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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Linear Systems And invertible Sheaf

Hello Fellow Mathematicians/Algebraic Geometer , This is question has two parts one which is more conceptual and the other more straight forward: i) Let $X$ be a non-singular protective variety ...
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About the ramification locus of a morphism with zero dimensional fibers

This question arises from my somewhat frustrating attempts to understand what etale means (in the world of algebraic varieties for now) and marry the more advanced algebraic geometry references and ...
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1answer
44 views

What is the class group of the complement of three lines in the projective plane?

I have a straightforward question : Let $ Y$ be the union of the three lines $ L_1:x=0 , L_2 :y=0$ and $L_3:z=0$ in the Projective plane $\mathbb{P }^2$. What is the Class group of the Complement ...
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Etale locally free sheaf is always locally free in Zarissky topology.

I'm trying to solve exercise III.10.5 from Hartshorne "Algebraic geometry". Let $\mathcal F$ be a coherent sheaf on a scheme $X$ locally free in 'etale topology, namely for any $x \in X$ there is an ...
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1answer
40 views

Taking module sheaf commutes with tensor product

I'm trying to prove proposition II.5.2.b in Algebraic Geometry by Hartshorne. The proposition states that for $ A $-modules $ M $ and $ N $ and $X=\text{Spec}\ A$ there is an isomorphism $ ...
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31 views

About the construction of Quot-Schemes

I am reading in the paper of Nitsure (link: http://arxiv.org/pdf/math/0504590v1.pdf) about the construction of the Quot-scheme $\mathrm{Quot}_{E/X/S}^{\Phi,L}$. After Lemma 5.4. they reduce to the ...
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22 views

group scheme of prime order p is killed by p

In the article "Group Schemes of Prime Order" by Tate and Oort (see here) it is proved that a group scheme of prime order $p$ over the base $S$ is killed by $p$ (Theorem 1). The authors state that it ...
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47 views

Can we recover étale, fpqc etc. morphisms of schemes from the affine versions?

$\DeclareMathOperator{\Spec}{Spec}$ Consider the following procedure for defining a class of morphisms of schemes: Take a suitable class of homomorphisms of rings (e.g., canonical maps to ...
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26 views

Why is the scheme associated to a product of quasi-projective varieties naturally isomorphic to the product of the associated schemes?

What I mean by this is, suppose $X$ and $Y$ are quasi-projective varieties over some arbitrary field $k$. Then $X\times Y$ is again a quasi-projective variety. I've seen this a few times, but what is ...
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58 views

Hartshorne notation in section III.12

I am reading section III.12 in Hartshorne, the one about the Semicontinuity Theorem. For $f:X \rightarrow Y$, where $Y=\mathrm{Spec}A$ and $\mathcal{F}$ a coherent sheaf on $X$, he writes ...
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34 views

If $X$ is quasi-projective but the scheme $\tilde{X}$ is affine, is $X$ necessarily affine?

I'm curious if the following works as a criterion to determine when a quasi-projective variety is actually affine. If $X$ is a quasi-projective variety, and the scheme $\tilde{X}$ is affine, does ...
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1answer
50 views

Punctual Hilbert scheme of four points

I am looking at $\text{Hilb}^4(\mathbb{C}^2)$, which is the Hilbert scheme of four points on $\mathbb{C}^2$. In particular, I am just looking at four points collided (at the origin), and want to know ...
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1answer
44 views

Exercise 6.3.N. on Ravi Vakil's notes (on morphism from $S$ scheme to $\mathbb{P}_B^n$)

I am working on Exercise 6.3.N. on Ravi Vakil's notes (on morphism from $S$ scheme to $\mathbb{P}_B^n$) and I would like some assistance. The exercise states: Let $B$ be a ring. If $X$ is a ...
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16 views

Morphism of schemes determined by their induced maps of $Z$ valued points

I am doing an exercise that states: morphism of schemes $X \rightarrow Y$ is determined by their induced maps of $Z$ valued points, as $Z$ varies over all schemes. I am a bit confused with this ...
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1answer
112 views

Interpretation of sheaf flat over a base

I am trying to get an interpretation of what means for a sheaf to be flat with respect to a base. The definition is that, given $f:X \rightarrow Y$ morphism of schemes, $\mathcal{F}$ is flat over $Y$ ...
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1answer
73 views

Did I use axiom of choice in my proof?

