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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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Hausdorff Property for Preschemes in Mumford's Red Book

Let $f,g: K \to X$ two morphisms between preschemes. In order to "compare" these two morphisms in David Mumford's "Red Book of Varieties and Schemes" there is suggested (see page 118) for a $x \in K$ ...
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50 views

Hartshorne, Exercise III 4.2 (a): A morphism $\mathcal{O}^r \to f_* \mathscr{M}$ that is an iso over the generic point.

I'm having some trouble with Exercise III 4.2 a) in Hartshorne's Algebraic geometry. It is Let $f: X \to Y$ be a finite surjective morphism of integral noetherian schemes. Show that there is a ...
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Geometric interpretation of the ideal class group?

When one thinks of the ring of integers in an algebraic number field as defining a one-dimensional scheme, what would the ideal class group correspond to geometrically please? Is it maybe the group of ...
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Closed points in the closure of a point on a scheme?

Let $X$ be the underlying space of an irreducible scheme. In particular, $X$ is non-empty. Note that the closure of any point is a closed irreducible subset. We say that a point has codimension $n$ ...
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29 views

Why are morphisms of finite type defined in the usual way?

We say that a morphism of schemes $f : Y \to X$ is finite if $X$ may be covered by affine open sets $\text{Spec}(B)$ such that each $f^{-1}(\text{Spec}(B))$ is affine, say of the form $\text{Spec}(A)$,...
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Surjection from a scheme of finite type

Let $S$ be a scheme, $X$ a scheme locally of finite type over $S$, $Y$ a scheme over $S$. Let $f:X\rightarrow Y$ a surjective morphism of $S$-schemes. Under what conditions is $Y$ locally of finite ...
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Smooth curves as iterated smooth hyperplane sections

Let $k$ be an algebraically closed field. Can every connected smooth projective one-dimensional $k$-scheme be obtained by taking iterated smooth hyperplane sections of the image of a projective ...
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Structure sheaf and residue field

Hartshorne defines the structure sheaf of $X=$ Spec$A$ as, for $ U \subset X$, $$ O_X(U)=\{s: U \rightarrow \dot\cup A_{\mathfrak{p}} \}$$ with 2 additional requirements, i.e. $s({\mathfrak{p}}) \in ...
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40 views

Universal mapping property for projective schemes and globalising it to projective bundles

Let $Y = \text{Spec}A$ be a noetherian affine scheme. Let $S$ be a graded $A$-algebra which is finitely generated by $S_{1}$ as an $S_{0}$ algebra. In other words, $S$ looks like $$ S = A[x_{0}, x_{1},...
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Smooth morphism into a normal scheme

I'm looking for an example of a smooth morphism $f:X\to Y$ of schemes with $Y$ normal and $X$ integral but not normal. Extra points when $X$ is a geometrically integral variety, and $Y$ as ...
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Geometric invariant theory Affine Quotients

Let $V=Spec(R[V])$ an affine variey and $G$ a linear reductive group action on $V$. (I'm not sure if the linear reductive condition is neccessary for the question). According to GIT the "quotient" $V/...
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Hartshorne Exercise III 3.2: $X$ is affine iff every component is affine

I'm trying to solve the following exercise frome Hartshorne's Algebraic Geometry: Exercise III 3.2. Let $X$ be a reduced noetherian scheme. Show that $X$ is affine if and only if each irreducible ...
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Non-flat locus for smooth schemes

Let $X$, $Y$ be connected smooth schemes of finite type over an algebraically closed field of characteristic 0. Let $f:X\rightarrow Y$ be a non-birational morphism surjective on underlying topological ...
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Not able to make sense of gluing via the functor of points in EGA

I am trying to use a well known result of Grothendieck to show that if $S$ is a scheme, and $\mathcal{B}$ is a quasi coherent sheaf of $\mathcal{O}_{S}$-algebras, then there is a relative affine ...
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What's in a Noetherian $\mathbb{A}$-Module Ephemeralization?

Just kidding, it's not Noetherian. And "Emphemeralization" implies it is a physical construct, or that if it is, due to knowledge heretofore unbeknownst but recently gained by visualization of black ...
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If $\mathcal{F}$ and $\mathcal{G}$ are locally free sheaves, then $\mathcal{Hom}_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})$ is a locally free sheaf.

If $\mathcal{F}$ and $\mathcal{G}$ are locally free sheaves on $X$ of rank $m$ and $n$ respectively, then $\mathcal{Hom}_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})$ is a locally free sheaf of rank $mn$....
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Belyi's Theorem for Proper Normal Curves

I have a question about an argument used in Szamuely's "Galois Groups and Fundamental Groups" in the excerpt below (or look up at page 127): In the proof of Belyi we beginn with the reduction step to ...
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Is a locally free sheaf of finite rank finitely presented?

