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 ...

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Tensor product of coherent sheaf with the stalk of structure sheaf

If $A$ is a commutative ring, $M\in A\text{-mod}$ and $\mathfrak{p}$ is a prime ideal of $A$ then it is an easy fact from commutative algebra that $M\otimes_{A}A_{\mathfrak{p}}\cong M_{\mathfrak{p}}$. ...
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3answers
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The support and the non-vanishing set of a function on a scheme

I have some confusion regarding the two concepts: Let $(X,\mathscr{O}_X)$ be a scheme, let $f\in \Gamma(\mathscr{O}_X,X)$ and define the support of $f$ to be $$\operatorname{Supp}(f) : = \{p\in X: ...
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45 views

Are embedded points where the nonreducedness is?

I know that if Spec $A$ is reduced, then there are no embedded points. I was wondering, if I know that $p$ is an embedded point of some Spec $B$, does that imply $B_{p}$ is non-reduced? Thanks!
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Functions on reduced schemes are determined by their values at each point.

This is an exercise in Vakil's Foundations of Algebraic Geometry, namely 5.2.A. Let $X$ be a reduced scheme. If $a\in \mathscr{O}_X(X)$ is such that its image in $\mathscr{O}_{X,p}$ lies in the ...
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37 views

Associated points of Spec $\mathbb{C}[x,y]/ I$

Suppose we know that the only associated points of Spec $\mathbb{C}[x,y]/ I$ were $[(y-x^2)]$, $[(x-1,y-1)]$ and $[(x-2,y-2)]$. Is there enough information to deduce if this scheme is reduced or not? ...
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126 views

Exercise 5.5.F. on Ravi Vakil's Notes related to associated points

Let $A$ be a Noetherian ring and $M$ a finitely generated $A$ module. In Ravi Vakil's notes he first states that the associated points of $M$ satisfy the following: (A) The associated points of $M$ ...
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52 views

Irreducible closed subsets of a scheme corresponds to points

I have posted an answer here for the case of an affine scheme, but I got stuck when I tried to generalize the argument to schemes. My thoughts Consider a point $p$ in the scheme, its closure in the ...
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27 views

A non-smooth subscheme of $\mathbb A^2_\mathbb R$

Consider the following subscheme $X$ of $\mathbb A^2_\mathbb R$ which is made by a curve minus a point and "plus" an isolated point $p$: Which is the simplest way to show that $X$ is singular at the ...
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1answer
58 views

$\operatorname{Spec}\mathbb{K}[x,y,z]/\langle x^2-yz \rangle$ is normal and singular

Let $X=\operatorname{Spec}\mathbb{K}[x,y,z]/\langle x^2-yz\rangle$ an affine scheme. It is singular because only at the rational point $0$ corresponding to the ideal $\langle x,y,z\rangle$, the ...
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29 views

The image of $f$ in $A$ in its resdue field

Let $A = k[x,y]/(xy,y^2)$, where $k$ is a field, and take $x+1+I \in A$. I know that $Q = (x+2+I, y+I)$ is a prime (maximal) ideal in $A$. Could someone please show me exactly how the image of ...
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136 views

What's With The Diagonal Morphism?

Given a morphism $X \to Y$ of schemes, we can construct a diagonal morphism $\delta: X \to X \times_Y X$ via the universal property of the fiber product applied to the identity map $X \to X$. ...
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46 views

Support of $f \in k[x,y]/(xy,y^2)$

Let $A = k[x,y]/(xy,y^2)$, where $k$ is a field, and take $f \in A$. I am working on an exercise that says prove that the support of $f$ (as a global section of the structure sheaf on Spec $A$) is ...
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33 views

A quotient of a scheme that isn't a quotient of a ringed space

In his book, prof. Qing Liu says that a quotient of a scheme can not be a quotient of ringed spaces. In order to prove this, he proposes an exercises (which I have slightly modified) 1) Let ...
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36 views

Construction of the quotient scheme

I have to construct the quotient of a scheme. Let $G$ is a finite group of automorphisms of a ring. 1) Let $p: \operatorname{Spec} A \mapsto \operatorname{Spec}(A^G)$ the morphism induced by the ...
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2answers
82 views

Understanding Hartshorne's proof that every projective morphism is proper.

