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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 ...
onRiv's user avatar
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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 ...
Ezio Greggio's user avatar
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Uniqueness of the $A$-scheme structure on $\operatorname{Proj} B$

Let $B$ be a graded $A$-algebra, and consider the topological space $X:=\operatorname{Proj} B$. Call $(X,O_X)$ the usual structure of scheme on $X$. It is a structure of $A$-scheme as well, because $...
Ezio Greggio's user avatar
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If the base change is a fiber product, is the original variety a fiber product too?

Assume we have an extension of algebraically closed fields $L/K$ and varieties $V_1$ and $V_2$ over $K$. Let $W_1\subseteq (V_1)_L$ and $W_2\subseteq (V_2)_L$ subvarieties, so $W_1\times_L W_2\...
Siegmeyer of Catarina's user avatar
2 votes
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155 views

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 ...
mathfan24's user avatar
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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 $\sigma: \operatorname{Spec} K\to X$ ...
Ezio Greggio's user avatar
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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....
user128787's user avatar
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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 $D\cdot C = 0$. ...
konoa's user avatar
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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 ...
maxo's user avatar
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Projection Morphism of Blowup

I'm currently reading the article A Short Course on Geometric Motivic Integration, by Manuel Blickle. In his proof of Theorem 3.3, the author considers the following. Let $X'=\operatorname{Bl}_{0}\...
LiminalSpace's user avatar
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1 answer
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morphisms between the sheaves restricted to open subset

Let $\mathscr F,\mathscr G$ be both sheaves on $X$ and $\varphi$ be morphisms between sheaves $\mathscr F,\mathscr G$. Let $U\subset X$ be an open subset( maybe we do not even need that $U$ is open in ...
Ziqiang Cui's user avatar
2 votes
0 answers
76 views

Serre duality for arbitrary sheaves?

Let $X$ be a connected proper Cohen-Macaulay scheme over a field $k$ of dimension $n$. Is there an example of an $\mathcal{O}_X$-module $\mathcal{F}$ on $X$ such that Serre duality $$ \operatorname{...
fish_monster's user avatar
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Is every open substack of $Y$ of the form $[U/G]$ where $U$ is an open subscheme of $X$? [closed]

Let $G$ be a finite group acting on a scheme $X$. Consider the algebraic stack quotient $Y=[X/G]$: Is every open substack of $Y$ of the form $[U/G]$ where $U$ is an open subscheme of $X$?
Angry_Math_Person's user avatar
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Glueing of morphism between schemes

On page 70 of Wedhorn& Görtz's Algebraic Geometry I:Schemes, there is one result which I do not know how to prove. Let $X,Y$ be schemes and let $X=\cup_i U_i$ be an open covering. Then morphisms $...
Ziqiang Cui's user avatar
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definition of closed immersion of schemes

The definition that I found on books for definition of closed immersion of schemes is the following: A closed immersion $i:Z\hookrightarrow X$ is a morphism which satifies: (1) The underlying ...
user11695417's user avatar
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Locality of closed immersions of schemes

Let $(\varphi,\varphi^\#):(X,O_X)\to (Y,O_Y)$ be a morphism of schemes. Let $V\subset Y$ be an open set, and $U:=\varphi^{-1}(V)$. Choosen $x\in U$, the diagram below commutes: $\require{AMScd}$ $$\...
Ezio Greggio's user avatar
1 vote
1 answer
79 views

Dimension of the Hilbert Scheme of conics

I am trying to compute the dimension of the Hilbert Scheme of conics in $\mathbb{P}^4$ $Hilb_{2T+1}(\mathbb{P}^4)$. I started with conics lying on a plane, so, taking the ideal $I=(Q,H_1,H_2)$ for two ...
Gowexx's user avatar
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For a scheme $Z$, let $z\in Z, t\in \mathscr O_Z(Z)$, what does $t(z)$ mean?

For a scheme $Z$, let $z\in Z, t\in \mathscr O_Z(Z)$, let $Z_t:=\{z\in Z:t(z)\not= 0\}$. The author claims $Z_t$ is open subset of $Z$. But what does $t(z)$ mean?
user11695417's user avatar
1 vote
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Correspondence between sheaves of ideals and closed immersion

Let $(X,\mathcal O_X)$ be a locally ringed space. Let $\mathcal J$ be a sheaf of ideals over $\mathcal O_X$. Then we can construct a closed immersion of locally ringed spaces: $$(Z(\mathcal J),i^{-1}(\...
Ezio Greggio's user avatar
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Accessible result motivating Arithmetic Geometry

My understanding of arithmetic geometry is that it applies concepts originating from algebraic geometry to schemes that are quite different from these initially studied in this field (those associated ...
Weier's user avatar
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Projective varieties contained in dense open subsets

Let $X$ be a smooth irreducible projective variety over the complex numbers. Let $U$ be a nontrivial dense open subset of $X$. Does there exist a projective curve $C$ inside $U$? My attempt: Let's ...
cupoftea's user avatar
  • 103
4 votes
2 answers
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Calculating the fibres of a scheme morphism are proper but the morphism is not proper

$\newcommand{\Spec}{\operatorname{Spec}}\newcommand{\C}{\mathbb C}\newcommand{\A}{\mathbb A}$ Let $f:\mathbb A^1_\mathbb C\rightarrow \mathbb A^1_\mathbb C$ be induced by the ring homomorphism $t\...
Chris's user avatar
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4 votes
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How can I check that for a quasi-compact morphism $f:Z\to X$, the kernel of $\mathcal{O}_X\to f_*\mathcal{O}_Z$ is quasi-coherent?