I have two different affine open covers for a scheme $X$, say $X = \cup_{i \in I} U_i$ and $X = \cup_{j \in J} V_j$. For each $p \in X$, we know there exist some $i(p)$ and $j(p)$ such that $p \in ...
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Set-theoretic intersection of affine open subschemes.

Let $X$ be a separated scheme, $U,V \subseteq X$ open affine subschemes, $\Delta \colon X \to X \times X$ the diagonal morphism and $\pi_1, \pi_2 \colon X \times X \to X $ the natural projections, so ...
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Is the push-forwad of a quasi-coherent sheaf under open immersion still quasi-coherent?

My question is related to this question: When is the pushforward of a quasi-coherent sheaf quasi-coherent? Hartshorne proof In Hartshorne page page 115 Proposition 5.8 it has been proved that if $X$ ...
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20 views

Equivalent definitions of noetherian scheme.

A scheme $X$ is locally noetherian iff there is a cover $X = \bigcup_i \text{Spec}(R_i)$ with noetherian $R_i$. When $X$ is also quasicompact, it is called noetherian. Question: Why is (as Hartshorne ...
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38 views

Properties of dominant morphism of schemes

I am trying to solve the following exercise 4.11, p.67 from Qing Liu's book "Algebraic geometry and arithmetic curves". Let $f:X\to Y$ be a morphism of irreducible schemes with respective generic ...
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68 views

Does dominant morphism of integral schemes is injective on sheaves?

Let $f:X \to Y$ be a dominant morphism of integral schemes. Is it true that it is equivalent to the fact that $\mathcal O_Y \to f_* \mathcal O_X$ is injective? Or does one imply another? It's quite ...
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1answer
37 views

On a canonical morphism from Spec $O_{X,p} \rightarrow X$

Let $X$ be a scheme. I am doing an exercise: Let $p \in X$. Describe a canonical (choice-free) morphism from Spec $O_{X,p} \rightarrow X$, with hint that says to make sure that the morphism is ...
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How to imagine the difference between the following schemes?

Consider $A=\operatorname{Spec} k[x]_{(x)}[t]$ and $B=\operatorname{Spec} k[x,t]_{(x)}$ for a field $k$ (Vakil, note 11.3.8). For me, both are infinitesimal neighborhoods of an affine line - the ...
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123 views

Showing closed immersions are stable under base extension without using that they are affine.

This question is based on question $3.11$ from chapter $2$ of Hartshorne, found on page $92$. Part $a)$ of said question asks to show that closed immersions are stable under base extension. In other ...
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2answers
81 views

If local rings are Noetherian, is scheme locally-Noetherian?

If all the local rings of a scheme are Noetherian, is the scheme locally-Noetherian?
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1answer
27 views

Equivalent conditions for a closed immersion of schemes

In Hartshorne, a closed immersion of schemes is defined to be a scheme morphism $\Phi \colon Y \to X$ such that $\Phi$ is a homeomorphism onto $\Phi(Y)$, $\Phi(Y)$ is closed in $|X|$ and $$ (*) ...
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1answer
51 views

Hartshorne III 9.5 confused about base extension.

Hartshorne III Proposition 9.5 states: Let $f:X\to Y$ be a flat morphism of schemes of finite type over a field $k$. For any point $x\in X$ let $y=f(x)$. Then $$\dim_x(X_y)=\dim_x X-\dim_y Y$$ ...
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0answers
38 views

Morphism between projective schemes induced by injection of graded rings

Let $A$ be a graded ring and $d>0$ be an integer. Define the graded ring $B$ such that $B_i=A_i$ if $d$ divides $i$ and $B_i=0$ otherwise. Is it true that a homomorphism of graded rings ...
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2answers
44 views

Basic question related to stalk of the structure sheaf of a scheme

Let $X$ and $Y$ be schemes. Suppose I have a morphism of schemes $(\pi, \phi)$, where $\pi: X \rightarrow Y$, thus $\pi$ is continuous, and $\phi: O_Y \rightarrow \pi_*O_X$. Let $p \in X$ and $\pi(p) ...
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2answers
49 views

Quasi-Finite + Affine -> Finite?

