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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Quotient of an affine scheme under the action of a finite group

I am trying to solve Exercise 2.3.21 from Liu’s book. I have a finite group $G$ acting on an affine scheme $\operatorname{Spec}A$, and I want to show that the quotient scheme $X/G$ exists and is isomorphic to $\operatorname{Spec}A^G$. I have defined…
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Global sections of projective rational curve are constants: easy proofs wanted

Suppose $C$ is a projective rational irreducible dimension $1$ Noetherian scheme over field $k$ and I would like to collect some of most usual common preferably as elementary as possible ways to prove that the $k$-space $\Gamma(C, O_C)$ of global…
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Is the spectrum of a stalk a subscheme?

Given a scheme $X$ and a point $p \in X$, I know that $\text{Spec}(\mathcal{O}_{X,p})$ consists of the point $p$ together with all generic points of irreducible closed subsets containing $p$. Can we thus view the spectrum of the stalk,…
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Affine-local Computation of Scheme-Theoretic Image for Reduced Schemes

The following is part of an argument in Vakil's Rising Sea, p. 238. Let $\pi : X \to \operatorname{Spec}{B}$ be a morphism of schemes with $X$ reduced and $g \in B$. Consider the morphism $$\pi^{\#}_{B_g} : B_g \to \Gamma(\operatorname{Spec}{B_g},…
Qi Zhu
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A criterion for irreducible topological subspace

Let $X$ be a non-empty spectral space and $P$ be a closed subset of $X$. If $U_1,U_2$ are two arbitary quasi-compact open subsets satisfying $P\cap U_1\neq\emptyset$ and $P\cap U_2\neq\emptyset$, then $P\cap U_1\cap U_2\neq\emptyset$. Can we deduce…
user832207
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What is the relation between formal group and formal scheme

What is the relation between formal group and formal scheme? Formal group is power series, which behaves like ' a group law without any group elements'. Is former one is special case of latter one?
dandelion
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Does completion preserve openness?

Let $\hat{A}$ be the completion of $A$ along the ideal $I$. Let $f$ be some non-zero divisor in $A$ that is not in $I$. $\text{Spec}(A_f)$ is an open subscheme of $\text{Spec}(A)$. Does that imply that $\text{Spec}(\widehat{A_f})$ is an open…
user127776
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Example of a locally Noetherian space $X$ and a point $x$ such that $\dim_xX=\infty$

There is a well-known example that a Noetherian ring $A$ with infinite Krull dimension due to Nagata, which gives a Noetherian space $\operatorname{Spec}A$ with infinite Krull dimension. Given the definition $\dim_xX=\inf_{x\in U}\dim U$ where $U$…
user832207
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$\dim _ x(X) = \dim _ y(Y) + d$ for a morphism of relative dimension $d$

Here is a lemma in tag01FE. Let $f: X→Y$ be a morphism of locally Noetherian schemes which is flat and locally of finite type such that all fibers are equidimensional of dimension d. Then for every point x in X with image y in Y we…
user832207
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Extensions of scalars and tangent map of morphism of schemes

Suppose I have a morphism of rings $\varphi: A \to B$, $M$ a $B$-module, $N$ an $A$-module and $f: M \to N$ a homomorphism of $A$-modules (here, $M$ is considered an $A$-module by restriction of scalars). How do I properly "extend" $f$ to a morphism…
user480840
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Are morphisms between projective curves finite?

For convenient we discuss about projective curves. Hartshorne II 6.8 proved a non-trivial morphism from a nonsingular curve to another curve is finite. So I'm wondering if we can prove the consequence for any curves and non-trivial morphism. I tried…
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Few questions regarding formal vector bundles on formal completions.

This question is regarding formal vector bundles on formal schemes. The formal schemes that I'm interested are formal completion of a scheme along a closed sub-scheme. The category of formal coherent sheaves form an abelian category. 1)Given a short…
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does there exist a finite morphism that is a retraction on a dense open subset?

Does there exist a finite morphism $f: X \to X$ from an integral scheme of finite type over a field to itself such that $f$ induces an identity map of topological spaces on a dense open $U \subset X$?
Dima Sustretov
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Points in the fiber product.

Some definitions use here: Definition. Let $\mathcal C$ be a category and $S\in \mathcal C$, then we can define the $S$-category to be morphisms $X\to S$ as objects, and morphisms between objects $X\to S$ and $Y\to S$ as $\mathcal C$-morphisms…
Hyacinth
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von Neumann Stability help

Using the forward time centered space scheme, I transformed the equation: $u_t-2u_{xx}-u_{yy}=0$ to $u_{i,j}^{n+1}=(1-2s_{x}-2s_{y})u_{i,j}^{n}+s_{x}(u_{i+1,j}^{n}+u_{i-1,j}^{n})+s_{y}(u_{i,j+1}^{n}+u_{i,j-1}^{n})$ letting $s_x=2 \Delta t/(\Delta…
user75269
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