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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1answer
53 views

Rational map is not defined on a subset of codimension $\geq 2$

I have the following Lemma (http://www.jmilne.org/math/xnotes/AVs.pdf, Lemma 3.2): A rational map $f:V\dashrightarrow W$ from a normal variety to a complete variety is defined on an open subset $U$ ...
0
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0answers
64 views

Reduced and not reduced points of a scheme.

Here is a small paragraph of a course of "Anwar Alameddin" that i found on the net accidently, and that i try to understand clearly, about intersection theory : Let $ k $ be an algebraically ...
1
vote
1answer
63 views

There exists a divisor linearly equivalent to $P$ not containing $P$

Let $X$ be a non-singular projective curve over $\mathbb C$ and let $P\in X$ be a closed point. Which is, in your opinion, the easiest way to prove that there exists a divisor $D$ of $X$ satisfying ...
1
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1answer
66 views

Generic point and pull back

Let's work over $\mathbb C$, Consider a finite morphism between two integral curves $f: X\rightarrow Y$, let $\eta_Y$ (resp $\eta_X$) be the generic points Questions: What is $X_{\eta_Y}=f^{-1}(\...
7
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1answer
131 views

Intersection of curves on surfaces and the sheaf $\mathcal O_{C\cap C'}$

Let $S$ be a (complex) projective surface and let $C,C'\subset S$ two closed irreducible curves (namely two prime Weil divisors). It is well defined the scheme $C\cap C'$ (as fibered product of $C\...
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0answers
39 views

A special presheaf of rings over the spectra $ \mathrm{Spec} A $.

Let $ A $ be a commutative ring with unity, and $ \mathrm{Spec} A $ the spectra of $ A $. We define over $ \mathrm{Spec} A $ the presheaf of rings as follows : If $ U $ is an open subset of $ \...
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0answers
34 views

An open subset of a scheme, schematically dense.

Let $U$ be an open subset of a scheme $X$, and let $Y$ be its complement. $U$ is called schematically dense, if for any other open set $V$ of $X$ the restriction map : $ \rho : \Gamma ( V , \mathcal{...
3
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0answers
78 views

Sheaves with surjective map from the ring of global sections to stalks

Let $M$ be a smooth manifold, $\mathscr O_M$ the sheaf of vector fields. Then, by an argument involving partition of unity, we can show that the map $$\mathscr O_M(M)\to\mathscr O_{M,x}$$ is ...
6
votes
1answer
87 views

Functoriality of Morphisms to Affine Scheme

It is known that $\text{Mor}(X,Y)$ is in bijective correspondence with $\text{Hom}(\mathcal{O}_Y(Y), \mathcal{O}_X(X))$, provided $Y$ is an affine scheme. I do not understand a small but crucial part ...
1
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1answer
98 views

What does the Grothendieck's period conjecture mean?

I would like to know how is the Grothendieck's period conjecture about algebraic cycles, defined explicitly ? and, what link has it with the Hodge conjecture for smooth complexe algebraic projective ...
1
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0answers
38 views

Projective general linear group over a ring

For $n$ a positive integer, I would like to define $PGL_n$ as a group scheme. One candidate is $X:=\mathrm{Proj}\big(\mathbb{Z}[x_{ij}:1\leq i,j\leq n]\big)\,\backslash \,\mathbb{V}(det)$, but I am ...
2
votes
1answer
33 views

subschemes and subobjects

In scheme theory, there are terms "open subscheme" and "closed subscheme", and in category theory, there is a term "subobject". I want to know relation between them. Are open subschemes and closed ...
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0answers
73 views

Dimension of integral schemes of locally finite type over a field

In Exercise 3.20 of Algebraic Geometry, Hartshorne makes several claims about the dimension of an integral scheme of finite type over a field. For instance, he claims that the dimension is equal to ...
1
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0answers
38 views

Isomorphism (?) of polynomial rings with different gradings and their Proj

It's a (relatively) well-known fact that, if $a_0,\ldots,a_n\in\mathbb{N}$ share a common factor $d\in\mathbb{N}$ then $$\operatorname{Proj}k_a[x_0,\ldots,x_n] \cong \operatorname{Proj}k_{a/d}[x_0,\...
0
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1answer
89 views

map between projective schemes induced by rational points

Given a map between $\mathbb P^n_{\mathbb C}$ and itself, given by $$ [x_0:\dots:x_n] \mapsto [p_0(x):\dots:p_n(x)]$$ where $p_i$ are homogeneous polynomials of the same degree, how do I find the ...
2
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0answers
24 views

Is this always quasi-projective?

Let $\phi:(X,x)\to (T,t)$ be a morphism of germs of algebraic schemes over $\mathbb{C}$. How can we always (without "quasi-projective") have an open embedding of the germ $(X,x)$ into a closed ...
1
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0answers
42 views

Is it true that $\mathrm{codim}(Z \cap Y, Y) \leq\mathrm{codim}(Z,X)$ for closed subsets $Z,Y$ of a scheme $X$?

