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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How to show $\operatorname{Pic}(X)=0$? Exercise $14.2$.Q Vakil's notes

I'm reading Vakil's notes and I'm struggling with the exercise $14.2$.Q. I've been able to prove everything except $\operatorname{Pic}(X)=0$ with $$ X=\operatorname{Spec}\frac{k[x,y,z]}{(xy-z^2)}. $$ ...
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What is a generically reduced scheme?

I am reading the book "3264 & All That Intersection Theory in Algebraic Geometry". In the following definition (see page 30) Definition 1.22. Let $f:Y\rightarrow X$ be a morphism of ...
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Quasi-separatedness of diagonal of DM stacks and group stabilizer of a point

I was studying Jarod Alper's book on stacks (https://sites.math.washington.edu/~jarod/moduli.pdf) and after the definition of stacks and some basic discussion about them, he proposes the following ...
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deformation to the normal cone in homotopy purity

I'm reading the proof of homotopy purity theorem. A key step in the proof is \textbf{Deformation to the normal cone}: Let $Z\xrightarrow{i}X$ be a closed immersion in $Sm/S$, where $S$ is noetherian ...
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Transition functions for $\mathcal{O}(1)$ on $\mathbb{P}^1$

I am trying to understand the transition functions for $\mathcal{O}(1)$ on $\mathbb{P}^1$. I've been stuck on this while reading these notes (Proposition 1.8, page 3) on divisors and invertible ...
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Surjective étale morphism between normal schemes

(1) If I have $X,Y$ noetherian schemes, locally of finite type over a field $k$, normal and connected, can I somehow conclude that a surjective étale morphism $X \to Y$ is finite? I have seen some ...
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Counterexample to one version of Chevalley's theorem?

Chevalley's theorem says that if $f \colon X \to Y$ is a morphism of finite presentation of schemes and $C \subset X$ is constructible, then $f(C)$ is constructible. Here, constructible means $C$ is ...
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Reference request for perfection of schemes over finite fields

I am currently reading a paper from 2021 which uses "perfection" of schemes over finite fields. If $X$ is such a scheme over $\mathbb F_q$, the associated perfection is denoted by $X^{\...
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Is that section of a smooth morphism is smooth?

This is Cisinski & Deglise's Triangulated categories of mixed motives. In this section. $\mathscr P$ is assumed be smooth morphisms. As the picture shows, $f,g$ are smooth separaed morphism of ...
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Factorization of a morphism of LRS

The questions, in particular, are the two written in italics, but any other correction is welcome. Thank you in advance. Let $(f,f^\#):(Y,\mathcal O_{Y})\to (X,\mathcal O_{X})$ be a morphism of LRS, ...
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Open set of the cuspidal curve is not principal

I'm struggling with this problem: Let $k$ a field and $X=\operatorname{Spec} k[x,y]\big/(x^3-y^2)$ and $U=X\setminus\{(x-1,y-1)\}$. Show that $U$ is not a principal open set in $X$. I tried to show ...
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Closed immersions of schemes vs closed immersions of LRS

In my course (that vaguely follows Liu's Algebraic Geometry) the definition of closed (resp. open) immersion $f:Y\to X$, is that $f$ induces a homeomorphism onto $f(Y)$, $f(Y)$ is closed (resp. open) ...
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Dimension of the fiber product $X\times_S Y$ of schemes over $k$.

Consider two morphisms $X\to S$ and $Y\to S$ where all the schemes involved are smooth algebraic varieties over a field $k$. Is it true that $$\dim X\times_S Y=\dim X+\dim Y-\dim S?$$ The case $S=\...
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Is there a hypothesis missing in this statement of the See-Saw principle?

In Huybrecht's 'Fourier-Mukai Transforms in Algebraic Geometry', the see-saw principle is stated in Proposition 9.4 as follows. Let $X$ be an irreducible complete variety over a field $k$, $T$ an ...
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Describing all endomorphisms of elliptic curves

Let $k$ be an algebraically closed field. A curve is a separated integral $k$-scheme of finite type over $k$ and of dimension one. An elliptic curve $E$ is a smooth projective curve of genus one (a $k$...
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Increasing sequence of reduced closed subschemes.

Let $k$ be a perfect field, $V$ be an irreducible smooth $k$-scheme and $W$ is an open dense subset of $V$. Let $F$ be the complement of $W$. In this lemma, Morel states that there is an increasing ...
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Why is $H^0(X_n, \mathcal{O}_{X_n})$ a local artin ring.