In chapter $II$ of Hartshorne, theorem $4.9$ shows that every projective morphism is proper, using the valuative criterion for properness. I understand how the required morphism is constructed, but I ...
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51 views

Construction of the quotient space of a ringed space

Let $G$ be a group acting on a ringed topological space $(X,\mathcal{O}_X)$ and let $p: X \mapsto Y=X/G$ endowed with the quotient topology. Clearly, the action is $x g=g^{-1}(x)$ It's also clear that ...
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104 views

Cuspidal curve realized from $\mathbb{P}^1$ adding a fat point

let me ask you a question which will show my poor understanding of stalks and ringed spaces.. I hope that this example will help me clarifying the subject. So here we go: I've read (in particular from ...
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0answers
32 views

Notational Confusion in a definiton of Hartshorne about Smooth of relative dimension

This the definition from Hartshorne's Algebraic Geometry of Smooth of relative dimension n I want to understand what is meant by $\Omega_{X/Y}\times k(x)$. $\Omega_{X/Y}$ is the sheaf of relative ...
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47 views

The diagonal morphism of the fibered product

In remark 3.3 pag. 100 of the book "Algebraic geometry and arithmetic curves", prof. Qing Liu say that if $p,Q$ are the projections from the fibered product $X \times_Y X$ onto $X$ and $s \in ...
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69 views

Utility and meaning of the relative setting in Scheme theory

I'm sorry if my question is rather trivial, but I'm starting to learn scheme theory and I have a very basic question. When talking about schemes I see that very often, instead of taking "a point of a ...
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1answer
151 views

Blow-up and base change

Consider a complex smooth (projective) surface $X$ and a blow-up $\epsilon:S\longrightarrow X$ at a point $x\in X$. Let $\sigma\in\text{Aut}(\mathbb C)$ be a field automorphism and moreover let ...
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33 views

Closed immersions and base change

Consider a field extension $L\subseteq K$ and two $L$ schemes $X_L$ and $Y_L$ with an embedding $j:X_L\longrightarrow Y_L$. Now take the base changes $$X:=X_L\times_{\text{Spec L}}\text{Spec} K$$ ...
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33 views

Valuative criteria at closed points

Let $X \to S$ be a morphism. In the valuative criteria for properness, is it enough to test morphisms $\text SpecK \to X$ from spectra of fields to $X$ such that the image is a closed point of $X$? ...
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271 views

Studying Deformation Theory of Schemes

Versal Property Local Deformation Space Mini-versal deformation space I came across these words while studying these papers a) Desingularization of moduli varities for vector bundles on curves, ...
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48 views

About the smoothness of a non-reduced variety.

Consider a $k$-scheme $X$ with the following properties: $k$ is algebraically closed, $X$ is irreducible, non-reduced, separated and of finite type; moreover $X$ has dimension $1$ ($X$ is a non ...
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2answers
50 views

Checking normality for quasi compact schemes

Let $X$ be a quasi compact scheme. We know that any point on $X$ is a generization of a close point. Could someone possibly explain me why it then follows that to check if $X$ is normal, it suffices ...
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35 views

Definition of Formally Smooth from Stack Project

$T, T'$ are affine schemes. What is meant by $F\leftarrow T$ (or $G \leftarrow T')$
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72 views

Closed sets flat morphism

Consider the sets $Y_{k}:=\{y\in Y\mid |f^{-1}(y)|\leq k\}$, where $f:X\to Y$ is a flat quasi-finite morphism between smooth irreducible affine varieties and $k$ a natural number. Are these sets ...
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2answers
150 views

Regular sequence of sections of line bundles over a coherent sheaf

I am reading the first chapter from the book by Huybrechts and Lehn, where I encountered the following definition. I have the following doubts regarding this definition : What is the map ...
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325 views

When is the pushforward of a quasi-coherent sheaf quasi-coherent? Hartshorne proof

In the following we have $f:X \rightarrow Y$ morphism of schemes, $\mathcal{F}$ a quasi-coherent sheaf on $X$ and I am referring to proposition 5.8 page 115 in Hartshorne. To prove that the ...
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1answer
150 views

Properties of fibers of a morphism of varieties

In this question, all varieties are supposed to be over an algebraically closed field $k$. Hypothesis: X is a smooth projective surface and $f:X\longrightarrow \mathbb P^1$ is a morphism with we ...
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1answer
61 views

closed point of a locally finite $k$-scheme

Let $X$ be a locally finite $k$-scheme, where $k$ is a field. Suppose I have $Spec B \subseteq X$ such that $B$ is a finitely generated $k$-algebra, and $p \in Spec B$ a closed point inside $Spec B$ ...
3
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1answer
47 views

Bertini's theorem for surfaces: informations about singular fibers.

