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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Closed sets flat morphism

Consider the sets $Y_{k}:=\{y\in Y\mid |f^{-1}(y)|\leq k\}$, where $f:X\to Y$ is a flat quasi-finite morphism between smooth irreducible affine varieties and $k$ a natural number. Are these sets ...
3
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1answer
90 views

Morphisms from spectra to schemes

Let $X$ be a scheme. Show that for any $x \in X$ there exists a canonical morphism $\textrm{Spec}\, \mathcal{O}_{X,x} \rightarrow X$. If $k(x)=\mathcal{O}_{X,x}/\frak{m}_{x}$ is the residue field at ...
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2answers
199 views

Scheme over S and morphisms

Quoting from Hartshorne Let $S$ be a fixed scheme. A scheme over $S$ is a scheme $X$, together with a morphism $X \to S$. If $X$ and $Y$ are schemes over $S$, a morphism of $X$ to $Y$ as schemes ...
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1answer
70 views

degree of morphism of schemes

Let $\phi: Y \to X$ be a finite etale morphism of proper smooth connected schemes over a field $K$ and suppose that the induced morphism $\phi: \overline{Y} \to \overline{X}$ has degree $n$, where ...
3
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1answer
64 views

Morphisms of schemes

I know that there is a morphism of schemes between $\mathbb{A}_k^{n+1}-\{0\}$ and $\mathbb{P}_k^n$, where $k$ is a field, given by $$(x_0,...,x_n) \to [x_0,...,x_n].$$ But, how can I stricly prove ...
3
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1answer
152 views

Graph morphism for a separated morphism of schemes

I want to prove the following: Let $f: X \rightarrow S$ be a separated morphism of schemes. Show that any section $g: S \rightarrow X$ of $f$ i.e. a morphism such that $f \circ g=\textrm{id}_{S}$ is ...
2
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1answer
163 views

Separated Morphisms of Schemes

(a) Let $f:X \rightarrow S$ be a separated morphism of schemes. Show that for any subscheme $U \subset X$, the restriction $f\mid_{U}:U \rightarrow S$ is separated. (b) Let $R$ be a commutative ring ...
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1answer
585 views

Finite morphisms of schemes are closed

I want to prove that finite morphisms of schemes are closed, but I cannot prove the affine case, namely: Given a finite morphism of rings $\varphi :B \to A$ prove that the induced morphism of ...
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1answer
115 views

“This property is local on” : properties of morphisms of $S$-schemes

I am learning schemes theory at school and I have for now only lectures notes that I am taking during the course. The professor is quite often using following expressions, without having defined them ...
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47 views
+50

Regular sequence of sections of line bundles over a coherent sheaf

I am reading the first chapter from the book by Huybrechts and Lehn, where I encountered the following definition. I have the following doubts regarding this definition : What is the map ...
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11 views

A property P of morphisms of $S$-schemes $f : X \rightarrow Y$ is local on $X$, or $Y$, or $S$ or [duplicate]

I have asked the same question on math.stackexhange here, but thought that is was a good idea to post it here also. I am learning schemes theory at school and I have for now only lectures notes that ...
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3answers
91 views

Is $Spec(R_p)$ to $Spec(R)$ is an open immersion?

Let $R$ be a ring. $p$ be a prime ideal of $R$ . Let $R\rightarrow R_p$ be the canonical. Consider the map of schemes $Spec(R_p) \rightarrow Spec(R)$. Is it an open immersion. $Spec(R_p)$ is an open ...
4
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2answers
126 views

When is the pushforward of a quasi-coherent sheaf quasi-coherent? Hartshorne proof

In the following we have $f:X \rightarrow Y$ morphism of schemes, $\mathcal{F}$ a quasi-coherent sheaf on $X$ and I am referring to proposition 5.8 page 115 in Hartshorne. To prove that the ...
2
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1answer
55 views

Properties of fibers of a morphism of varieties

In this question, all varieties are supposed to be over an algebraically closed field $k$. Hypothesis: X is a smooth projective surface and $f:X\longrightarrow \mathbb P^1$ is a morphism with we ...
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1answer
35 views

closed point of a locally finite $k$-scheme

Let $X$ be a locally finite $k$-scheme, where $k$ is a field. Suppose I have $Spec B \subseteq X$ such that $B$ is a finitely generated $k$-algebra, and $p \in Spec B$ a closed point inside $Spec B$ ...
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44 views

Stalk of a locally finite type $k$-scheme, where $k$ is a field.

I think there is something I am not understanding and I am a bit confused at the moment. I would appreciate any help! Let $X$ be a locally finite type $k$-scheme, where $k$ is a field. Say $p \in ...
3
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1answer
38 views

Bertini's theorem for surfaces: informations about singular fibers.

