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 ...
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+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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2answers
125 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 ...
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1answer
54 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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43 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 ...
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1answer
34 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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1answer
37 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 $H\subseteq\mathbb P^n$ ...
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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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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 ...
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 ...
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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 ...
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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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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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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. ...
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1answer
39 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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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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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 ...
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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 ...
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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!
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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 ...
4
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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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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 ...
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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 ...
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1answer
79 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
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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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 ...
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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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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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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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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 ...
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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 ...
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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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67 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 ...
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0answers
32 views

Closed subscheme vs open subscheme in ring of dual numbers

If we take the ring of dual numbers $R=k[x]/(x^2)$ for algebraically closed field $k$, we note that by the ideal correspondence theorem, the only prime ideal in $R$ is $(x)$. Thus the scheme $spec ...
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1answer
37 views

Question about composition of morphisms of schemes (Mumford's)

On section 7 of Chapter 2 of Mumford's Red Book, there is the following statement: Suppose $X\stackrel{f}{\rightarrow}Y\stackrel{g}{\rightarrow}Z$ are morphisms of schemes, such that $g$ is of finite ...
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1answer
37 views

Simple questions about morphisms of finite type and proper morphisms

I'm studying algebraic geometry (it is my first course), by following Mumford's Red Book, and now I'm stacked in some (probably silly) questions about schemes, more precisely in the section 7 of ...
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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 ...
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20 views

Hilbert series computation for Hilbert scheme of $n$ points on $\mathbb C^2$

How can we show that $$\sum_{n = 0}^\infty q^n \operatorname{character}_T S^n(\mathbb C[x,y])= \prod_{p_1,p_2\geq 0}\frac{1}{1-t_1^{p_1}t_2^{p_2}q}$$ where $T$ acts on $x,y$ as $(t_1,t_2)$? ...
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104 views

Hartshorne's Exercise II. 2.15 (fully faithful functor)

I'm struggling with the last part of the exercise. Namely, let $V,W$ be any two varieties over a field $k$. We build the functor $t$, which induces a natural map $$ ...
2
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1answer
47 views

integral ring homomorphism

Consider a homomorphism $f: A\to B$ of commutative rings and let $b\in B$. Let $g\colon A[X]\to B[X]$ be defined by $g(X) = X$. Put $I = g^{-1}((bX-1))$ (contraction of the ideal $(bX-1)\subseteq ...
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53 views

Point at infinity of the scheme $\mathbb{P}^1$

Let $\mathbb{P}^1_k$ be the scheme defined as either $\text{Proj }k[t]$ or obtained by gluing two affine lines appropriately. What is the point at infinity which in topology usually is given by ...
3
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1answer
61 views

Questions about tangent and cotangent bundle on schemes

In differential geometry, for a smooth manifold $M$ we have the definition of the tangent bundle and the cotangent bundle and then $k$-forms are defined to be (smooth) sections of the $k$-exterior ...
2
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1answer
58 views

When the fibers of a flat morphism are varieties.

Notations: As in Hatshorne's book. Suppose that $f:X\longrightarrow Y$ is a flat morphism between two non-singular projective varieties over an algebraically closed field. Are the fibers of $f$ ...
4
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1answer
61 views

A particular example of a non-reduced scheme (with a reduced ring of global sections)?

I am searching for a scheme $X$ which can be obtained by gluing two affine schemes (along open subsets) such that: 1) X is non-reduced; 2) $\Gamma(X,\mathcal O_X)$ is a reduced ring. Any ideas? ...
4
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1answer
64 views

Is Spec ($k[x_1,x_2,\ldots])$ a smooth $k$-scheme?

Let $k$ be a field and let $A = k[x_1,x_2,\ldots]$. Note that $A$ is not of finite type over $k$. Is $\operatorname{Spec} A\to \operatorname{Spec} k$ a smooth morphism of schemes? I think it is ...