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Questions tagged [schemes]

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 commutative ring with unit.

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41 views

Unnecessary premises in proposition about base change (Görtz-Wedhorn)

Here is proposition 4.20 from Görtz and Wedhorn's Algebraic Geometry I. It seems to me that the right square in (4.5.1) is completely unnecessary: We can always choose $S = X$ making the right square ...
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Flat Morphism between Schemes

My inquisitive search for a nice geometric interpretation for flat morphism between schemes $f:X \to Y$ lastly almost succeeds in following MO-thread presenting an interesting approach: https://...
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43 views

What is a rational section of an invertible sheaf?

I am studying Cartier divisors, and I am confused about exactly how they correspond to rational sections of a line bundle, or what a rational section of a line bundle even is. Let $X$ be an integral ...
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Geometric Intuition behind Flatness

I'm looking for a nice geometric interpretation for flat morphisms between schemes. In wiki's article https://en.wikipedia.org/wiki/Morphism_of_schemes#Intuition is stated that flat families ...
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1answer
44 views

Affine $n$-space over a scheme

In an exercise of Eisenbud-Harris The Geometry of Schemes, they ask to prove the following: Let $S$ be any scheme. Let $\mathbb{A}_{\mathbb{Z}}^n = \mathrm{Spec}\mathbb{Z}[x_1, \dots , x_n]$ be ...
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Do we have $f=g$ as continuous maps on the underlying topological spaces? [closed]

Suppose $(f,f^\sharp),(g,g^\sharp):(X,\mathcal O_X)\to (Y,\mathcal O_Y)$ are two separated morphisms of schemes, and $A$ is an open dense subset of $X$, if $(f|_A,(f|_A)^\sharp)=(g|_A,(g|_A)^\sharp):(...
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19 views

Is the composite of $X\times_SY\to X$ with $X\to Y$ the same as $X\times_SY\to Y$? [closed]

Suppose $X\to Y\to S$ are morphisms of schemes, is the composite of $p:X\times_SY\to X$ with $X\to Y$ the same as $q:X\times_SY\to Y$? $p$ and $q$ are projections.
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1answer
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A doubt on Proposition 5.1.12 of Liu's Algebraic geometry and arithmetic curves.

Let $X$ be a scheme. Let $0 \to \mathcal{F} \to \mathcal{G} \to \mathcal{H} \to 0$ be an exact sequence of $\mathcal{O}_X$-modules. If two of them are quasi-coherent, then so is the third. This is ...
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2answers
27 views

Let $k$ a field, $k'$ a subfield, and $A$ any associative $k$-algebra. Can a quotient of $A$ ever yield $k'$?

I am trying to learn some basic Scheme theory out of Eisenbud's book "Schemes: the Language of Modern Algebraic Geometry." I'm trying to understand how elements of a ring can be treated as functions ...
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1answer
46 views

$Spec(R)$ is irreducible if and only if $Spec(R[T])$ is irreducible

I want to show that for an arbitrary ring $R$ the following equivalence holds: $Spec(R)$ is irreducible if and only if $Spec(R[T])$ is irreducible. I have tried to show this by using the ...
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1answer
21 views

How to show the image of $U$ in $X'$ is dense?

In Proposition 1.4.18(Chow's Lemma), how to show the image of $U$ in $X'$ is dense?
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Is $U\times_SY$ dense in $X\times_SY$?

Given two morphisms of schemes $X\to S$ and $Y\to S$, suppose $U$ is an open subscheme of $X$ and $U$ is dense in $X$, is $U\times_SY$ dense in $X\times_SY$?
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1answer
36 views

Restriction section of a sheaf to a closed set?

I am doing an exercise from Hartshorne (II Ex 6.2) on divisors and I have come across an abuse of notation that I am not entire sure how to interpret. Let $X \hookrightarrow \mathbb{P}_{k}^{n}$ be a ...
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30 views

Scheme-theoretic Preimage and Fibered Product of Schemes

Following Eisenbud-Harris The Geometry of Schemes, and I'm having trouble understanding a specific part of their proof that fibered products exist in the category of schemes. The affine case is okay,...
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35 views

Is there a definition of morphism of gluing data of schemes that makes gluing an adjoint functor?

Tag 01JA of the Stacks Project is about gluing schemes. There, a gluing data is defined as a collection of schemes, and isomorphisms between open subsets of those schemes satisfying the cocycle ...
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1answer
29 views

Multiplicity and degree of irreducible projective subschemes.

Suppose $X \subset \mathbb{P}^n$ is an irreducible projective scheme. Then its multiplicity $\mu_X$ is defined as the length of the local ring $\mathcal{O}_{X,\eta}$ over itself, where $\eta$ is the ...
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1answer
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The kernel of a morphism of quasi-coherent sheaves on a scheme $(X,\mathcal{O}_X)$ is quasi-coherent.

Let $(X,\mathcal{O}_X)$ be a scheme. I know that an $\mathcal{O}_X$-module $\mathcal{F}$ is quasi-coherent if for each $x \in X$ there exists an open neighborhood $U$ of $x$ and an exact sequence of $\...
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41 views

Families of genus zero curves vs Family of deformations of $\mathbb{P}_k^1$.