This might be a standard thing but I'm not so sure. Say $X$ is an irreducible affine scheme, $Y$ is an irreducible closed subset of $X$, and $Z$ a closed subset of $X$. If $Z \cap Y \neq \...
1
vote
1answer
123 views

normalization of a curve with a node is not flat

Given the ring $$A = \frac{K[x,y]}{y^2-x^2(x+1)}$$ I know that its normalization is $K[t]$, where $$x\mapsto t^2-1\qquad y\mapsto t^3-t$$ I have to show that the normalization map is not flat. I know ...
1
vote
0answers
130 views

When does a principal divisor have degree 0?

We know that on a smooth curve over an algebraically closed field principal divisors have degree 0. This holds true also for the projective space over any field (in this case equality holds too). My ...
0
votes
1answer
23 views

cardinality of fiber in a finite morphism of schemes

Given $f:X\to Y$ a finite morphism of schemes, with $Y$ locally noetherian, let's take a point $q\in Y$, and an affine noetherian open set $$q\in U=Spec(B)\subseteq Y$$ Then $$f^{-1}(U)=V=Spec(A)\...
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0answers
63 views

Tangent space of an algebraic variety $ X $ in a point $ x \in X $.

Let $X$ be an algebraic variety, and $ \mathcal{O}_{X,x} $ its local ring at a point $x \in X$, and $ \mathfrak{m}_x $ its maximal ideal. Let set $ k_x = \mathcal{O}_{X,x} / \mathfrak{m}_x $ the ...
3
votes
0answers
101 views

When is “number of points in the fiber” semicontinuous?

Let $f:X \rightarrow Y$ be a finite morphism of schemes. For $y \in Y$ let $n(y)$ be the cardinality of the set-theoretic fiber $f^-1(y)$. This is finite since $f$ is finite. I think that the ...
0
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0answers
42 views

Global section on product of sheafifications

Say I have two presheaves $\mathcal F$ and $\mathcal G$ of $\mathcal O_X$-modules on a scheme $X$. Let $\mathcal F^+\!, \mathcal G^+\!$ be their respective sheafifications. We then have a commitative ...
0
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0answers
48 views

Simple question about closed immersions

I would like to know if I understand closed immersions correctly. Let $f:X \rightarrow Y$ be a morphism of schemes (or locally ringed spaces in general). The morphism $f^\sharp:\mathscr{O}_Y \...
2
votes
1answer
96 views

Every commutative ring is a quotient of a normal ring?

In the book Étale cohomology by Milne I found on p. 37 (in the context of constructing the henselization of a local ring) the following claim: "Every ring is a quotient of a normal ring". The same is ...
2
votes
0answers
85 views

About the relation between the categories $\text{Sch}$, $\text{LRS}$ and $\text{RS}$

I've been trying to find some useful categorical facts about the category of schemes, locally ringed spaces and ringed spaces (that I shall denote by $\text{Sch}$, $\text{LRS}$ and $\text{RS}$ ...
1
vote
1answer
108 views

Spectrum of the Ring of Continuous Functions on a Space

I was wondering when exactly we can recover the topological space, $X$, from its ring of continuous functions into $\mathbb C$ (or some sort of sufficient topological group). For any topological ...
0
votes
1answer
134 views

Are Monomorphisms injective?

In the categories of topological spaces, rings, groups and sets I know that a morphism is a monomorphism iff it's injective. Things are different for schemes. In fact I know that a scheme injective ...
1
vote
1answer
48 views

Lifting of partial section

Does someone know if the following is true, and if so could you provide a reference. Let $X$ be a smooth $S$-scheme, let $Z$ be a closed subscheme of $S$, and let $t : Z \to X$ be an $S$-morphism. ...
5
votes
1answer
101 views

Uniqueness of the structure sheaf

Given a ring $A$ and his Spectrum $X=Spec(A)$ seen as a topological space with the Zariski topology, it's possible to build a sheaf on $X$ satisfying the conditions $O_X(X)=A$ $O_X(X_f)=A_f$ where $...
1
vote
0answers
49 views

Elementary transformation of vector bundles - equivalent definition

The background is as follows: Let $S$ be a locally noetherian scheme, $E$ a vector bundle over $S$ of rank $N+1$. Let $X=\mathbb{P}(E)$ be the projective bundle and $\pi:X\longrightarrow S$ be the ...
0
votes
1answer
58 views

Irreducible component of a scheme over a DVR

Let $\mathcal M$ be a (reduced) quasi-projective scheme over a DVR (of mixed caracteristics), $R$. Suppose that the generic fiber $\mathcal M_{\eta_R}$ is (nonempty) smooth and irreducible of ...
0
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0answers
61 views

Are Prevarieties irreducible?