Let $V$ be regular, proper and of dimension 2 over $S = $ spec $R$, for $R$ a complete discrete valuation ring with uniformizer $t$, maximal ideal $\mathfrak{m}$, and algebraically closed residue ...
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Existence of a model for $(X,L)$

Recently I was reading Moriwaki's work about the heights on arithmetic varieties, and I came across the following doubt about the setting. Let $(X,L)$ be a couple where $X$ is a projective nonsingular ...
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Deformations of finite schemes

I am reading some texts about the tangent space to the Hilbert scheme. Apparently, $T_{[Z]}(Hilb (X)) = H^0(Z, N_{Z/X})$ for any $Z$ is a consensus which I agree with, but then regarding the hilbert ...
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Can $\nu_Y (f) = 0$ for every prime divisor containing $Z = \overline{\{z\}}$, where $f \notin \mathcal{O}_z$?

This is about the argument in Hartshorne exercise III.6.8(a), which is supposed to show $X_s$ form a base for the topology of a noetherian, integral, separated, locally factorial scheme $X$, where $s$ ...
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Group of units in local Artinian ring with algebraically closed residue field

My question is: why is the group of units in a local Artin ring with algebraically closed residue field $q$-divisible (i.e. the multiplication by $q$ endomorphism is surjective) for $q$ prime to char $...
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For a family of short exact sequences of coherent sheaves, can we define the splitting subscheme?

Let $k$ be an algebraically closed field of characteristic zero. Let $X$ be a projective scheme over $k$. We can talk about short exact sequences of coherent sheaves on $X$. Suppose we have a family ...
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Global sections are finite product of global sections of irreducible comoonents

I would like to extend the question here. Assume that $X\rightarrow Spec(K)$ is proper and X is reduced. Since is proper and $Spec(K)$ is Noetherian, $X$ is Noetherian and hence it will be finite ...
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What is the $\mathbb{R}$-scheme $\operatorname{Spec} \mathbb{C}$?

I want to study $\operatorname{Spec} \mathbb{C}$ in the category of $\mathbb{R}$-schemes. I know that the structural morphism is derived from the ring morphism $\mathbb{R} \to \mathbb{C}$, which must ...
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In the category of schemes, what conditions on a closed monomorphism make it a closed immersion?

If the question can be simplified, we can work on smaller categories, such as the category of varieties or schemes of finite type over a field, etc. By a closed morphism of schemes I mean it is closed ...
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The splitting locus in a Quot scheme, is it closed or locally closed?

Assume we work over $\mathbb C$. Let $X$ be a projective scheme over $\mathbb C$ with an ample line bundle $\mathcal L$. Let $\mathcal F_1,\mathcal F_2$ be coherent sheaves on $X$. Let $P_1,P_2\in\...
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Where is the error in this proof that all morphisms of schemes are quasi-compact?

Lemma 1. All preimages of affine open subschemes are affine (Inverse of open affine subscheme is affine). Lemma 2. All affine schemes are quasi-compact (Proof that an affine scheme is quasi compact). ...
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Confusion regarding definition of Grassmannian

I had been thinking of Grass$(p,r)$ as the set of $r$ dim subspaces of $k^p$ where $k$ is whatever ground field we take in the context. We then topologize this set. Now in one of his papers, Nitsure ...
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1 answer
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Global functions on arithmetic varieties

Let $f:X\to\operatorname{Spec} O_K$ be an arithmetic variety where $K$ is a number field and $O_K$ is its rings of integers. We assume that $X$ is integral, projective, regular and $f$ is flat. If ...
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2 votes
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$Hom$ vector bundle of (geometric) vector bundles over a scheme

In an analytic context, given two vector bundles $E\rightarrow X$ and $F\rightarrow X$ over a complex manifold $X$ we can define $\textit{Hom}(E,F)\rightarrow X$ in the natural way $Hom(E,F)=\coprod_{...
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When is the reflexive hull of the conormal sheaf locally free?

Let $X\subseteq Y$ be a (singular) complex analytic subspace or closed subscheme with defining ideal $I$, where $Y$ is smooth. Stalks of the structure sheaf of $X$ are assumed to be integral ...
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1 answer
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Transition functions on invertible sheaves

I'm studying "Foundations of algebraic geometry" by Ravi Vakil. Chapter 14.1 is about invertible sheaves on $\mathbb{P}^1_k$. It says between two affine subsets $\operatorname{Spec}k[x_{1/0}]...
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Hartshorne II.3.10

I'm working on Exercise II.3.10 from Hartshorne and I'm baffled by what should be a relatively simple exercise on schemes. The exercise states If $f:X\to Y$ is a morphism (of schemes), $y\in Y$ a ...
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pullback of schemes

So we had the following statment in the lecture: I just dont get why this equation is true, even though it probably just follows immediately from the universal property of the pullback. Thanks in ...
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1 answer
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Semicubical parabola is not isomorphic to the affine line (module of differentials)

