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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Is every affine variety an affine scheme?
I have seen that the notion of affine scheme is a generalization of the notion of affine varieties where the coordinate ring is replaced by any commutative unit ring, and the variety with the Zariski ...
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Two definitions of scheme theoretic dual projective space
Vakil’s FOAG gives the definition of the dual peojective space via introducing new indeterminates: (sorry that I have to quote it as a screenshot)
And the answer in Scheme theoretic dual of $\mathbb ...
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Hartshorne proposition II.5.4
I am really confused on the proof of Proposition II.5.4 on Hartshorne's algebraic geometry book. Especially, I am not sure how does the previous Lemma II.5.3 work on this proposition. Let me state the ...
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Realizing a finite group as a scheme
Suppose $G$ is a finite group. I have seen in various sources, without explanation, that we can interpret $G$ as a scheme by letting $G:=\coprod_{g\in G}\operatorname{Spec}\mathbb{Z}$. Why and how can ...
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Understanding Pullback of Cartier Divisors
Let $g\colon X\to Y$ be a morphism of schemes, and let $D=(U_i,f_i)$ be a Cartier Divisor on $Y$. I've seen the following definition of the pullback of $D$: $g^*D\colon=(g^{-1}(U_i),f_i\circ g)$, as ...
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What ring's spectrum corresponds to the affine scheme that is the inverse image of a morphism between two affine schemes?
Giving morphism of schemes $\pi$ : $\operatorname{Spec}A \rightarrow \operatorname{Spec}B$, by definition we have $\pi^{-1}\mathscr{O}_{\operatorname{Spec}B}$ is a sheaf of rings on $\operatorname{...
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Gluing step in the construction of the fiber product of $S$-schemes
I am reading through the construction of $X \times_S Y$, where $X$ and $Y$ are $S$-schemes in Liu's Algebraic Geometry and Arithmetic Curves (Proposition 3.1.2), and I am somewhat stuck justifying a ...
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Shortest path from undergrad to the (co)tangent complex?
After reading the first two answers to this question, I've become interested in understanding the concept of (co)tangent complex as a way to get some intuition about homotopical algebra, being ...
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For a family of extensions of sheaves, does nonsplit over generic fibre imply nonsplit over special fibre?
Let $A\subseteq K$ be discrete valuation ring in its field of fractions. Let $\kappa:=A/\mathfrak m$ be the residue field. Let $X$ be a scheme, flat and projective over $A$. Consider a short exact ...
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$\mathbb{Z}$-scheme from $\mathbb{Q}$-scheme
Let $I$ be an ideal of $\mathbb{Q}[T_1,\dotsc,T_n]$, and let $J=I\cap\mathbb{Z}[T_1,\dotsc,T_n]$.
Consider $X=\operatorname{Spec}(\mathbb{Q}[T_1,\dotsc,T_n]/I)$ and $Y=\operatorname{Spec}(\mathbb{Z}[...
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Is a nonconstant morphism of projective varieties necessarily finite?
Let $k$ be a field, and let $X$ and $Y$ be projective varieties over $k$.
Do there exist morphisms $X \to Y$ that are neither finite nor constant? I know this cannot happen for $X$ and $Y$ curves (as ...
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Computing $H^0(\mathbb{P}^1,nK)$
I am trying to understand why $\dim H^0(\mathbb{P}^1,nK)=0$ for $n\geq 1$ and $K=-2\infty$, a canonical divisor on $\mathbb{P}^1$.
By Riemann-Roch, we have that $h^0(nK)=h^0(K-nK)+\deg(nK)+1$, which ...
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Noetherian condition of Serre's theorem on affineness
I am currently reading the proof of Hartshorne Theorem 3.7, which is a famous result of Serre. It stated the following:
Let $X$ be a noetherian scheme, then the following are equivalent:
$X$ is ...
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Smooth morphisms of schemes
I am trying to understand smooth morphisms of schemes as in https://stacks.math.columbia.edu/tag/01V4
All the definition seems to only take in considerations the source of the morphism, as the ...
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Spectrum is ringed space
Hello I just started studying theory about affine schemes and I am now studying ringed spaces. I see everywhere the Spec(R) is a ringed space but I can’t find a proof. I think it’s obvious since the ...
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Question about tensor product
Let $T\to A$, $T\to B$ be homomorphisms of (finitely generated) $k$-algebras where $k$ is an algebraically closed field of characteristic $0$. Assume moreover that $R$ is another $k$-algebra (reduced, ...
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What is the map $f^*: Br(Y)\to Br(X)$?
Let $f:X \to Y$ be a morphism of schemes.
Let $Br(X)$ be Brauer group of $X$.
I heard $Br(-)$ is functrial, that is, there is induced map $f^*: Br(Y)\to Br(X)$.
How $f$ induces $f^*$, in other words, ...
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A question in exercise 5.8 in hartshorne, chapter3.