If $f:X\to Y$ is an affine morphism of schemes, say with $Y$ irreducible, that is quasi-finite - all of the fibers, including the generic fiber, are finite - is it true that $f$ is finite? If not, ...
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69 views

Prerequisite to start learning the Fulton's book about : Intersection theory. [duplicate]

Good evening everyone , Could you tell me please, what to have as a prerequisite to learn the following course here [link removed by a moderator, because at least two users expressed their concern ...
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1answer
42 views

Proof of proposition 5.3.1 of Ravi Vakil's notes on algebraic geometry

I am reading the proof of proposition 5.3.1 of Ravi Vakil's notes on algebraic geometry, and I have a problem with the last sentence : "If $g' = g''/f^n$ ($g''\in A$) then $\textrm{Spec}((A_f)_{g'}) = ...
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1answer
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Why are finite group schemes usually assumed flat?

I am learning about group schemes at the moment. When it comes to finite group schemes, every author I have read so far restricts himself to the case of schemes which are also flat over the base, ...
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on what morphism of schemes look like locally

I am working on the following exercise (Exercise 6.3.C. on Ravi Vakil's online notes), which has two parts: Given a morphism of schemes $\pi: X \rightarrow Y$, show that if Spec $A$ is an affine open ...
3
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1answer
67 views

Isomorphism of Proj schemes of graded rings, Hartshorne 2.14

This question is based on exercise $2.14$ of chapter $2$ of Hartshorne. Suppose $\varphi:S\rightarrow T$ is a graded homomorphism of graded (commutative, unital) rings such that $\varphi_d := ...
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1answer
31 views

dimension of a scheme and degree of an L-function

Disclaimer: I first asked this question on Mathoverflow but I was told it was off-topic for that site, so I ask it here. I try to understand correctly the notion of scheme, as Serre in the second ...
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0answers
42 views

Existence of Harder-Narasimhan filtration

I am trying to understand the proof of the existence of Harder-Narasimhan filtration from Huybrechts and Lehn. Let $X$ be a projective scheme with a fixed ample line bundle. Then the theorem says ...
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13 views

Scheme theoretic definition of field extensions

You can think about a number field $K$ as the spectrum of its ring of integers. Is there anything equivalent for a field extension?
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1answer
41 views

The intersection of affine subscheme and preimage of affine on separated scheme is affine

Let $f:X \to Y$ be a morphism of separated schemes and $V,U$ be affine open subschemes of $X$ and $Y$ respectively. Why is the intersection $U \cap f^{-1}(V)$ affine? It is easy to prove that ...
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1answer
43 views

Characters of group scheme represented by Cartier dual

For a commutative group scheme $\pi \colon G \to S$ finite locally free over a base scheme $S$ we let $A := \pi_* \mathcal{O}_G$ and $A^* = \mathcal{Hom}_\mathcal{O_S}(A, \mathcal{O}_S)$. Then $A^*$ ...
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1answer
43 views

scheme of finite type - geometric interpretation

I'm using the definitions given in Qing Liu's Book: A morphism $f : X \to Y$ is said to be of finite type if $f$ is quasi-compact, and if for every affine open subset $V$ of $Y$, and for every ...
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1answer
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How to see $\Gamma(\mathscr{O}_S,\operatorname{Proj} S)$ as a ring?

Let $S$ be a finitely generated graded $A$-algebra. For each homogeneous $f\in S_+$, we have a scheme structure $D(f)\cong \operatorname{Spec} S_{(f)}$ where $S_{(f)}$ denotes the zeroth piece of the ...
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When the inverse image of a hypersurface is equicodimensional?

Let $K$ be a field of characteristic zero and let $f\in K[x_1,\ldots,x_m]$. Let $h:Y\rightarrow \mathbb{A}_K^m$ be a resolution of singularities for $f$, that is, $Y$ is a closed subscheme of some ...
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18 views

Given a surjective morphism of complete alg. schemes and an integral subvariant on the target find corresponding on the source

The following is an exercise in scheme theory. I am really rusty and I'd like to get some help and full solution. Let $f: X \rightarrow Y$ morphism of complete alg. schemes. Assume $f$ surjective. ...
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1answer
80 views

Rational maps from noetherian $k$-scheme to projective $k$-scheme extend over regular codimension 1 sets

I am trying to solve and am currently stuck on Vakil's problem 16.5.B which is asked as follows: Suppose $X$ is a Noetherian $k$-scheme and $Z$ is a irreducible codimesion $1$ subvariety whose ...
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1answer
54 views

Geometric connectedness and geometric fiber

Let $X$ be an algebraic variety over a field $k$ (i.e. a scheme of finite type over $\text{Spec}(k)$. According to Remark 3.2.11 of Qing Liu's book Algebraic geometry and arithmetic curves, we have ...
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1answer
41 views

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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1answer
43 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 ...