Let $S$ be a complex non-singular projective surface embedded in some $\mathbb P^n$. Thanks to the Bertini's theorem (Hartshorne theorem II.8.18) there exists a hyperplane $H\subseteq\mathbb P^n$ ...
2
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1answer
50 views

affine morphism of schemes

Let $X$ be a smooth projective variety over a field $k$ and $\mathcal{E}$ a locally free sheaf of finite rank. Consider the total space ...
3
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1answer
44 views

Projective dimension of a coherent sheaf in a short exact sequence

Let $X$ be a noetherian integral scheme. We define the projective/homological dimension of a torsion free coherent sheaf $E$ to be $\mathrm{dh}(E)= \sup\{dh(E_x)|x\in X\}$, here dh$(E_x)$ denotes the ...
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2answers
40 views

Why is dh$(k(x))=$dim $X$?

Let $X$ be an integral Noetherian scheme. Let $x\in X$ be a regular closed point of $X$. Then Huybrechts and Lehn in his book, says that dh$(k(x))=$dim $X$. Here dh$(k(x))$ refers to the ...
4
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0answers
78 views

Eisenbud/Haris, Exercice I-53: morphism between global spectra

Let $X=\operatorname{Spec}(\mathscr{F})$ and $Y=\operatorname{Spec}(\mathscr{G})$ two global spectra over a scheme $S$ and let $f:X\to Y$ be a morphism. I want to show that for all prime ideal sheaf ...
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1answer
38 views

Are the fibers of this morphism reduced?

Let $X$ be a non singular complex projective surface (Hartshorne notation!) and consider a morphism $f:X\longrightarrow\mathbb P^1_{\mathbb C}$ with the following properties: $f$ is flat $f$ is ...
3
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83 views

Equivalent definition of Schemes

I recall seeing that the category of schemes can be captured by a general construction as follows. Let $\mathbf{Spec}\colon \mathbf{CRing}^{op}\to \mathbf{LRS}$ be the usual functor from the ...
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1answer
99 views

Why can schemes of finite type over $\mathrm{Spec}\left(k\right)$ be considered to be affine?

Let $k$ be a field (not necessarily algebraically closed). We call $k$-variety a scheme of finite type over $\mathrm{Spec}\left(k\right)$. Let $X$ be a geometrically reduced $k$-variety and $Y$ a ...
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73 views

Do schemes help us understand elliptic curves?

I'm reading Silverman and Tate's "Rational Points on Elliptic Curves" and I'm very much enjoying learning about these objects, and in particular doing a bit of number theory. It's different to what ...
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0answers
13 views

Complexifying Lie group actions

In Atiyah-Pressley's paper Convexity and Loop Groups, it is claimed that the Bruhat cells $C_\lambda$ are invariant under the natural $S^1 \times T$-action and that the authors will prove this claim. ...
5
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2answers
164 views

function field of an integral scheme

Suppose $X$ is an integral scheme, and let $\eta \in X$ be its generic point. Then the local ring $\mathcal{O}_{X,\eta}$ is a field, called the function field of $X$ and denoted $K(X)$. Why is $K(X)$ ...
4
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1answer
85 views

The Theorem on Formal Funtions - Harthshorne Theorem 11.1

Let $f:X \rightarrow Y$ a projective morphism of noetherian schemes, let $\mathcal{F}$ be a coherent sheaf on $X$, and let $y\in Y$ be a point. For each $n\geq 1$ we define $X_n=X \times_Y Spec ...
2
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1answer
60 views

A rigorous characterization for a ringed spaces to be isomorphic to an affine scheme.

On page 21-22 of the book The Geometry of Schemes by Eisenbud and Harris there is a characterization for when a ringed spaces $(X,\mathcal{O}_X)$ is isomorphic to an affine scheme ...
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1answer
41 views

Equality of two $K$-valued points for reduced $K$

I'm reading "Red Book of varieties and schemes". There is definition 2, page 118. Let $f,g:K \to X $ be to $K-$valued points of scheme $X$, we say that they are equal at $x\in K$ $(f(x) \equiv g(x))$ ...
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2answers
93 views

Spectrum of $\mathcal{O}(U)$

Let $X=\operatorname{Spec}(A)$ be the spectrum of the comm. ring $A$ and let $\mathcal{O}$ be the associated sheaf of rings, i.e. for $U \subseteq X$ open, $\mathcal{O}(U)$ is the ring of all ...
4
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1answer
69 views

What is the algebraic tangent cone really?

Let $A$ be a (commutative unital) ring, let $\mathfrak{a} \subseteq A$ be an ideal, and let $B = A / \mathfrak{a}$. Then we have a descending filtration $$\cdots \subseteq \mathfrak{a}^3 \subseteq ...
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1answer
12 views

open subset of scheme with zero section

Let's take a scheme $X$. Is it possible to have an open non-empty subset $U$ of $X$ such that $\mathcal{O}_X(U)=0$? I can't find an argument against it, since there could exist some open set ...
4
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1answer
92 views

