# Questions tagged [schemes]

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

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### two definitions of degree of invertible sheaf on projective curve

Hi I saw two defs from Vakil's FOAG and Görtz&Wedhorn’s AG for the degree of invertible sheaves on projective curve: (Vakil's) 18.4.2. Important definition: degree of a line bundle on a ...
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### Clarify on Propositions 3.36 and 3.38 in Qing Liu's algebraic geometry book

I have two questions, that I put together as they are related to the same topic, the projective scheme of a graded $A$-algebra $B$. Question 1. Do you confirm that points (a), (b) and (c) of ...
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### Is this morphism finite

Let $f:X \to Y$ and $g:Y \to Z$ be morphisms of projective schemes such that $f$ is finite; $g$ is surjective; $\dim Y = \dim Z$; and $g \circ f$ is finite. Is $g$ finite in this case? I'm really ...
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### Bijection between sections of a scheme over $K$ and points with residue field isomorphic to $K$

Let $X$ be a scheme over the field $K$, i.e. there is a morphism $\pi: X\to \operatorname{Spec} K$. Recall that a section for $\pi$ is by definition a morphism $\sigma: \operatorname{Spec} K\to X$ ...
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### Ex. 3.2.2 of Qing Liu: Open immersion into locally Noetherian scheme is of finite type

The definitions are as follows: A scheme is locally Noetherian if the sections over every affine open subset form a Noetherian ring. A morphism $f:X\to Y$ is of finite type if: It is quasicompact (i....
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### Consequences of intersection product equal to $0$

We work over $\mathbb{C}$. Let $X$ be a smooth projective variety, let $D$ be a nef prime divisor and let $C$ be a smooth irreducible curve. I know that, if $D\cap C=\emptyset$, then $D\cdot C = 0$. ...
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### Tangent space to locus of varieties containing a given subvariety (Hilbert schemes)

Suppose that I have a variety $X$ in $\mathbb{P}^r$ of a given type. Then I know that the tangent space to the irreducible component $\mathcal{H}_X$ of the Hilbert scheme containing $[X]$ can be ...
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### Hartshorne Theorem II.4.3

I am trying to understand the proof of Theorem 4.3 in Hartshorne's Algebraic Geometry (the valuative criterion for separatedness). I'm having trouble justifying the highlighted line below: Conversely,...
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### Eisenbud and Harris - Exercise II-12

Exercise II-12 in The Geometry of Schemes says: Show that the subscheme of $\mathbb{A}_K^2$ given by the ideal $(y - x^2, xy)$ arises as the limit of three points on the conic curve $y = x^2$ and is ...
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### Construct schemes like toric varieties on manifolds

I'm reading a book about toric varieties, and some thoughts occured to me. Toric varieties are constructed by fans which are the union of rational cones in $\mathbb{R}^n$. Is it possible to construct ...
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### Chapter 2 proposition 2.6 Hartshorne Algebraic Geometry

This may be dumb but I don’t really understand what we are saying. Hartshorne claims: Let $V$ be a variety over an algebraically closed field $k$ and let $\mathcal{O}_V$ be its sheaf of regular ...
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### Finite type ring homomorphisms and finitely many localizations

Let $\phi : B \to A$ be a homomorphism of commutative rings with identity. Let $f_1, \ldots, f_n$ be elements of $A$, such that $(f_1, \ldots, f_n) = (1)$. I want to prove that if each of the ...
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### coherent sheaves annihilated by ideal sheaves and morphisms between them

Let $X$ be a Noetherian scheme and $\mathcal I\subseteq \mathcal O_X$ be a coherent ideal sheaf defining a closed subscheme $Z$ of $X$. Let $i:Z\to X$ be the closed immersion. I have the following ...
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### Characterization of reduced closed subschemes of a scheme

I'm proving the fact stated below in italics. I couldn't find a detailed proof anywhere, as it seems that it is a ordinary fact to check; the hint I managed to find suggested to reduce to the affine ...
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### Projection from scheme-theoretic fibre is homeomorphism onto the fibre. [duplicate]

$\newcommand{\Spec}{\operatorname{Spec}}$ $\require{AMScd}$ Let $f:X\rightarrow Y$ be a morphism of schemes, and suppose $y\in Y$. Let $X_y=\Spec k_y\times_YX$, where $k_y$ is the residue field at $y$....
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### Is this a correct way of defining the scheme morphism $\operatorname{Spec}k_y\rightarrow Y$?

Let $Y$ be a scheme and $y\in Y$ a point. Let $k_y$ be the residue field of the stalk $(\mathcal{O}_{Y})_y$. I am trying to define the scheme morphism $\operatorname{Spec}k_y\rightarrow Y$, but was ...
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### Showing that $\mathbb A^n_\mathbb C\rightarrow \operatorname{Spec}\mathbb C$ is not proper.

$\newcommand{\Spec}{\operatorname{Spec}}\newcommand{\P}{\mathbb P}\newcommand{\C}{\mathbb C}\newcommand{\A}{\mathbb A}$ Let $\A^n_\C=\Spec \C[x_1,\dots,x_n]$, and let $f:\A^n_\C\rightarrow \Spec \C$ ...
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### Why is properness a good analogue of compactness in scheme theory?

Let $X$ be a $Z$-scheme, i.e. equip $X$ with a morphism $f:X\rightarrow Z$. Then $X$ is proper over $Z$ if it is separated over $Z$, of finite type over $Z$, and if $f$ is universally closed. Why is ...
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### How to verify a scheme is a fibre product

Above is one proposition that I found in Wedhorn& Görtz's Algebraic Geometry I in page 103. I do not understand two parts: (1) He uses the assumption II that the induced maps on stalk at any ...
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### surjective Ox-mod endomorphism on quasi-coherent finite type sheaf is bijective

Exercise 7.22 in Görtz-Wedhorn "Algebraic Geometry I: Schemes" goes as follows: "Let $X$ be a scheme, $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module of finite type. Show that ...
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### Regarding Vakil's Exercise $10.7.A$

$\newcommand{\O}{\mathscr{O}}\newcommand{\Spec}{\operatorname{Spec}}\newcommand{\Frac}{\operatorname{Frac}}$ In exercise 10.7.A of Vakil's Rising Sea we are tasked with showing that if $A$ is an ...
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