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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About definition of finite presentation morphism

Simple question: why finite type morphisms of scheme are required to be only quasi-compact while finitly presented morphisms are asked to by quasi-compacts and quasi-separated?
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Equality of morphisms $f,g:K\rightarrow X$ of schemes, where $K$ is a reduced scheme.

Suppose that $f,g:K\rightarrow X$ are morphisms of schemes, where $K$ is a reduced scheme. I want to show that $f=g$ if and only if for all $x\in K$, $f(x)\equiv g(x)$. Here $f(x)\equiv g(x)$ means ...
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Motivation for separated and proper schemes

Hartshorne mention at the beginning of section 4 in chapter 2 that the definition of separated is similliar to hausdorff. We all can see that. That is also what I found in google. Again - we all can ...
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Is the sheafified Cech complex for an étale cover a resolution?

Suppose I have an étale cover of a scheme $X$ by etale $X$-schemes $U_{i}$, let $U:=\coprod_{i=1}^{n} U_{i}$ and write $f:U\rightarrow X$ for the induced map. Then for every quasicoherent sheaf $\...
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necessary and sufficient condition for Normalization of Integral schemes

Let $f:Y \longrightarrow X$ be a morphism of integral schemes. I was wondering if the following is true? $f$ is the normalization morphism $\Leftrightarrow$ $Y$ is normal and $f$ is birational and ...
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Example of Picard number in family of smooth variety jumping

For a scheme or formal scheme $X$, let $\mathrm{Pic} X$ be its Picard group. If $X$ is a smooth proper variety over an algebraically closed field, let $\mathrm{Pic}^{0}(X)$ be the subgroup consisting ...
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When two Algebraic vector bundles on a Noetherian quasi-affine scheme are equal in $K_0$ of the scheme

Let $X$ be a (connected) Noetherian scheme and $K_0(X)$ denote the Grothendieck group of the category of Algebraic vector bundles (coherent sheaves that are locally free and of constant rank ( as $X$ ...
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Frame bundle and $GL_n-$-principal bundle

I am not an expert in algebraic geometry, so my question maybe be trivial for those who are. All schemes are over $\mathbb C$. Let X be a scheme and $\mathcal E\rightarrow X$ be a vector bundle of ...
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Affine scheme which is connected but not irreducible and not reduced.

Are there any examples of affine schemes which is connected but not irreducible and not reduced?. Reducedness and irredubility doesn't deduces connectedness, so I think there should be examples, but I ...
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Global-section functor distributes over tensor product of vector-bundles on punctured spectrum?

Let $(R, \mathfrak m)$ be a Noetherian local ring of depth $\ge 2$. Let $\mathcal F, \mathcal G$ be Algebraic vector bundles on the punctured spectrum $U=\operatorname{Spec}(R)\setminus \{\mathfrak m\}...
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Nillradical is prime ideal, then the ring is not a product ring.

Let $A$ be a commutative ring and $nill(A)$ is not a prime ideal. This is just a characterization of $SpecA$ to be irreducible. Then, according to the argument of general topology, irreducible ...
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Affine scheme which is irreducible but not connected.

Could you let me know the example of affine scheme which is irreducible but not connected? I know affinescheme SpecA is irreducible only if nillradical of A is prime ideal. But I have trouble with ...
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Punctured spectrum of a (reduced) Noetherian local ring of dimension $1$ is an affine- scheme?

Let $(R, \mathfrak m)$ be a Noetherian local ring of dimension $1$. Then the affine- scheme $X=\operatorname {Spec}(R)$ can be written as a set-theoretic union $\operatorname{Spec}(R)=Min(R)\cup \{\...
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Why is the ring of local-ring valued points of a ring scheme a local ring.

I'm confused on a supposedly easy claim: Let $T$ be a base scheme, and let $\mathbf{R}$ be a ring scheme over $T$, i.e. a scheme $\mathbf{R} \to T$ such that for all $E \in \operatorname{Sch}_{/T}$ $\...
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Vanishing of cohomology of affine scheme

In EGA I 5.1, more specifically the proof of 5.1.9, which states that $X$ is affine iff the closed subscheme defined by a quasi-coherent sheaf of ideals $\mathscr{I}$ such that $\mathscr{I}^n = 0$ for ...
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Are affine morphisms with coherent direct image finite?

Let $f:X \longrightarrow Y$ be a morphsim of Noetherian schemes. I was doing excersise 5.5 of Hartshorn Algebraic Geometry and in (c) i showed that finite morphisms preserve coherence (i.e. if $\...
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Is $\theta^\sharp:\mathscr O_{\operatorname{Spec}f_*\mathscr O_X}\to \theta_*\mathscr O_X$ isomorphic?

Suppose $f:X\to Y$ is a quasi-compact and separated morphism of schemes, $\pi: \operatorname{Spec}f_*\mathscr O_X\to Y$ is the canonical affine morphism such that $\pi_*\mathscr O_{\operatorname{Spec}...
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Morphism $f:X\rightarrow\text{Spec}(B)$ is quasi - affine, if $X$ is quasi - affine.