Suppose I am looking to classify isomorphism classes of connected, compact, projective curves of genus zero. We usually start this moduli problem by considering a scheme $S$ and families of genus ...
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26 views

Is subgroup scheme of smooth group scheme flat over base scheme?

Let $\mathcal{A},\mathcal{B}$ be smooth separated commutative group scheme over $S$, where $S$ is Noetherian. Let $\iota:\mathcal{A}\rightarrow\mathcal{B}$ be a morphism of group scheme which is ...
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31 views

How to understand the closure of a point?

I encounter the definiton of generic point in Hartshorne's as follows: A generic point for an irreducible closed subset $Z$ is a point $P$ such that $Z=${$\overline{P}$}, where {$\overline{P}$} ...
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Tangent space of a scheme and subschemes of length two

I found in Huybrecht's book Fourier Mukai transforms in Algebraic Geometry the following statement A tangent vector $v$ at $x \in X$ is the data of a length two subscheme $Z$ concentrated at x Here ...
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30 views

Motivation for Smoothness and Regularity of Schemes

I'm looking for motivation for the property of smoothness (of morphisms) and regularity (of objects) in algebraic geometry, from an algebraic viewpoint. I understand that requiring this makes the ...
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Existence of $k$-valued Points [closed]

Let $X$ be a $S$-scheme. (For sake of simplicity assume $S = Spec(\mathbb{Z})$) I have a quite general question about a proper about a property of schemes and advantages of it. I read often that for ...
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52 views

Is the pull-back of the structure sheaf the structure sheaf?

Maybe this is a stupid question, but I got irritated by it: Suppose $f: X \rightarrow Y$ is a morphism of schemes. That comes with a map of sheaves $f^\#: \mathcal{O}_Y \rightarrow f_* \mathcal{O}_X$. ...
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1answer
91 views

Hypersurfaces of degree $d$ in $\mathbb{P}^n_k$ that contain a given closed $X$

Let $k$ be an algebraically closed field and consider $\mathbb{P}^n_k$, the be the $n-$dimensional projective space over $k$. It is known that, for any integer $d>0$, there is a bijection between ...
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1answer
24 views

Dimension of a closed subset and transcendental degree

Let $X=Spec\;A$ be a scheme of finite type over a field $k$ and $x\in X$. Let $q\subset A$ be an ideal corresponding to $x$. How one can show that $\dim \overline{\lbrace x \rbrace} = \text{tr.deg} (...
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48 views

Constructing an invertible sheaf from a Cartier divisor?

Let $X$ be a scheme. By definition, a Cartier divisor $D$ is a global section of the sheaf $\mathcal{M}_X^{\times}/\mathcal{O}_X^{\times}$, where $\mathcal{M}_X^{\times}$ is the sheaf of ...
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1answer
50 views

Clarification on a proof that the rank of a locally free sheaf is the same everywhere if $X$ is connected.

I have seen the answer in this previous post. My question is as follows. Given a locally free sheaf $F$ over a connected scheme $X$. Why is it true that if $U$ and $V$ are two open sets in $X$ such ...
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Definition of $going-up$ map

The exercise 5.10 of Atiyah's Commutative Algebra gives the definition of $going-up$ map: A ring homomorphism $f:A\rightarrow B$ is said to have the $going-up$ (resp. the $going-down$ property) if ...
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extension of discrete time process to the unit interval

I am reading a proof and i don't understand some lines of cumputations. We have a Euler Scheme SDE given by $X_{n}(\frac{k+1}{n}) =X_{n}(\frac{k}{n}) +b(X_{n}(\frac{k}{n}))/n + \sigma(X_{n}(\frac{...
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1answer
33 views

Affine charts are dense in projective space

Given a field $k$, we define the scheme-theoretic $n$-th affine space over $k$ by $\mathbb{A}^n_k=\text{Spec}(k[X_1,\dots,X_n])$ and the $n$-th projective space over $k$ by $\mathbb{P}^n_k=\text{Proj}(...
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Why is there a correspondence between $ |\text{Proj}\;S| - V(f) $ and $ \text{Spec}\; (S_{f})_{0}$?

