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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56 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 ...
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1answer
57 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 ...
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1answer
57 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 ...
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1answer
115 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 ...
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36 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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26 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 , ...
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71 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 ...
5
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1answer
78 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 ...
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1answer
76 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 ...
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30 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
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1answer
28 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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62 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 ...
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28 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 ...
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1answer
86 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
23 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 ...
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41 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 ...
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1answer
82 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 ...
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0answers
79 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 ...
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1answer
19 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 ...
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57 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 ...
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84 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 ...
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37 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 ...
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42 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
84 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
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0answers
78 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 $\text{RS}$ ...
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1answer
86 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
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1answer
96 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
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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
95 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 ...
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0answers
45 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 ...
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1answer
54 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 ...
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58 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
551 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 ...
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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 ...
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40 views

Discriminant ideal and short exact sequence of finite group schemes

Let $0 \to G' \to G \to G'' \to 0$ be a short exact sequence of finite flat commutative group schemes over a Dedekind domain $\mathcal O$ with field of fractions of characteristic zero. Let ...
4
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50 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 ...
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0answers
76 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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0answers
30 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
76 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 ...
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0answers
108 views

Intersection of affine open subschemes

If X is separated scheme, then for any affine opens U, V, the intersection is affine. If X is quasi-separated, then the intersection can be covered by finitely many affine opens. Is there some ...
1
vote
1answer
63 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 ...
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0answers
56 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?
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1answer
80 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 ...
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1answer
61 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 ...
3
votes
2answers
145 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 $\eta$ ...
2
votes
1answer
94 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 ...
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0answers
90 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
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0answers
62 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
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1answer
53 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 ...