Questions tagged [schemes]

The concept of scheme mimics the concept of manifold obtained by gluing pieces isomorphic to open balls, but with different "basic" gluing pieces. Properly, a scheme is a locally ringed space locally isomorphic (in the category of locally ringed spaces) to an affine scheme, the spectrum of a commutative ring with unit.

An affine scheme $X$ is a locally ringed space that is isomorphic to $\mathrm{Spec}R$, which is the spectrum of a commutative ring $R$. That is, for our commutative ring $R$, the closed subsets of $X$ correspond to the ideals of $R$, with the points of $X$ corresponding to prime ideals. Then $X$ being a locally ringed space means that it's equipped with a structure sheaf $\mathcal{O}_X$ that assigns to each open set $U$ the ring of regular functions on $U$.

A scheme then, is a locally ringed space that admits a open covering $\{U_i\}$ such that each $U_i$ is an affine scheme.

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For a morphism of affine schemes, the inverse of an open affine subscheme is affine

This seems ridiculously simple, but it's eluding me. Suppose $f:X\rightarrow Y$ is a morphism of affine schemes. Let $V$ be an open affine subscheme of $Y$. Why is $f^{-1}(V)$ affine? I noted that $V$ is quasi-compact and wrote it as a finite union…
Potato
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Finite morphisms of schemes are closed

I want to prove that finite morphisms of schemes are closed, but I cannot prove the affine case, namely: Given a finite morphism of rings $\varphi :B \to A$ prove that the induced morphism of schemes $f:X \to Y$ is closed. For this I'm trying…
Radu Titiu
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Fibres in algebraic geometry: multiplicity

Currently I am studying varieties over $\mathbb{C}$ and I know some scheme theory. My professor mentioned the other day that given a morphism of varieties over an alg. closed field $k$: $f: X \rightarrow Y$, one should count fibres "with…
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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 $\eta$ generic point of Y, is called generically finite, if…
Chris
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On a *ringed space*, show that the non vanishing set of $f$ is open, and that it is invertible there

This is an exercise of Ravi Vakil that I solved by a very trivial argument without using the hint. For this reason I'm worried that I might have missed something. If $f$ is a function on a locally ringed space $X$, show that the subset of $X$ where…
Rodrigo
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Exercise 1.24 from Joe Harris' Algebraic Geometry: First Course

I have a question about solvability of Exercise 1.24 (p. 14) from Joe Harris' Algebraic Geometry: A First Course and correctness of following 'synthetic' construction which according to the book (or alternatively due to page 79 in this script)…
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morphism from a local ring of a scheme to the scheme

Let $X$ be a scheme, and $x \in X.$ Let $U=\text{Spec}(A)$ be an open affine subset containing $x,$ then we have the natural morphism $\mathcal{O}_X(U) \to \mathcal{O}_{X,x}$ inducing a morphism $ \text{Spec} \;\mathcal{O}_{X,x} \to U$ and by…
Ehsan M. Kermani
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Basic understanding of Spec$(\mathbb Z)$

So, I'm looking into schemes, and found that I have no intuition in the field, so I decided to look into some simple (as in affine and well-known) examples. As I like to dwell on the basics for a while, and texts on graduate level tend to move too…
Arthur
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What's With The Diagonal Morphism?

Given a morphism $X \to Y$ of schemes, we can construct a diagonal morphism $\delta: X \to X \times_Y X$ via the universal property of the fiber product applied to the identity map $X \to X$. Recently, I've been puzzled (shocked even) by how many…
Alexander
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divisor class group of a product of schemes

The first part of this question is quite general: let $X$ and $Y$ be noetherian integral separated schemes which are regular in codimension one. Is there any relationship between the divisor class group $\text{Cl}(X \times Y)$ and the groups…
Justin Campbell
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on the adjointness of the global section functor and the Spec functor

In fact, it is an exercise on Hartshorne, Ex 2.4 to the second chapter of it (p.79): Let $A$ be a ring and $(X,\mathcal{O}_X)$ be a scheme. Given a morphism $f:X\longrightarrow \operatorname{Spec} A$, we have an associated map on sheaves $f^\sharp…
user14242
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How to tell whether a scheme is reduced from its functor of points?

Suppose I have a scheme $X$ and I want to know if $X$ is reduced, but all I have access to is the functor $$ R\mapsto X(R)=Mor(\operatorname{Spec}(R),X) $$ from commutative rings to sets (rather than, say, an explicit covering by spectra of some…
KotelKanim
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Reality check: $\mathcal{O}_{\mathbb{P}_\mathbb{R}^1}(1)$ and $\mathcal{O}_{\mathbb{P}_\mathbb{R}^1}(-1)$ are both Möbius strips?

This question follows up this previous question, which has an accepted answer that I am having trouble believing. (Edit: the answer has now been fixed; thanks Georges!) I have been working on giving myself as concrete an understanding as possible of…
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Restricting a Weil divisor to a local scheme

Apologies if the title is garbage. I couldn't think up anything more pertinent, as it's really just a notational question. In Hartshorne, p.141, Proposition 6.11, it is shown that Weil divisors are Cartier (under some assumptions). My question is…
Tom H
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Why do we need noetherianness (or something like it) for Serre's criterion for affineness?

Serre's criterion for affineness (Hartshorne III.3.7) states that: Let $X$ be a noetherian scheme. Suppose $H^1(X, \mathcal{F})= 0$ for every quasi-coherent sheaf on $X$. Then $X$ is affine. There is a more general statement (EGA II.5.2.1) that…
Akhil Mathew
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