I need to prove that a locally free sheaf of finite rank is finitely presented. Being homework, I'm not asking for a proof but rather for an explanation of why my counterexample is wrong. Here is the ...
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Locally Closed Immersion

My question refers to a step in the proof of Prop. 7.4.1 (pages 312-313) in Bosch's "Algebraic Geometry and Commutative Algebra". Here the excerpt: Let $X$ be relative $S$ scheme. The goal is to ...
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68 views

Zero-Dimensional Subschemes of Degree 21

I'm working on the following problem from Eisenbud and Harris' Geometry of Schemes. Consider zero dimensional subschemes of $\mathbb{A}^4_K$ of degree 21 such that $$V(\mathfrak{m}^3)\subset \Gamma \...
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are the global sections of a flat sheaf over a discrete valuation ring a free module?

Let $f:X\to \operatorname{Spec}\mathbf{Z}_p$ be a smooth proper $\mathbf{Z}_p$-scheme and $F$ a coherent sheaf on $X$ which is flat over $\mathbf{Z}_p$. Further suppose that $H^1(X_p, F_p)=0$ where $...
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Understanding the proof of the fact that the Chow group of a scheme $X$ is graded by dimension.

I would like to understand the proof of this fact: If $X$ is a scheme (separated, of finite type over $k=\overline{k}$) then the Chow group of $X$ is graded by dimension; that is, \begin{equation} A(...
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Automorphism inducing identity on closed points

Let $k$ be an algebraically closed field. Let $X$ be a $k$-scheme that admits a $k$-immersion into projective space. Is it true that any $k$-automorphism of $X$ that induces identity on closed points ...
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Do closed immersions preserve stalks?

This questions stems from an earlier confusion about the distinction between open and closed immersions between schemes. I understand that an open immersion $U\to S$ can simply be read as an open ...
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Sections of a locally free sheaf

Let $\mathcal{F}$ be a locally free sheaf of rank $n$ on a scheme $X$. We know that we can associate to it a vector bundle $F$ on $X$ such that $F_x \simeq \mathcal{F}(x)$, where with $\mathcal{F}(x)$ ...
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Image of the diagonal map in a scheme and the set on which the projections agree

Let $f:X\to Y$ be a morphism of schemes, let $z\in X\times_Y X$ such that (1) $p_1(z)=p_2(z)$; (2) let $x = p_1(z) = p_2(z)$, then $(p^\#_1)z:\mathcal{O}_{X,x}\to \mathcal{O}_{X\times_Y X,z}$ is ...
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Cycle associated to a closed subscheme

Let $X$ be an algebraic variety (i.e. an integral $k$-scheme, such that $X \to \mathrm{Spec \;}k$ is separated and of finite type). Let $Y$ be a closed subscheme and $Y_1, \dots Y_n$ the irreducible ...
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example of algebraic variety with infinitely many singularities

Let $X$ ba an algebraic variety and $\mathrm{Sing}(X)$ be the set of all singular points. For a set $A$ , $|A|$ denotes the cardinality of $A$ . I konw examples of algebraic variety with finite ...
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Property of Unirational variety

Let $X$ be an algebraic variety over field $k$ snd $n=\mathrm{dim}(X)$ . We assume $X$ is unirational. There exists $m \in \mathbb{N}$ and a dominant rational map $\phi : \mathbb{P}_k^m \...
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Can an algebraic variety be embedded in projective space ???

Let $X$ be an algebraic variety over field $k$ . $X$ can be embedded in a complete variety by Nagata's compactification theorem. Moreover, can we embed $X$ in a projective space $\mathbb{P}_k^n$ $?...
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Can every maximal ideal of Dedekind domain be principal after restricting to a small enough distinguished open subset?

Let $S=\textrm{Spec }R$ where $R$ is a Dedekind domain, let $\mathfrak{p}$ be a maximal ideal of $R$, which is a closed point of $S$, can we find an distinguished open affine subset of $S$, say $\...
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A certain functor in Hakim's “Topos annellés et schemas relatifs” is a sheaf for the canonical topology

In M. Hakim's Topos annellés et schemas relatifs, page 43, (3.4.7), the Author wants to define a sheaf $f_{0A}^*(X)$ over a topos $T$, with respect to the canonical topology. A scheme $X$ is given, ...
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What is a $k$-scheme isomorphism?

In Example 2.4.2 of Chapter IV 2 of Hartshorne's Algebraic Geometry, it gives $\pi:X=Spec(k[t])\to Spec(k)$ a scheme over $k$ and $F:X\to X$ and $Spec(k)\to Spec(k)$ the morphism with identity on the ...
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When is “base changing morphisms of schemes” surjective?