Forgive me for the length of this post, but I feel this question is deserving of some detail. Suppose $ S $ is a positively graded $ A$-algebra, where $ A $ is a ring. That is $ S = \bigoplus_{i=0}^{\...
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1answer
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Criterion for showing that a morphism of schemes is an isomorphism

Suppose $\left(\phi,\phi^{\#}\right):X\longrightarrow Y$ is a morphism of schemes with the following properties: 1.$\phi$ is an isomorphism of topological spaces; 2.Every $\phi^{\#}_U:\mathcal{O}...
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Cech cohomology does not compute étale cohomology - Explanation of an example

In the first answer to this MO post the author says that the $H^2$ of $X$ can be compute using the Cech-to-derived functor spectral sequence, i.e. in that case the Mayer-Vietoris sequence. I'm having ...
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1answer
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Restriction of a sheaf of modules

Let $X$ be a scheme and $Y$ be a closed subscheme. For $\mathcal{F}$ a sheaf of modules on $X$ to be the pushforward of a sheaf of modules on $Y$ via the inclusion $i: Y \rightarrow X$ is necessary ...
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The big étale and Zariski topoi are generated by small sites

Grothendieck (SGA 4, Exposé VII, Corollaire 3.2) proves that if $X$ is a quasiseparated scheme then, although the étale site is not small, there exists a subsite of it which is small and has the same ...
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The small étale topos of a scheme is equivalent to the category of finite $\pi_1(X,x)$-sets for every scheme $X$ and every geometric point $x$

Recall Milne, Etale cohomology, Theorem I.5.3: Let $x$ be a geometric point of a connected scheme $X$. The functor $Hom(x,-):FEt/X\to Sets$ ($FEt/X$ being the category of $X$-schemes finite and ...
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1answer
45 views

Are sections of an injective sheaf of abelian groups themselves injective abelian groups?

Let $X$ be a topological space, $\mathcal I$ some injective sheaf of abelian groups on $X$ and let $U \subset X$ be open. Is $\mathcal I(U)$ an injective abelian group? If not what requirements would ...
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44 views

Proper Morphism with Finite Fibers is Finite

Let $f: X \to Y$ a morphism between schemes. I'm looking for a reference / sketch for the proof that is $f$ proper amd has finite fibers then $f$ is already finite. Can the proof redicaly simplified ...
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Galois Action on Scheme

Let $X$ be a $K$-scheme and $L \vert K$ be a Galois extension with Galois group $G= Gal(L,K)$. Let consider the base change $X_L := X \otimes_K L:= X \times_{Spec(K)} Spec(L)$. Since $X_L$ is a $L$-...
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Pullback of a Global Section

Set $f:Y \to X$ be a morphism between schemes and $s \in \Gamma(X, \mathcal{O}_X)$ be a global section. The map $f$ induces a functor $f^*$ (so called pullback functor) that pulls back $\mathcal{O}...
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1answer
58 views

proof in English of Proposition on morphisms into affine schemes

Does anybody know where to find an English proof of the following proposition Let $(X, \mathcal{O}_X)$ be a locally ringed space, $Y = \operatorname{Spec} A$ an affine scheme. Then the natural map ...
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36 views

Birational Morphism to a Normal Scheme Isomorphism

My question refers the answer of @user18119 in following thread: Are these two notions of Galois morphism the same We have $f:X\to Y$ a finite morphism of integral schemes and $G$ letautomorphism ...
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1answer
56 views

Geometry of the subalgebra $\Bbbk[x^2-1]\leq \Bbbk[x]$ (intuition for integral elements)

Given a field $\Bbbk$ consider the subalgebra $\Bbbk[x^2-1]\leq \Bbbk[x]$. This is an integral extension of algebras. Write $\mathfrak q= (x-1)\vartriangleleft \Bbbk[x]$ and $\mathfrak p=\Bbbk [x^2-1]\...
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0answers
34 views

Inverse image of a structure sheaf under the inclusion of a closed subset and why it isn't quasi-coherent?

This question is motivated by Example 5.2.4 in Algebraic Geometry by Hartshorne (page 112). As far as I understand, it claims that if $Y$ is a closed subscheme of $X$, and one considers the inverse ...
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69 views

Is constancy of fiber degree analytic-local on the source in a finite flat family?

Let $X \subset \mathbb{A}_{\mathbb{C}}^2$ be a closed subscheme, and let $\pi \colon X \to \mathbb{A}_{\mathbb{C}}^1$ be a finite flat map. For a point $p \in X(\mathbb{C})$, is it true that we can ...
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1answer
70 views

Why is $\mathbb{G}_m$ is a representable functor?

What does it mean that multiplication $\mathbb{G}_m$ is a representable functor, with $\mathbb{G}_m = \text{Spec}(\mathbb{Z}[x,x^{-1}])$ ? When I looked at the stacks project page on $\mathbb{G}_m$ ...
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54 views

Using the functor of points approach to show a scheme theoretic construction exists

T am trying to refram what I have learned in algebraic geometry in the context of the functor of points approach. In particular, I want to practice proving the existence of a scheme satisfying a ...
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49 views

Condition of pushward commutes with tensor product

Let $f$ be a morphism between schemes. Is there a sufficient and necessary condition on $f$ such that $f_*$ commutes with $\otimes$? i.e. $$f_*F\otimes f_*G\cong f_*(F\otimes G)$$ for all coherent ...
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1answer
91 views

When does a quasifinite surjective flat morphism have constant fiber multiplicity near a point?

Let $V \subset \mathbb{A}_{\mathbb{C}}^2 = \operatorname{Spec}\mathbb{C}[x,t]$ be a closed subscheme containing the point $x = t = 0$, and suppose we have a quasifinite flat surjective morphism $\pi \...