Here is exercise 1 from chapter 8.1 of Bosch - Algebraic Geometry and Commutative Algebra: (Exercise:) For a field $K$, consider the coordinate ring $A = K[t_1, t_2]/(t_2^2 − t_1^3)$ of Neile’s ...
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4 votes
3 answers
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Set of points where stalk is integral domain is open

I struggled to find a solution for the exercise 4.9 in the second chapter of Liu's book Algebraic Geometry and Arithmetic Curves. The first part is to show the set of points $x\in X$ such that $\...
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Quasi-compactness is a property of morphisms of schemes stable under base change

That's it, I'm trying to prove that if $f:X\to S$ is a quasi-compact morphism of schemes and $g:T\to S$ is any morphism, then the base change $f_T:X\times_ST\to T$ is also quasi-compact. The proof in ...
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4 votes
2 answers
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Two definitions of quasi-separated morphisms which may not be equivalent?

Definition. A topological space $X$ is said to be quasi-separated if the intersection of two quasi-compact open subsets $U,V\subset X$ is quasi-compact. In Definition 11.14 from these notes from a ...
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preimages of morphisms of schemes

So I have (again) a question regarding sheaves and their properties. Specifically, I have a problem with understanding a certain statement of the following proof: Just for completeness, I added the ...
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1 vote
1 answer
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Equivalence between connections on $\mathcal{E}$ and $\mathcal{P}^1$-linear isomorphisms that induce the identity modulo $\Omega^1_{X|S}$

In Berthelot and Ogus' book "notes on crystalline cohomology", I don't understand the proof of proposition 2.9: Given an $O_X$-module $\mathcal{E}$ on an $S$-scheme $X,$ a connection $\nabla$...
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Is there a way to prove that $\mathbb P^n \to \operatorname{Spec} \mathbb Z$ is universally closed without the valuative criterion?

The title is self explanatory. I was just wondering, since some books such as Hartshorne introduce the valuative criteria basically to show that projective space is proper.
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Preimage of a proper morphism

We know that proper  morphisms of  varieties take  closed subsets to closed subsets. What can I say of the preimage  of a proper morphism? For example, if I know  that $f(g(x))$  is an uncountable ...
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Defining morphism of sheaves [duplicate]

Suppose we have sheaves $\mathcal{F},\mathcal{G}$ on a topological space $X$ where $\mathcal{U}$ is a base of $X$. Then to define a morphism $\varphi:\mathcal{F}\rightarrow \mathcal{G}$, is it enough ...
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Flatness of a morphism for an infinitesimal deformation

I am trying to learn about infinitesimal deformations and I am particularly looking at Example 1.2.2 (i) from the book Deformations of Algebraic Schemes by Edoardo Sernesi which states the following: ...
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1 answer
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Quasi-projective A-schemes are locally of finite type over A, The Rising Sea, Ex.5.3.D

A projective $A$-scheme is a $\operatorname{Proj} S_{\bullet}$ where $S_{\bullet}$ is a finitely generated graded ring over $A=S_0$. A quasi-projective $A$-scheme is an open quasicompact subscheme of ...
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Cohomology of the formal fiber

Let $f: X \to Y$ be a morphism of schemes. Take $y \in Y$ and consider the map $i: \text{Spec}~\widehat{{\mathcal{O}}_{Y, y}} \to Y$ and denote $Z$ to be the fiber product of these two maps. I think ...
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Question about a proof that all étale morphisms are locally standard étale

In chapter 1 of Milne's Étale cohomology book from 1980, Theorem 3.14 states that : If $f: Y\longrightarrow X$ is étale in some open neighbourhood of a point $y\in Y$, then there are affine open ...
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4 votes
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Classification of coherent sheaves on $\Bbb P^1$ with a doubled point (nonseparated)

Let $X$ be the scheme obtained by gluing two copies of $\Bbb P^1_k$ along $D(x)$ (basically the projective version of the line with two origins). Is there a good classification of coherent sheaves on $...
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Constants of universal derivation on an $R$-algebra $A$ under localization ($d: S^{-1}A \to S^{-1}\Omega_{A/R}$)

Suppose that $A$ is an $R$-algebra, with both $R, A$ integral domains. Let $B = S^{-1}A$ be a localization of $A$. Let $$\partial_A: A \to M$$ $$\partial_B: B \to S^{-1}M$$ be an $R$-derivation on $A$ ...
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3 votes
2 answers
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Map from X to $\mathbb P^1_k$ of degree $3$ with $X$ curve of genus $3$.

Let $X$ be a curve over $k$ algebraically closed field. Here $X$ is a proper, smooth, connected, $\text{dim}(X)=1$ scheme over $k$. Suppose that $g(X) = 3$ with $g(X) = \text{dim}_k H^1(X,\mathcal O_X)...
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