In Goertz-Wedhorn, a prevariety is defined to be a connected space with functions that locally is an affine variety (were an affine variety is a space with functions that is isomorphic to the space ...
10
votes
2answers
632 views

What *is* affine space?

In my recent reading of various books and notes on algebraic geometry and scheme theory, I have come across three definitions of affine $n$-space over a field $k$: $\mathbb{A}_k^n$ is $k^n$ 'without ...
2
votes
2answers
89 views

Is defining an “$S$-scheme” this way just a stylistic choice?

I have a potentially pretty silly question, involving the definition of an "$S$-scheme". For a fixed scheme $S$, the definition of the category of $S$-schemes is a category with: objects: scheme ...
3
votes
1answer
45 views

Zero-section as homomorphism of rings

Let $s : X \to E$ be the zero section of a vector bundle $E$ over a scheme $X$. Zariski-locally this corresponds to a homomorphism $Sym_A(M) \to A$ of $A$-algebras where $M$ is a finitely generated ...
4
votes
0answers
62 views

Birational map between reduced schemes.

I am reading Ravi Vakil's notes "Foundation of Algebraic Geometry" and in Proposition 6.5.5., it states the following: Suppose $X$ and $Y$ are reduced schemes. Then $X$ and $Y$ are birational if and ...
1
vote
0answers
81 views

Gluing schemes: Tips and tricks.

Like many other people I have talked to, I always find checking the cocycle condition quite hard and messy. I notice that most books avoid showing explicitly that the cocycle condition is satisfied, ...
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vote
0answers
32 views

Invertible Prime in a Noetherian Scheme

What means the following: Let $\ell$ be a prime, and $X$ a Noetherian Scheme, on wich the prime $\ell$ is invertible. The reference is the appendix of "Weil Conjectures, Perverse Sheaves and l’adic ...
3
votes
1answer
78 views

Translation from schemes to varieties

At the moment, I know very little algebraic geometry (sadly!) so I apologise for the silliness/stupidity of these questions. Set up: Let $k$ be a field (not necessarily algebraically closed). Take ...
1
vote
1answer
71 views

Ideal sheaf is aquasi-coherent sheaf of ideals.

Let $X$ be a scheme, For any closed subscheme $Y$ of $X$, the corresponding ideal $I_Y$ given by the kernel of the morphism $i^{\#}:\mathcal{O}_x\rightarrow i_{*}\mathcal{O}_y$ is quasi-coherent sheaf ...
0
votes
0answers
59 views

Kodaira dimensions

I am learning about schemes, specifically abstrac varieties over $\mathbb{C}$ and I want to read a good book or reference where explain the kodaira dimensions, somebody can help me?
1
vote
1answer
90 views

Two definitions of coherent sheaf

There are at least two ways to define a (quasi)coherent sheaf on the scheme $(X,\mathcal{O}_X)$, namely the one given in Hartshorne's "Algebraic geometry" (I guess you know it: it is given in terms of ...
1
vote
1answer
68 views

Confusion about affine schemes

Let $\varphi:A \rightarrow B$ be a ring morphism. Consider the corresponding map of schemes, $f: Spec\ B \rightarrow Spec\ A$. This map $f$ is given by $\varphi ^{-1},$ since prime ideals pull back ...
4
votes
2answers
207 views

Hartshorne generically finite morphisms (II, 3.7)

I have a question concerning one of the exercises of Hartshorne, Ch. II. Namely: Exercise 3.7 about gerneically finite morphisms. A morphism $f: X \rightarrow Y$ with Y irreducible and $\eta$ ...
4
votes
1answer
138 views

Closed subscheme of an affine scheme.

This is a portion of the exercise 2.3.11 (b) of Hartshorne. Suppose $Y$ is a closed subscheme of $X=\operatorname{Spec}A$. I would like to show that there are $f_{i}\in A$ so $D(f_{i})$ cover $X$ and ...
2
votes
0answers
121 views

Computing the cotangent complex: what's the ring?

As far as I understand, deformation theory of schemes may be calculated via the cotangent complex. I have read that in general the cotangent complex may be difficult to compute. However, I have a ...
4
votes
0answers
74 views

Determining a scheme $X$ is affine from $Qcoh(X)$

My question is a subquestion of this question. I do not want to use full reconstruction theorems. The settings is the following. Let $X$ be a scheme. Assume that the adjunction in TAG01BH of the ...
5
votes
1answer
54 views

Quick question: could someone clarify me the notion of “$k$-points” of a scheme?

Let $X$ be a scheme, and $k$ be a field. What does it mean by $X(k)$? The reason I am asking this is that there seems to be several definition(or convention) on it, and I can't see why they lead to ...
3
votes
1answer
47 views

$\overline{\mathbb{Q}}$-valued points of a $\mathbb{C}$-scheme

I've just read the definition of a $T$-valued point on an $S$-scheme from Ravi Vakil's notes (i.e. the morphisms $T \to S$), and something about the definition doesn't seem to make sense intuitively. ...