# 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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### Is the scheme-theoretic image stable under taking products?

Let $f:X\to Y$ be a morphism of schemes and let $Z\subset Y$ be its scheme-theoretic image. If $T$ is any other scheme, consider the induced morphism $g=f\times 1_T:X\times T\to Y\times T$. ...
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### Stalks of the ideal sheaf of an irreducible subscheme

Suppose that $X$ is a noetherian scheme such that $Z\subseteq X$ is a closed subscheme. Clearly $Z$ define an ideal sheaf $\mathscr I\subset\mathscr O_X$. Now let $z\in Z$ be a point such that it is ...
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### Vakil's definition of smoothness — what happens at non-closed points?

The following is definition 12.2.6 in Vakil's notes. A $k$-scheme is $k$-smooth of dimension $d$, or smooth of dimension $d$ over $k$, if it is pure dimension $d$, and there exists a cover by ...
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### Geometric intuition for the Stein factorization theorem?

What is the intuition behind the Stein Factorization Theorem? I understand that it was originally a theorem in several complex variables, so I was wondering if there's some geometric explanation that ...
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### Geometric intuition for normalization as intersection of valuation rings?

Why should the normalization of a ring correspond to the intersection of valuation rings containing it? I am looking for a geometric explanation, if possible. I understand that normalization at a ...
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### Completion along locally closed subscheme

If $X$ is any scheme over $k$ then we know that the image of the diagonal $\Delta(X)$ is locally closed in $X \times_k X$, so that there is an open set $W$ of $X \times_k X$ with $\Delta(X)$ closed in ...
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### Action on algebraic variety and adjoint bundles

Let $X$ be a complex algebraic variety and let $G$ be a complex algebraic group; I mean that $X$ is a reduced, separated scheme of finite type on $Spec\mathbb{C}$, and the underlying set of $G$ is a ...
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### Classifying non-reduced points in noetherian schemes

Let $X$ be a noetherian scheme. So in particular $X$ is a finite, locally finite, union of its irreducible components $X = \bigcup^n_i K_i$. Non-reduced points in $X$ fall into 2 categories: Fat ...
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### If $X$ is a proper scheme over $k$, is $X/G$ separated?

Let $X$ a proper scheme over a field $k$ and let $G$ a finite group of its automorphism (as $k$-scheme). Let suppose that the quotient $X/G$ exists, is it is separated? How to prove it? If the ...
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### Finite type and finite fibers implies quasi-finite

I am trying to understand different finiteness conditions, in particular I am looking at the following exercise from Algebraic Geometry and Arithmetic Curves by Qing Liu: Let $f:X\rightarrow Y$ be a ...
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### Motivation of completions of schemes?

Sorry if this is a stupid question, but what are the uses of a completion of a scheme along a closed subscheme? Are there any nice universal properties it satisfies, or do certain morphisms factor ...
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### How much algebraic geometry is there in complex geometry (for example, Demailly)?

I wonder how much of modern algebraic geometry (schemes, etc) is there in complex (algebro-analytic) geometry. What I mean is complex algebraic (analytic) geometry in the sense of Demailly, Lasarsfeld ...
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### Schemes whose ring of global sections separate points

Let $X$ be a scheme over an algebraically closed field. We say that $\Gamma(X,\mathcal{O}_X)$ separates points iff for every $x,y \in X$ there's an $f \in \Gamma(X,\mathcal{O}_X)$ with $f(x)\ne f(y)$. ...
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### Example of a module which is free at an isolated point

I'm looking for the most simple example of a quasicoherent sheaf $\mathcal{F}$ over a scheme $X$ (preferably affine for simplicity) which has a free stalk $\mathcal{F}_x$ at a point $x \in X$ and yet ...
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### $\Gamma(X,-)$ for Quasi-Coherent/Coherent Sheaves Maps to R-modules/Finitely Generated R modules

I'm trying to show that the functor $\Gamma(X,-)$ from the category of quasi-coherent sheaves maps a quasi-coherent sheaf to an R-Module, and also that for coherent sheaves the same functor takes the ...
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I am trying to fill in the details of the proof of Proposition 2, II. Preschemes, §6, in Mumford's Red Book (page 114, 2nd edition). For any two preschemes $X_1,X_2$, $$\operatorname{hom}(X_1,... 0answers 28 views ### A characterization of normal schemes (clarification of a statement of proposition) The following is taken from 4.1 in Liu's book. Definition: A scheme X is normal at x \in X if O_{X,x} is normal. X is normal if it is irreducible and normal at every point. ... 1answer 304 views ### When can stalks be glued to recover a sheaf? Let \mathcal{F} be a sheaf over some topological space. The stalks are \mathcal{F}_x= \underset{{x\in U}}{ \underrightarrow{\lim}} \mathcal{F}(U). Is there a special name for a sheaf that ... 0answers 32 views ### Construction of relative projective space via glueing I would like to gain further practice on glueing schemes by constructing projective space over a ring. I am considering the following: I wonder how we get that \mathcal{O}_{X_i} (X_{ij}) = \... 0answers 24 views ### Pulling back the sheaf of ideals under diagonal embedding Let X be a scheme (over a field k) and \Delta:X\to X\times X be a diagonal embedding. Let I be a sheaf of ideals of the diagonal \Delta(X) in X\times X. For positive integer n how could ... 0answers 13 views ### Classification of group schemes of order 2: Why is sx=x\otimes 1+1\otimes x-cx\otimes x? In their paper Group Schemes of Prime Order, Tate and Oort state Suppose$$G=\operatorname{Spec}(A)\text,\quad S=\operatorname{Spec}(R) \text, and suppose the augmentation ideal $I=\... 1answer 97 views ### An algebraic variety as a scheme If we take the definition of a variety as a reduced integral scheme of finite type over an algebraically closed field$k$, then a variety is in particular a scheme over$k$, so is a scheme$X$with a ... 0answers 36 views ### Milne's definition of$G$-torsors I am trying to understand the proof of proposition 4.1 of Milne's book Étale Cohomology (p.120) and I am getting really confused with some points of the reverse implication: if I understand correctly ... 1answer 88 views ### Inverse function theorem for etale morphisms Looking around stackexchange, it seems there are many related questions, but I'm a beginner and I can't find a proof on the internet (without going through the more general results in stacks project). ... 0answers 51 views ### Pushforward of a quasicoherent sheaf on a noetherian scheme is quasicoherent I'm reading through and trying to understand the proof of Proposition 5.8 in Chapter II of Algebraic Geometry by Hartshorne that the pushforward of quasicoherent sheaf$\mathcal{F}$by a morphism$\...
Let $X$ be a scheme over a field $k$. For an extension field $K$ of $k$, we can change the base to obtain the scheme $X_K$. Supposedly it is possible that $X$ is irreducible while $X_K$ is reducible, ...