As in the picture, I have a question on the proof of part (b).I think I have found an effective divisor $D=\sum P_i$ on $\widetilde X$ with $L(D)\cong L$ and such that $f(P_i)$ is nonsingular point on ...
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Problem $I-24$ from Eisenbud and Harris's The Geometry of Schemes
The following is a statement from Eisenbud & Harris geometry of schemes
Let $(X, \mathcal{O})$ be any ringed space, and let $R = \mathcal{O}(X)$. For any $f \in R$ we can define a set $U_f \...
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There is birational and integral morphism between glued curve and original curve
Let $C_0:$ be a projective closure of affine curve $y^{2}=x^4-7$. $C_0$ has singular point $[0:1:0]$ at infinity.
Let another affine curve be
$C_1: v^{2}=u^{4}(1/u^4-7)=1-7u^4$.
To make smooth ...
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A complete variety is proper over $k$: why does it suffice to check universally closed on varieties instead of all schemes?
A complete variety $X$ over $k$ is defined to be one where the projection map $X\times_{\operatorname{spec} k} Y\to Y$ is closed for every variety $Y$. It is stated on Wikipedia and various places ...
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Vanishing of $\text{Ext}^i_{X}(E, F)$ vs. $\text{Ext}^i_{\mathcal O_x}(E_x, F_x)$ for $E,F \in \mathcal D^b(\text{Coh } X)$
Let $X$ be a Noetherian scheme and $E,F \in \mathcal D^b(\text{Coh } X)$. Let $x\in X$ be a closed point. Then, is there any connection between $\text{Ext}^i_{\mathcal O_x}(E_x, F_x)$ and $\text{Ext}^...
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Embedding of fiber products of schemes
Let $X\to Z$ and $Y\to Z$ be scheme morphisms (everything noetherian, of finite type over an algebraically closed field $k$ of char $0$), and consider a morphism to another scheme $X\times_ZY\to T$ ...
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Quasicompact morphism of schemes implies the closure of the set-theoretic image is the underlying set of the scheme-theoretic image
This is Corollary 9.4.5 in Vakil's notes. Suppose $\pi:X\to Y$ is a quasicompact morphism of schemes, the closure of the set-theoretic image of $\pi$ is the underlying set of the scheme-theoretic ...
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Let $C'$ be normalization of singular curve $C$. Does $C(K)\neq \emptyset$ imply $C'(K)\neq \emptyset$?
Let $K$ be a number field or finite field. Let $C$ be a singular curve defined over $K$.
Let $C'$ be normalization of $C$ and $\phi : C'\to C$ be normalization map.
It is well known that this map $\...
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Degree of the image of a curve given by canonical divisor
Suppose $C$ is a curve of genus $g$. Let $K$ be a canonical divisor on $C$, let $n\geq 3$ be an integer, and let $C\to\mathbb{P}^N$ be the map determined by the linear system $|nK|$, where $N$ can be ...
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Equivalent definition of geometric vector bundles over schemes
The following is Hartshonre's definition of geometric vector bundles:
Let $Y$ be a scheme. A (geometric) vector bundle of rank $n$ over $Y$ is a scheme $X$ and a morphism $f:X\to Y$ together with ...
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Degree of image of the canonical map [duplicate]
Suppose $X$ is a nonhyperelliptic curve of genus $g\geq 3$. There is an embedding $X\to\mathbb{P}^{g-1}$ determined by the canonical linear system. Hartshorne states that the image of this map is a ...
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fibre product with a scheme in terms of its glueing data
So heres the situation. Let S be a scheme. Imagine you have two S-schemes X,Y with restrictions U,V respectively that are glued via a homeomorphism $f:U\cong V$. lets call the sheaf we obtain by this ...
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If $X$ is a not a rational curve, then $\dim |P|=0$ for all $P$ in $X$.
I am trying to understand the following statement: If $X$ is a not a rational curve, then $\dim |P|=0$ for all $P$ in $X$. This is stated by Hartshorne while proving Lemma 5.1, that a canonical linear ...
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For a proper scheme $X,$ $X(\mathbb{Z})=X(\mathbb{Q})?$
Let $X$ be an integral model of a $\mathbb{Q}$-variety. If $X$ is proper, I found in a paper (page 1 in https://arxiv.org/pdf/2010.11763.pdf) that says $X(\mathbb{Z})=X(\mathbb{Q})$ from the valuation ...
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flatness of structural sheaf of scheme over ring of global sections
Let $X$ be an affine scheme. $U \subseteq X$ open. then I want to show that $\mathcal{O}_X(U)$ is flat over $\mathcal{O}_X(X)$.
But I want to prove it only by knowing the definition of structure sheaf ...
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When is the tangent space of a fiber product the fiber product of the tangent spaces?
If $X,Y,Z$ are schemes locally of finite type over an algebraically closed field of characteristic 0, and $X\to Z$ and $Y\to Z$ are morphisms, then is it true that $T_{(x,y)}(X\times_Z Y)\simeq T_x(X)\...