Suppose $X, Y$ are $S$-schemes and $S'\to S$ is a morphism. Every $S$-morphism $f:X\to Y$ gives rise to a $S'$-morphism $f':X'\to Y'$ (where $f'$, $X'$, $Y'$ are the base changes). Under which ...
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What does higher André--Quillen (co-)homology tell about deformation theory?

Let $f : X \to Y$ be a morphism of schemes, and denote by $L_{X/Y}$ the associated cotangent complex. It is often said that the cotangent complex controls the deformation theory of $X$. Several ...
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Cartier divisor example in Harthsorne

This question is about example 6.11.4 in Chapter II of Hartshorne. The example is about computing the Cartier divisor class group of the cuspidal cubic curve $y^2z = x^3$ in $\mathbb{P}_{k}^{2}$. He ...
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A question on the proof of Rigidity Lemma in birational geometry

I am reading the book, Birational Geometry of Algebraic Varieties, by J$\acute{\mathrm a}$nos Koll$\acute{\mathrm a}$r et al.. (Rigidity Lemma) Let $Y$ be an irreducible variety and $f:Y\to Z$ a ...
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What is transition function of scheme theoretic vector bundle ??

Let X be a scheme X and $\mathscr{F}$ be a locally free sheaf of rank $n$ over $X$. I want to consider a vector bundle over an algebraic variety $X$ , that is , the relative spec over $X$ of ...
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Characterization of torsion free sheaves

In "The Geometry of Moduli Spaces of Sheaves" by Huybrechts and Lehn a torsion free sheaf is defined as coherent sheaf $E$ on an integral Noetherian scheme $X$ s.t. for every $x\in X$ and every non-...
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Weil divisors on Noetherian local ring of dimension $1$

Ler $A$ be a Noetherian local ring of dimension $1$, with maximal ideal $\mathfrak m$ and minimal prime ideals $\mathfrak p_1,\dots, \mathfrak p_r$. In exercise 11.18 of the book "Algebraic Geometry 1"...
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Henselian Scheme Characterization

I have a question about an argument in a proof from Bosch's "Neron Models" (page 47): It's not clear to me how to see that (d) implies (e). According to the author's sketch it's a consequence of ...
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Can one obtain a generating set for a module of local sections of a coherent sheaf by finding generating sets at the stalks?

Question is possibly slightly different than posed if I am misunderstanding what coherency means. I do not know the correct term for "set of local sections corresponding to $F(U)$". Let $F$ be a ...
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Action on Group Scheme

I have question which arises from the answer of my former thread: https://mathoverflow.net/questions/324887/why-is-for-a-group-scheme-of-finite-type-smooth-resp-irreducible-equivale Let $G/K$ be a ...
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65 views

Abelian Variety Commutative

I have a question about a step in a proof from Lang's "Abelian Varieties" (page 20): By definition an abelian variety $A$ over field $k$ is a proper smooth $k$-group scheme that is irreducible. In ...
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$\mathbb{A}^n _R \to \operatorname{Spec}(R)$ is not Proper Morphism

Consider the affine $n$-space $\mathbb{A}^n _R= \operatorname{Spec}(R)(R[X_1,, ..., X_n])$ over arbitrary nontrivial ring $R$. (assume furthermore $n \ge 1$) I want to show that the related ...
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Map between stalks induced by a finite morphism is finite

Let $f:X\rightarrow Y$ be a finite morphism of scheme. For $x\in X$, $y=f(x)\in Y$. Is it true that the map $$f^\sharp_x:\mathcal{O}_{Y,y}\rightarrow \mathcal{O}_{X,x}$$ is finite? (i.e $\mathcal{O}_{...
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Confused about why the conormal exact sequence is what it is on a scheme

Consider a composition of morphisms of schemes, $$Z \stackrel{j}{\longrightarrow} X \stackrel{f}{\longrightarrow} Y,$$ where $j: Z \rightarrow X$ is a closed immersion with sheaf of ideals $\mathscr{I}...
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Question on functor of points formalism in algebraic geometry

The question is based on [Chapter $5$, Section $q$, Remark $5.65$ of] Milne's Algebraic Geometry notes. I will try to make the question self-contained and be consistent with Milne. Assume all schemes ...
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Valuative Criterion of Properness: Quasi Compact Morphism

I have a question about an argument from Bosch's "Algebraic Geometry and Commutative Algebra" (page 476): The author reduced in the excerpt above that to verify that (ii) $=>$ (i) (so that $f$ is ...
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40 views

Veronese Embedding for Schemes

I have a question based on following exercise from Bosch's "Algebraic Geometry and Commutative Algebra" (page 461): (*) Let $\mathbb{P}^m$ projective space and $\mathcal{L}:= \mathcal{O}_{P^m}(d)$ ...