# 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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### Vakil's FOAG, Exercise 9.2.K: Transcendental Complex Numbers

How does one realize a transcendental complex number as a maximal ideal of $\mathbb{Q}(t) \otimes_{\mathbb{Q}} \mathbb{C}$? This is the essence of Exercise 9.2.K in Vakil's FOAG. Here is what I've ...
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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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### Functions on Finitely-Generated, Nilpotent Free, k-Algebras Determined by Values on Closed Points

I am working (slowly and with much labor) through Vakil's Algebraic Geometry and came upon this problem. Suppose $k$ is an algebraically closed field, and $A = k[x_1,... ,x_n]/I$ is a finitely ...
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### Spec$(R)$ a scheme of finite type over $\mathbb{C} \implies R$ is a finitely generated algebra over $\mathbb{C}$.

Suppose $(\text{Spec}(R), \tilde{R})$ is a scheme locally of finite type over $\mathbb{C}$. We want to show that $R$ is a finitely generated $\mathbb{C}$-algebra. Since $(\text{Spec}(R), \tilde{R})$ ...
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### Multiplicity of Cartier divisor on locally noetherian scheme is only non-zero at generic point

I'm following chapter 7 in Qing Liu's book 'Algebraic Geometry and Arithmetic Curves' about 'Divisors and applications to curves'. My question concerns Definition 1.27: Let $A$ be a Noetherian ...
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### Comparing covers

Considering the Zariski topology, let $$V = \bigcup_{i \in I} U_i$$ be a maximal open cover of $V$ by basic open sets. Similarly, let $$V' = \bigcup_{j \in J} W_j$$ by the maximal open cover of $V'$ ...
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### Families as fibres of a morphism

In both Algebraic Geometry by Hartshorne and Geometry of Schemes by Eisenbud and Harris, the authors describe the notion of a family of schemes as being the fibres of a morphism $f:X\to Y$. Or as ...
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### Fibres of a morphism

Let $f:X\rightarrow S$ a proper morphism, and $s\in S$ a point. If $S$ is locally Noetherian, then what are the properties of the fibre scheme $X_s$ over the Spec of the residue field at $s$? Is this ...
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### Existence of scheme quotient

I have a morphism of schemes $X\to S$ which is very nice: flat, proper, finitely presented. I also have a finite group $G$ acting (faithfully, but not necessarily freely) on $X/S$. 1) I am quite ...
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### The non-existense of the fine moduli scheme of vector bundles. Why?

The reference I am using is this enter link description here. The question is about the moduli space of vector bundles. I am trying to understand why the fine moduli scheme does not exist. Let $C$ a ...
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### What does it mean for a scheme to be proper?

What exactly does it mean for a scheme to be proper? I can't seem to find an actual definition of this anyway despite the term being frequently used.
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### Strict transform of blow up

Let $X$ be a smooth projective variety over $\mathbb{C}$. Consider the blow up of $X$ about a closed subvariety $Z$. Let $X'=Bl_Z(X)$. Let $Y$ be a smooth irreducible divisor of $X$ properly ...
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### Diagonal morphism of regular variety is a regular embedding

Let $X$ be a regular $k-$variety (i.e. all of its local rings are regular) of pure dimension $d$. Then I would like to show that the diagonal morphism $X\rightarrow X\times_k X$ is a regular embedding ...
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### Cohomology Class of a Subvariety

I'm working on question 7.4 of Chapter III.7 in Hartshorne's Algebraic Geometry. The question is about the cohomology class of a subvariety. The setup is as follows: $X$ is an $n$-dimensional non-...
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### Cartier divisor and map to associated line bundle

Given a Cartier divisor $D$ on an integral, separated scheme $X$ of finite type over an algebrically closed field $k$. Does such a divisor always induce a map $\mathcal{O}_X \to \mathcal O_X(D)$? I ...
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### Push forward of an exact sequence of sheaves under blow up

Let $X$ be a smooth projective variety of dimension $n\geq 3$. Let $Z$ be a smooth subvariety of $X$ of codimension at least 3. Let $Y$ be the blow up of $X$ along $Z$ and let $f:Y\longrightarrow X$ ...
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### “This property is local on” : properties of morphisms of $S$-schemes

I am learning schemes theory at school and I have for now only lectures notes that I am taking during the course. The professor is quite often using following expressions, without having defined them :...
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### Synthetic differential geometry and formally étale morphisms?

Upon looking throug Kostcki's synthetic differential geometry notes, I stumbled upon the following definition. (Here $R$ is the geometric line, $W$ is a Weil algebra, and $\operatorname{Spec}_RW$ is ...
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### Problem in understanding the proof of lemma $7.2.5$ in Liu's book

Let's analyze the proof of the following lemma: Lemma: Let $X$ be an integral, Noetherian scheme and let $f\in K(X)^\ast$, then for all but finitely many points $x\in X$ of codimension $1$ we have ...
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### Rational sections of invertible sheaves and hermitian inner products

Notations: Let $X$ be a $\mathbb C$-scheme of finite type, projective, integral and of dimension $1$ (i.e. an algebraic curve) and with function field $K(X)$. The set of closed points is $X(\mathbb C)$...
I'm trying to make a scheme out of an affine variety $V\subset k^n$, when the base field $k$ is not algebraically closed. I wonder if $V$ is the spectrum of its ring of regular functions \$\mathcal{O}(...