by showing that the smooth locus is open on $X$, then transferring this to $Y$ using the ...
1
vote
0answers
34 views

Are $ \mathbb{A}^n (k ) = k^n $ and $ \mathbb{P}^{n} (k) = \mathrm{Proj} \ k[X_0 , \dots , X_n ]$ irreducibles?

Are $ \mathbb{A} (k ) = k^n $ and $ \mathbb{P}^{n} (k) = \mathrm{Proj} \ k[X_0 , \dots , X_n ]$ irreducibles when $ k $ is a domain ? Thanks in advance for your help.
2
votes
2answers
61 views

Sheafyness and relative chinese remainder theorem

The relative chinese remainder theorem says that for any ring $R$ with two ideals $I,J$ we have an iso $R/(I\cap J)\cong R/I\times_{R/(I+J)}R/J$. Let's take $R=\Bbbk [x_1,\dots ,x_n]$ for $\Bbbk $ ...
1
vote
1answer
37 views

Fiber product with diagonal morphism [duplicate]

Stacks tag 01KR states that the diagram of schemes is "by general category theory" "a fibre product diagram". I tried to show this using the universal property, but didn't obtain anything useful. ...