# Tagged Questions

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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### 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 $U$ ...
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### 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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### 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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### 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 ...
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### 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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### 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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### 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 ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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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### 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 ...
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### 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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### 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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### 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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### 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 ...
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### 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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### 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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### 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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### 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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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 ...
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### 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 $\eta$ ...
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### 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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### 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 ...
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### 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 ...
### 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 ...
### $\overline{\mathbb{Q}}$-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. ...