Question related to a stalk of a scheme

Suppose $X$ is a scheme and suppose $C$ and $C'$ are two irreducible components of $X$. Suppose also that $p \in C \cap C'$. Does is it then follow that $O_{X,p}$ is not an integral domain? Thanks!
2
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1answer
55 views

uniqueness of morphism $Spec(K) \to X$ of schemes

let $K$ be a field and $X$ a scheme. I'd like to understand the bijection $Hom_{Sch}(Spec(K), X) \cong \{x \in X | \exists \kappa(x) \to K \}$ That map is given by sending a morphism $f: Spec(K) \to ...
0
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0answers
19 views

higher order schemes or curvature to be smoothed

I want to calculate the radius of curvature of a meandering river. I have done a first estimated using a tool in arcgis. These are the results. The values of -99999 are recorded for points of no ...
2
votes
2answers
59 views

Restriction maps in an integral scheme are injective

Suppose $X$ is an integral scheme. I would like to show that the restriction maps $res_{U,V} : O_X(U) \rightarrow O_X(V)$ is an inclusion so long as $V$ is not empty. I was wondering if someone could ...
2
votes
1answer
80 views

Describing $Spec(\mathcal{O}_K[X])$

Let $K$ be an algebraic number field and $\mathcal{O}_K$ its ring of integers. I am trying to describe $Spec(\mathcal{O}_K[X])$ in terms of fibers of the map $g: Spec(\mathcal{O}_K[X]) \rightarrow ...
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1answer
53 views

Why is : $ \ \ V \times_U W = p^{-1} (V) \bigcap q^{-1} (W) $?

Let $ f: Y \to X $ and $ g : Z \to X $ be two morphisms of schemes. Suppose we know that $ Y \times_X Z $ exists, and let $ p $ dénote its projection on $ Y $, and $ q $ its projection on $ Z $. ...
4
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1answer
56 views

Definition of $\mathbb{A}^n_S$ by glueing

In Eisenbud and Haris (Geometry of schemes I.2.4), if $S=\cup_\alpha U_\alpha$ with the $U_\alpha$ affines, to define $\mathbb{A}^n_S$ one take $X=\cup_\alpha \mathbb{A}^n_{U_\alpha}$ with ...
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1answer
26 views

Closed subscheme of a projective scheme determined by homogeneous ideals

So in Ravi Vakil's notes Ex 8.2C, I have to prove that if $\pi:X\hookrightarrow\text{Proj}\ S_{\cdot}$ is a closed subscheme (here $S_{\cdot}$ is a graded ring finitely generated by elements of degree ...
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1answer
61 views

Why is the functor $S\mapsto\prod_{1\leq j\leq k}\mathcal{O}_{S}^{n}\left(S\right)$ representable by this scheme?

Let $S$ be a scheme and $\mathcal {O}_S$ the structure sheaf of rings over $S$. Question: Why is the functor $S\mapsto\prod_{1\leq j\leq k}\mathcal{O}_{S}^{n}\left(S\right)$ representable by a ...
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1answer
43 views

How to prove that a morphism of schemes preserves the surjectivity of ring endomorphisms?

Let $\left(T,\mathcal{O}_{T}\right)$ and $\left(S,\mathcal{O}_{S}\right)$ be schemes and $f:T\rightarrow S$. $U\subset S$ and $V\subset T$ are some open sets that correspond to the definition of ...
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2answers
54 views

examples of interpreting schemes (Eisenbud)

I am having trouble understanding the role primary decomposition plays in ``interpreting'' the geometric picture of a scheme. Here are the examples I am struggling with from Eisenbud's Commutative ...
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0answers
40 views

Chow lemma + resolution of singularities

I am trying to understand the following simple reduction in Deligne's Theorie de Hodge II. He works in the category of schemes of finite type over $\mathrm{Spec}(\mathbf{C})$. Let $f : X \to S$ be a ...
14
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2answers
636 views

History of Algebraic Geometry: Motivation behind definition of schemes

I am trying to read an article by Jean Dieudonne which talks about development of Algebraic Geometry. The article was being published in the journal "Advances in Mathematics" Volume 3, Issue 3, Pages ...
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1answer
48 views

Does it hold $\mathcal{O}_X (U) =\bigcap_{x \in U} \mathcal{O}_{X, x} \in K (X)$?

maybe this is a stupid question, but I'm not seeing if this is true for some spaces (affines at least). Let $K (X) = \varinjlim\limits_{\emptyset \neq U \in \text{Open}(X)} \mathcal{O}_X (U)$ be the ...
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41 views

Alternative proof of Noether Normalization Lemma

On Mumford's Red Book (end of section 7 of chapter 2, pg 126, 127), there is an alternative proof of Noether's Normalization Lemma that goes like this: For an affine variety $X$ over an algebraically ...
2
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1answer
63 views

What is the canonical morphism of $\mathbb{P}^n_A$ to $\text{Spec }A$?

On Liu's book Algebraic Geometry and Arithmetic Curves, he gives a general definition of the projective scheme $\mathbb{P}^n_A$, for $A$ a ring, as $Proj (B)$, for $B=A[x_0,\dots,x_n]$. Later, he ...
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68 views

Some questions about Qing Liu's proof of Valuative Criterion of Properness

I'm studying algebraic geometry by following Mumford's Red Book, but recently I found Qing Liu's book "Algebraic Geometry and Arithmetic Curves", which seems to be more careful and detailed in its ...