Let $S$ be a complex non-singular projective surface embedded in some $\mathbb P^n$. Thanks to the Bertini's theorem (Hartshorne theorem II.8.18) there exists a hyperplane $H\subseteq\mathbb P^n$ ...
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A property P of morphisms of $S$-schemes $f : X \rightarrow Y$ is local on $X$, or $Y$, or $S$ or [duplicate]

I have asked the same question on math.stackexhange here, but thought that is was a good idea to post it here also. I am learning schemes theory at school and I have for now only lectures notes that ...
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1answer
163 views

“This property is local on” : properties of morphisms of $S$-schemes

I am learning schemes theory at school and I have for now only lectures notes that I am taking during the course. The professor is quite often using following expressions, without having defined them ...
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1answer
57 views

Projective dimension of a coherent sheaf in a short exact sequence

Let $X$ be a noetherian integral scheme. We define the projective/homological dimension of a torsion free coherent sheaf $E$ to be $\mathrm{dh}(E)= \sup\{dh(E_x)|x\in X\}$, here dh$(E_x)$ denotes the ...
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2answers
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Why is dh$(k(x))=$dim $X$?

Let $X$ be an integral Noetherian scheme. Let $x\in X$ be a regular closed point of $X$. Then Huybrechts and Lehn in his book, says that dh$(k(x))=$dim $X$. Here dh$(k(x))$ refers to the ...
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1answer
54 views

Are the fibers of this morphism reduced?

Let $X$ be a non singular complex projective surface (Hartshorne notation!) and consider a morphism $f:X\longrightarrow\mathbb P^1_{\mathbb C}$ with the following properties: $f$ is flat $f$ is ...
3
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1answer
102 views

Equivalent definition of Schemes

I recall seeing that the category of schemes can be captured by a general construction as follows. Let $\mathbf{Spec}\colon \mathbf{CRing}^{op}\to \mathbf{LRS}$ be the usual functor from the ...
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145 views

Do schemes help us understand elliptic curves?

I'm reading Silverman and Tate's "Rational Points on Elliptic Curves" and I'm very much enjoying learning about these objects, and in particular doing a bit of number theory. It's different to what ...
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31 views

Complexifying Lie group actions

In Atiyah-Pressley's paper Convexity and Loop Groups, it is claimed that the Bruhat cells $C_\lambda$ are invariant under the natural $S^1 \times T$-action and that the authors will prove this claim. ...
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1answer
46 views

Equality of two $K$-valued points for reduced $K$

I'm reading "Red Book of varieties and schemes". There is definition 2, page 118. Let $f,g:K \to X $ be to $K-$valued points of scheme $X$, we say that they are equal at $x\in K$ $(f(x) \equiv g(x))$ ...
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1answer
80 views

A rigorous characterization for a ringed spaces to be isomorphic to an affine scheme.

On page 21-22 of the book The Geometry of Schemes by Eisenbud and Harris there is a characterization for when a ringed spaces $(X,\mathcal{O}_X)$ is isomorphic to an affine scheme ...
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102 views

Spectrum of $\mathcal{O}(U)$

Let $X=\operatorname{Spec}(A)$ be the spectrum of the comm. ring $A$ and let $\mathcal{O}$ be the associated sheaf of rings, i.e. for $U \subseteq X$ open, $\mathcal{O}(U)$ is the ring of all ...
5
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1answer
135 views

What is the algebraic tangent cone really?

Let $A$ be a (commutative unital) ring, let $\mathfrak{a} \subseteq A$ be an ideal, and let $B = A / \mathfrak{a}$. Then we have a descending filtration $$\cdots \subseteq \mathfrak{a}^3 \subseteq ...
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1answer
15 views

open subset of scheme with zero section

Let's take a scheme $X$. Is it possible to have an open non-empty subset $U$ of $X$ such that $\mathcal{O}_X(U)=0$? I can't find an argument against it, since there could exist some open set ...
4
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1answer
121 views

Question related to a stalk of a scheme

Suppose $X$ is a scheme and suppose $C$ and $C'$ are two irreducible components of $X$. Suppose also that $p \in C \cap C'$. Does is it then follow that $O_{X,p}$ is not an integral domain? Thanks!
3
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1answer
80 views

uniqueness of morphism $Spec(K) \to X$ of schemes

let $K$ be a field and $X$ a scheme. I'd like to understand the bijection $Hom_{Sch}(Spec(K), X) \cong \{x \in X | \exists \kappa(x) \to K \}$ That map is given by sending a morphism $f: Spec(K) \to ...
5
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85 views

Eisenbud/Haris, Exercice I-53: morphism between global spectra

Let $X=\operatorname{Spec}(\mathscr{F})$ and $Y=\operatorname{Spec}(\mathscr{G})$ two global spectra over a scheme $S$ and let $f:X\to Y$ be a morphism. I want to show that for all prime ideal sheaf ...
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29 views

higher order schemes or curvature to be smoothed

I want to calculate the radius of curvature of a meandering river. I have done a first estimated using a tool in arcgis. These are the results. The values of -99999 are recorded for points of no ...