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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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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64 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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56 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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67 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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39 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
53 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
57 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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18 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
120 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
77 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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57 views

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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1answer
47 views

Is the push-forward 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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1answer
43 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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1answer
73 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
40 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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1answer
79 views

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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1answer
142 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
85 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
36 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
60 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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39 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
46 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
54 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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1answer
44 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
108 views

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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43 views

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 ...
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1answer
78 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
37 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
80 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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17 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
57 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
49 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
48 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
53 views

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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21 views

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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19 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
86 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
62 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
48 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
56 views

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
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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41 views

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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1answer
31 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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1answer
112 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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1answer
49 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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1answer
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 ...
2
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1answer
52 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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1answer
27 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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1answer
116 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 $X \to X$. ...