I have a question regarding a proof from Bosch's Algebraic Geometry book, namely section 9.5, Proposition 3, part (ii): Let $f:X\rightarrow Y=\text{Spec}(B)$ be a morphism of schemes. $f$ is quasi-...
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What is the $k^{al}$-points of a variety over $k$?

Let $X=\mathrm{Spec}(\mathbb{Q}[x,y]/(y^2-x^3-x-1))$. So $X$ is an elliptic curve over $\mathbb{Q}$ given by function $y^2=x^3+x+1$. Now I wonder what is meaning of a $\mathbb{C}$ points of $X$? I ...
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Refinement of two stratifications of an algebraic variety

Let $(X,\mathscr{O}_{X})$ be a separated $\mathbb{C}$-scheme of finite type. A locally finite decomposition $X=\coprod_{i\in I}X_{i}$ into locally closed subschemes is said to be a stratification of $...
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Is the affine cone of a flat projective scheme again flat?

I'm trying to solve Hartshorne Chap.III Ex.9.5.(c): The biggest problem is to show the existence of a closed subscheme $\tilde{X}\subset \mathbb{P}_{T}^{n+1}$ such that $\tilde{X}_{t} = \operatorname{...
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When is relative Hilbert scheme $\text{Hilb}_{X/S}^{p(t)}$ flat over $S$?

Let $S$ be a scheme of finite type over the complex numbers $\mathbb{C}$ and let $X\subset S\times\mathbb{P}^r$ be a projective family over $S$. In the book "Geometry of Algebraic Curves Volume II" ...
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Morphism between quotient sheaf ($K_{X}/\mathcal{O}_{X}$) and $\bigoplus_{x\in\mathcal{X}}i_{x,*}(K(X)/\mathcal{O}_{X,x})$

Consider an integral scheme $X$ of finite type over $k$ of dimension $1$. Then $X$ consists of one generic point $\eta$ and the rest of the points are closed points. We call the set of closed points $\...
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A question about an example of integral notherian scheme that is not a finite type over K

While reading up on 'finite type over $k$' on Hartshorne, algebraic geometry, I have tried to understand the following example : Example ) If $P$ is a point of a variety of $V$, with local ring $\...
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Concerning the spectrum of a quasi coherent $\mathcal{O}_{X}$ algebra

I am reading chapter 11 (on vector bundles) of algebraic geometry I by Wedhorn/Görtz. There they have the following proposition 11.1: I understand what they write in equation 11.2.1. However, I do ...
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The quotient of the constant sheaf by the structure sheaf is flasque.

Let $X$ be an integral scheme of finite type over $k$ with $\dim(X)=1$. Let $\mathcal{K}_{X}$ be the constant sheaf on $X$ with value $K(X)$, where $K(X)$ is the function field of $X$. I want to ...
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Geometric points in fibre of finite étale morphism $\phi : Y \rightarrow X$ is independent of fibre

I am reading the following notes http://www.math.toronto.edu/~jacobt/Lecture6.pdf and trying to understand the conclusion of Lemma 2.1. We are trying to show that the number of geometric points above ...
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29 views

Classification of points on an irreducible reduced scheme of finite type over $k$ of dimension $1$.

Suppose that $X$ is an irreducible reduced scheme of finite type over $k$ of dimension $1$. Meaning that the biggest chain of distinct irreducible closed subsets $X_{0}\subset X_{1}\subset ... \subset ...
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59 views

Why is the pullback (between affine varieties) of a quasi coherent sheaf quasi coherent?

Let $\phi:A\to B$ be a ring homomorphism inducing $f :\operatorname{Spec}(B) \to \operatorname{Spec}(A)$ on spectra. Let $M$ be an $A$-module and $\widetilde{M}$ be the corresponding quasi coherent ...
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Pullback along absolute Frobenius

Consider the following diagram where $X^{(p)}$ is the pullback of $X$ along the absolute frobenius $F_S$ of S. $$\require{AMScd} \begin{CD} X^{(p)} @>{}>> X\\ @VVV @VVV \\ S @>{F_S}>&...
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Result on étaleness of Group schemes

I have a question about a proof from Arithmetic Geometry (edited by Cornell & Silverman) on page 51: The proof starts with "According to [6], we may and do assume $S = Spec \ k$, ... [6] refers ...
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Computing local dimensions of affine and projective space

Again I am stuck trying to solve an exercise in Bosch's Algebraic Geometry. I apologize for this rather lengthy post. For a discrete valuation ring $R$, consider the scheme $S=\rm{Spec}(R)$ . ...
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What can we say about a restriction to a closed subset of an affine scheme?

Let $X=\operatorname{Spec}(A)$ and $(X,\mathscr O_X)$ the affine scheme and $Z \subset X$ a closed subset. By definition we have $\alpha \lhd A$ s.t $Z=\operatorname{Spec}(A/\alpha)$ (at least up to ...
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If $S\to T$ is a closed immersion, then $X\times_S Y\simeq X\times_T Y$.

