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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Irreducible components of the closure of a locally closed subset

I would like to ask for a reference of the following fact: the irreducible components of the closure of a locally closed subset are the closures of the irreducible components of the subset. Thank you very much.
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Understanding geometric visualisation of a double point (fat point)

This question comes from Example 12.21 (a) of Gathmann's 2019 notes, here. In the example, take $R = K[x]/(x^2)$ for some field $K$, so that $\operatorname{Spec} R$ is a single point $\mathfrak{p} = (x)$. In the third paragraph of the example, he…
mi.f.zh
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Short Exact Sequence of Tangent Space of smooth morphism

I'm reading the proof of Proposition 4.3.39 in Qing Liu's book. Proposition 4.3.39 Let $f:X\rightarrow Y$ be a morphism of finite type to a locally Noetherian scheme. Let us suppose $f$ is smooth at $x\in X$. Let $y = f(x)$. Then we have an exact…
Hydrogen
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Isomorphism of schemes restricts down to local isomorphism

I just want to make sure I am understanding the definitions I'm learning correctly. Suppose we have an isomorphism of schemes $\pi: X \to Y$, and suppose we have some affine cover of $X$, $\{U_i\}$. Is it true that $\pi|_{U_i}$ is an isomorphism for…
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How to describe the stalk on fiber $X_y$?

Let $f:X\rightarrow Y$ be a morphism of schemes. (May assume more if necessary) Let $x\in X$ and $y\in Y$ such that $f(x) = y$ and $X_y$ be fiber over $y$. My question: Is there any relation in general between the stalk $\mathcal O_{X_y,x}$ and…
Hydrogen
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A Theorem of Dimension of Fiber in Qing Liu (Theorem 4.3.12)

I'm reading Qing Liu. I feel confused about the proof of theorem 4.3.12 and corollary 3.14. Theorem 4.3.12. Let $f$ be a morphism of locally Noetherian schemes. Let $x\in X$ and $y= f(x)$. Then $\dim (\mathcal{O}_{X_y,x})\geq \dim…
Hydrogen
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Estimate dimension of fibers for a morphism of schemes under 'very' weak assumpions

Let $f: X \to Y$ be an arbitrary dominant morphism of locally noetherian schemes over any field $k$. Assume that $X,Y$ are also irreducible with dimensions $\operatorname{X}=n $ and $ \operatorname{Y}=m$. We don't make any additional assumptions…
Isak the XI
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Algebraic fundamental group of $X = Spec\mathbb{Z}\left[\frac{1}{2}\right]$

I'm quite new to the notion of the fundamental group of schemes and I'm reading Lenstra's notes on Galois theory for schemes. One of the exercise asks to prove that the fundamental group $\pi(Spec\mathbb{Z}\left[\frac{1}{2}\right])$ is topologically…
Avenavolo
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If $(X, \mathcal O_X)$ is a scheme, what is $\mathcal O_X(U)$ where $U$ is any open subset (not necessarily affine open)?

If $(X, \mathcal O_X)$ is a scheme, what is $\mathcal O_X(U)$ where $U$ is any open subset (not necessarily affine open)? Below is what I currently know. Let $A$ be a ring. Hartshorne defines the structure sheaf on $\operatorname{Spec}(A)$ as: For…
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Locally noetherian schemes

I'm reading Algebraic Geometry written by R. Hartshorne. There is a proposition in section 3, I have few problems with a part of its proof A scheme $X$ is locally noetherian iff for every open subset $U=\operatorname{Spec} A$, $A$ is…
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Suffices to check a morphism of schemes is unramified on closed points.

I am starting to read Milne's Etale Cohomology notes and in it they mention: A morphism $\varphi :Y \rightarrow X$ is unramified if it is of finite type and if the maps $\mathcal{O}_{X,f(y)} \rightarrow \mathcal{O}_{Y,y}$ are unramified for all…
H_Hassan
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Countable algebraically closed field inside an incountable algebraically closed of characteristic zero

I would like to know if the following claim is true. I found this claim in a paper without proof :(. Let $k$ an uncountable algebraically closed field of characteristic $0$. Let $S=\operatorname{Spec}(k[X_1,...,…
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Does pushout of schemes along formal neighborhoods exist in the category of schemes?

I have a question about gluing specific types of schemes which doesn't fit into any well-known gluing situation. Assume $C$ is an algebraic curve and $p$ a point on it. The formal completion of $C$ at that point is $k[[t]]$. Now imagine…
user127776
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Sheaves $\mathcal{F}$ such that $\mathcal{F} \otimes\mathcal{G} \cong \mathcal{O}_X^{\oplus n}$: what are they called?

If $X$ is a scheme then the Picard group of $X$ is the collection of invertible sheaves, so the collection of those sheaves $\mathcal{F}$ such that there's an $\mathcal{F}^{-1}$ satisfying $\mathcal{F} \otimes \mathcal{F}^{-1} \cong \mathcal{O}_X$.…
Jim
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Closed subschemes and quasi-coherent sheaves of ideals

Let me quote two results from Qing Liu's Algebraic Geometry and Arithmetic Curves: Lemma 2.2.23: Let $X$ be a ringed space, $\mathcal{J}$ be a sheaf of ideals on $X$, $V(\mathcal{J}) = \{x\in X\;:\;\mathcal{J}_x \neq \mathcal{O}_{X,x}\}$, and…
Nico
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