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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6
votes
1answer
76 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 ...
5
votes
1answer
138 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 ...
1
vote
2answers
83 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?
1
vote
1answer
30 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 $$ (*) ...
2
votes
1answer
56 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 $$\dim_x(X_y)=\dim_x X-\dim_y Y$$ ...
1
vote
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 ...
2
votes
2answers
45 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) ...
2
votes
2answers
50 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, ...
0
votes
0answers
70 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 ...
1
vote
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'}) = ...
7
votes
1answer
95 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, ...
3
votes
0answers
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 ...
3
votes
1answer
73 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 := ...
0
votes
1answer
35 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 ...
0
votes
0answers
54 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 ...
0
votes
0answers
15 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?
1
vote
1answer
48 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 ...
4
votes
1answer
47 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 $A^*$ ...
0
votes
1answer
45 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 ...
0
votes
1answer
49 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 ...
0
votes
0answers
19 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 ...
0
votes
0answers
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. ...
1
vote
1answer
83 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 ...
1
vote
1answer
58 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 ...
0
votes
1answer
45 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 $M\otimes_{A}A_{\mathfrak{p}}\cong M_{\mathfrak{p}}$. ...
2
votes
3answers
53 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: ...
1
vote
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!
0
votes
0answers
39 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 ...
0
votes
1answer
30 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? ...
0
votes
1answer
109 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 $M$ ...
1
vote
1answer
47 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 ...
2
votes
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
votes
1answer
51 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 ...
1
vote
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 ...
4
votes
1answer
99 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 $X \to X$. ...
0
votes
1answer
43 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 ...
0
votes
0answers
32 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 ...
0
votes
0answers
34 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 ...
2
votes
2answers
74 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 ...
1
vote
0answers
45 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 ...
4
votes
1answer
101 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 ...
1
vote
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 ...
0
votes
0answers
44 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 ...
4
votes
0answers
68 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 ...
5
votes
1answer
138 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 ...
3
votes
1answer
32 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 $$X:=X_L\times_{\text{Spec L}}\text{Spec} K$$ ...
1
vote
0answers
32 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 $X$? ...
7
votes
1answer
202 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, ...
2
votes
0answers
40 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 ...
0
votes
2answers
41 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 ...