If $f:Z\rightarrow X$ is a quasi compact morphism of schemes and $\mathcal J :=\ker(\mathcal O_X \rightarrow f_* \mathcal{O}_Z)$, then $(\operatorname{Supp}( \mathcal O_X/ \mathcal J)$,$i^{-1}(\...
user11695417's user avatar
1 vote
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53 views

For any open set $U\subset X$, $X$ an integral scheme, $p\in U$, $\mathscr O_{X}(U)\rightarrow \mathscr O_{X,p}$ is injective

In the case that $U$ is affine, the statement is trivial because the stalk at $p$ is the localization of $\mathscr O_{X}(U)$ except for elements in prime ideal $p$ and the map identifies with the map $...
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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,...
Frank's user avatar
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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 ...
stillconfused's user avatar
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The closed points of $\mathbb{A}^2_\mathbb{R}$ do not correspond to ordered pairs of closed points of $\mathbb{A}^1_\mathbb{R}$

In Eisenbud & Harris's The Geometry of Schemes, after a discussion of the closed points in $\mathbb{A}^1_\mathbb{R}$ and in $\mathbb{A}^2_\mathbb{R}$, the authors note that The closed points in $\...
stillconfused's user avatar
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Equivalence of the Definition of closed immersion and closed subscheme

In wedhorn and Görtz‘s algebraic geometry book Algebraic Geometry I:Schemes page 86 I found the following definition. (1)A closed subscheme of $X$ is given by a closed subset $Z\subset X$ and an ideal ...
张耀威's user avatar
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3 votes
2 answers
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Are these stalks isomorphic?

If $X$ is a scheme with structure sheaf $\mathcal{O}_X$ and $h: X \to \text{Spec }\mathcal{O}_X(X)$ is the unique morphism of schemes such that $h^\sharp(\text{Spec } \mathcal{O}_X(X)): \mathcal{O}_X(...
soggycornflakes's user avatar
3 votes
0 answers
37 views

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 ...
Siyuan Yin's user avatar
3 votes
1 answer
67 views

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 ...
Francesco's user avatar
3 votes
0 answers
26 views

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 ...
Adelhart's user avatar
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1 vote
1 answer
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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 ...
Alex's user avatar
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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 ...
Ezio Greggio's user avatar
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0 answers
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Local Action by Group Scheme (Milne's Algebraic Groups)

I have a question about following proof from Milne's book "Algebraic Groups: the theory of group schemes of finite type over a field", Chapter 8, proposition 8.9: PROPOSITION 8.9. Let $G \...
user267839's user avatar
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1 vote
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Stalk of coherent sheaves and pushforward/pullback

Let $X$ be a Noetherian scheme. Let $\mathcal F$ be a coherent $\mathcal O_X$-module. Let $x\in X$. There is a natural morphism Spec $\mathcal O_{X, x}\xrightarrow{j} X$. Define $\mathcal F(x):=j_*(...
Alex's user avatar
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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 $y$....
Chris's user avatar
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1 vote
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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 ...
Chris's user avatar
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2 votes
2 answers
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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 $f:\A^n_\C\rightarrow \Spec \C$ ...
Chris's user avatar
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3 votes
0 answers
98 views

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 ...
Chris's user avatar
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3 votes
0 answers
158 views

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 ...
Ziqiang Cui's user avatar
2 votes
0 answers
51 views

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 ...
Absent mind's user avatar
1 vote
0 answers
74 views

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 ...
Chris's user avatar
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1 vote
1 answer
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Liu lemma 3.3.17: Proper + affine = finite

This has been asked and answered before here. I'm not satisfied with the answer (probably I don't understand it). Also I don't want a proof of "proper+affine=finite" (which is here) but of ...
user128787's user avatar
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2 votes
2 answers
114 views

What is the stalk map for a morphism of affine schemes?

$\newcommand{\Spec}{\operatorname{Spec}} \newcommand{\O}{\mathscr{O}}$ Let $X=\Spec A$, and $Y=\Spec B$, and suppose that $f:X\rightarrow Y$ is a morphism coming from the ring homomorphism $\phi:B\...
Chris's user avatar
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0 votes
1 answer
74 views

Projective morphisms are quasi compact and quasi separated or not

I am very new to algebraic geometry. I was reading about different kinds of morphisms between schemes. I am wondering about the following question: Is every projective morphism quasi-compact and ...
KAK's user avatar
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1 vote
0 answers
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The reason why scheme theory emerged.

In any history, there is a cause-and-effect relationship. So I became curious about the situation in which the scheme theory came to appear. In other words, I'm curious about what problem was left ...
jhzg's user avatar
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1 vote
1 answer
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The symbol of de Rham cohomology on Stacks Project

In the chapter of de Rham cohomology on Stack Project, there is a symbol $H^{i}(R\Gamma(X, \Omega_{X / S}^{\bullet}))$. What does $R\Gamma$ mean?
jhzg's user avatar
  • 301
3 votes
0 answers
119 views

Is every normalization a blowup?

Is the normalization of a variety always a blowup along some coherent ideal sheaf? If not, I would like to see a concrete counter-example. Let $Y \to X$ be the normalization. The answer is positive in ...
SeparatedScheme's user avatar
0 votes
1 answer
73 views

Compute $Spec(\mathbb{Z}_{(p)})\times_{\mathbb{Z}}Spec(\mathbb{Z}_{(q)})\cong Spec(\mathbb{Q})$

Let $p$, $q$ be two different prime numbers. Now want to compute fiber products as below: $Spec(\mathbb{Z}_{(p)})\times_{\mathbb{Z}}Spec(\mathbb{Z}_{(q)})\cong Spec(\mathbb{Q})$ $Z_{(p)}$ is discrete ...
Un peti mensonage's user avatar

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