Let $X,Y$ be $S$-schemes and $S\to T$ a closed immersion of schemes. Prove that we have a natural $T$-isomorphism $X\times_S Y\simeq X\times_T Y$. Let $f:X\to S$ and $g:Y\to S$ the structural ...
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Proof with valuations in Liu's book

I have a question about a claim I found in Liu's book Algebraic Geometry and Arithmetic Curves in the Proof of Theorem 8.3.26 (b) page 356: In (b) uses the notation of a "center". This can be looked ...
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$X_K$ is Reduced iff $X_F$ is reduced for every finite extension $k \subset F \subset K$

I have a couple of questions about a proof given in the answer for this question: Base change and irreducibility/reducedness/connectedness in Qing Liu's book (3.2.7 and 3.2.11 using 3.2.6) Let $X$...
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Isomorphism between schemes

I'm trying to show that every closed subscheme $Y$ of an affine scheme $X = \operatorname{Spec}(R)$ is isomorphic to $\operatorname{Spec}(R/I)$ for some ideal $I$ of $R$. I try to show that there ...
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Proof of Chow's Lemma, where is proper necessary?

I am looking at Wikipeda for the proof of Chow's Lemma. My impression is that the proof uses $X$ is a separated scheme over a Noetherian scheme $S$ (throughout where it is argued that some graph map ...
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How much information about $R$ is encoded in its local rings OR in its tangent spaces if I know the topology of $\operatorname{Spec}(R)$?

Assume I know the topology of the spectrum $\operatorname{Spec}(R)=(X,\mathcal{O}_{X})$ of a reduced ring $R$, and I know what the local rings $\mathcal{O}_{X,x}$ look like for every $x\in X$. How ...
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Clarification about Riemann-Roch for non-reduced curves

This is exercise 18.4.S in Vakil's Foundations of Algebraic Geometry. Let $C$ be a projective curve over a field $k$ (possibly singular), with irreducible components $C_1, ... C_n$, with generic ...
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Affine open for arbitrarily chosen 2 points on a scheme [duplicate]

Let $X$ be a reduced scheme. If there are arbitrarily chosen two points $p_1, p_2 \in X$, does the following always hold? Q. There exists some affine open neighbourhood $U$ such that $U \ni p_1, p_2$....
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Question about the function field of a noetherian, integral, locally factorial scheme

Let $X$ be a scheme. Assume $X$ is noetherian, integral, and locally factorial. Let $\eta$ be the generic point of $X$, then the function field of $X$ is $K(X):=O_{X, \eta}$. Let $Y$ be a prime ...
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Pushforward of tangent bundle. Can it be $0$?

Let $f:X\to Y$ a smooth flat map between, integral projective varieties such that $\dim(Y)<\dim(X)$. Let's assume by semplicity that everything is over $\mathbb C$. So we are in the situation of "a ...
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First (historically) solved problems using schemes theory

tonight I was wondering during a pause on my studies how the mathematical community reacted to Grothendieck's EGA books in the sense of searching for the first problems solved using schemes, for I ...
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Direct image with the absolute Frobenius

Let $X$ be a smooth projective curve over a characteristic $p$ scheme $S$. Let $\mathcal E$ be a vector bundle on $X$. Let $\Psi:\operatorname{Der}_S(\mathcal O_X,\mathcal O_X) \to \operatorname{End}...
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The ideal sheaf of a closed subscheme of the projective $\mathbb{C}$-scheme.

Consider the closed subscheme $\text{Spec}(\mathbb{C})\sqcup\text{Spec}(\mathbb{C})$ of $\mathbb{P}_{\mathbb{C}}^{1}$. Let $i: \text{Spec}(\mathbb{C})\sqcup\text{Spec}(\mathbb{C}) \rightarrow \mathbb{...
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Deligne generic base change theorem $l$-adic sheaves

Let $f:X \to Y$ be a finite type morphism of noetherian schemes and $F$ a constructible etale sheaf on $X$. Deligne shows in SGA 4 1/2 that there exists a dense open subset $U \subseteq Y$ such that $...
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The affine scheme $\text{Spec}(\mathbb{C}\times\mathbb{C})$ is a projective $\mathbb{C}$-scheme.

I want to show that $X:=\text{Spec}(\mathbb{C}\times \mathbb{C})$ is a projective scheme. So we have to find a closed immersion from $X$ to $\mathbb{P}^{1}_{\mathbb{C}}$. Brief explanation of my ...
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Construction of the reduced scheme $X_{red}$ out of a scheme $X$

If $X$ is a scheme, we define the sheaf $(\mathcal{O}_X)_{red}$ as the sheafification of the pre-sheaf defined by $U\mapsto \mathcal{O}_X(U)/\sqrt{0}$, where $\sqrt{0}$ is the nilradical of the ring $\...
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Extendability criteria for Morphisms of Schemes

Let $X,Y$ over a say locally noetherian scheme $S$. (later we have to discuss if the locally noetherian scheme condition for $S$ is really important.) Let $p \in X$ be a point and we have a morphism $...

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