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Elimination property on scheme morphisms locally of finite type
I was trying to prove the following scheme morphism exercise. If $f: X \rightarrow Y, g: Y \rightarrow Z $ are such that $g \circ f $ is locally of finite type then $f$ is locally of finite type. I ...
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If special fiber of $S$ is smooth, can we say $S$ itself is regular?
Let $A$ be a DVR with maximal ideal $P$ and $S$ the scheme over $A$ defined by $$S=\operatorname{Spec} A[X,Y,Z]/(Y^2Z-X^3-XZ^2).$$
If the special fiber of $S$ is smooth, can we say $S$ is regular ?
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The degree of a morphism of schemes
Let $X$ and $Y$ be schemes over a field $k$, and let $\phi :X \to Y$ be a morphism of schemes.
What is the most general situation where one can define the degree of $f$? Is $X$ and $Y$ being ...
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Is the change of base scheme $\operatorname{Spec} K \to \operatorname{Spec} k$ relevant?
Let $X$ be a $K$-scheme (affine if necessary), $K|k$ an extension of fields (separable, finite, ... as necessary).
Then $k \hookrightarrow K$ is unique as a homomorphism of $k$-algebras and so $\...
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Görtz-Wedhorn Proposition 7.46. Coherence in sheaves over locally noetherian schemes.
I am trying to undestand the following proposition in Görtz-Wedhorn Algebraic Geometry I. What I do not understand is the reduction to the affine case he does in order to prove $(iii) \Rightarrow (i)$....
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Is there a notion of inverse image for schemes?
Let $f: X \to Y$ be a morphism of schemes, and let $Z$ be a subscheme of $Y$. Is there any reasonable way to talk about an "inverse image" of $Z$?
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Sheaf morphism from closed subscheme is a closed immersion
For $K=\bar{K}$ a field consider $X=\mathbb P^1_K$, $Z=\{P_1,\dots,P_n\}\subseteq X$ closed points. Give $Z$ the reduced induced closed subscheme structure and write $\iota:Z\to X$ the closed ...
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Monomorphism of schemes which is not a an open or closed immersion
I have showed that every open/closed immersion of schemes is a monomorphism of schemes, but I know that the converse is not true, i.e., not every monomorphism of schemes is an open/closed immersion.
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Do restrictions preserve ring structure of ringed space?
Let $(X, \mathcal{O})$ be a ringed space, i.e. $X$ is a topological space and $\mathcal{O}$ is a sheaf of rings on the open subsets of $X$.
I would like to show that for two global sections $a, b\in \...
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Diagonal of immersion is an isomorphism
Let $f:X \rightarrow Y$ be a morphism of schemes. We say that $f$ is an immersion if it can be decomposed as $f=g \circ h$, where $h$ is a closed immersion and $g$ an open immersion. Then, the ...
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$V$ is locally closed subscheme (i.e. a closed subscheme of an open subscheme of $X$) do not implies that it's the open subscheme of closed subscheme
I was reading Professor Vakil's FOAG, and in exercise 9.2B there is a statement that :
If $V$ is an open subscheme of a closed subscheme will implies that $V$ is a closed subscheme of an open ...
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scheme theoretic support of a finitely generated $A$-module
I was doing Professor Vakil's FOAG, in exercise 9.1M which defines an scheme- theoretic support of a finite generated $A$- module $M$ to be:
the scheme-theoretic intersection of all closed subschemes $...
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How do I interpret the intersection of a variety with a "non-closed hyperplane?"
I am trying to understand Vakil's statement and proof of Bertini's theorem, which has been updated since many of the questions related to it were posted on this website (for what it's worth, I'm not ...
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detail in proof that existence of ample invertible sheaf implies separatedness
Let $X$ be a scheme and $\mathcal{L}$ be an ample invertible sheaf on $X$, then lemma 28.26.7 in stacks project says that $X$ is separated.
The proof uses the valuative criterion and goes as follows. ...
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quasicoherent sheaf of ideal defines a closed subscheme
I was doing Professor Vakil's FOAG in exercise 9.1 F needs to prove:
that quasicoherent sheaf of ideals on $Y$ produce an closed subscheme on $Y$.
Here is my attempt, to construct it locally is not ...
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Question on Hartshorne exercise II 5.17(e)
I'm trying to solve exercise II 5.17(e) of Hartshorne:
Let $f:X\to Y$ be an affine morphism between schemes (i.e. preimage of every open affine subscheme of $Y$ is still affine), and let $\mathcal{A}=...
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A question on Vakil's Rising Sea (Exercise 4.1.A, version 2022)
Problem description
In the exercise in question we are supposed to prove that the natural map
$$A_f\to\mathcal O_{\text{Spec}(A)}(D(f))$$
is an isomorphism. Here $A_f$ is the localization of the ring $...
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