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.

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On the openness of the map $\mathbb A^{n+1}_k\smallsetminus 0\rightarrow \mathbb P^n_k$.

In this question, it was asked whether the map $\pi:\mathbb A^{n+1}_k\smallsetminus 0\rightarrow \mathbb P^n_k$ is open, and I tried to draft to an answer but ran into a snag. First, we reduce to the ...
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Show two definitions of open subfunctors are equivalent

I have come across two definitions of "open subfunctors", which I would like to show are equivalent. This amounts to solving the exercise below (marked problem). In Eisenbud & Harris' ...
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Closed subscheme of an affine scheme.

This is a portion of the exercise 2.3.11 (b) of Hartshorne. Suppose $Y$ is a closed subscheme of $X=\operatorname{Spec}A$. I would like to show that there are $f_{i}\in A$ so $D(f_{i})$ cover $X$ and ...
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rational section non vanishing everywhere at any irreduicble component implies regular?

Let $X$ be a locally Notherian and reduced scheme, $\mathscr{L}$ be an invertible sheaf on it and $s$ be a rational section on $\mathscr{L}$. Suppose $s$ is not vanishing everywhere at any irreducible ...
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Görtz-Wedhorn corollary 5.33 : $\operatorname{codim}_X V(f_1, \dotsc, f_r) \leq r$ if $X$ is a $k$-scheme of finite type

I don't understand how to conclude the proof of the corollary 5.33 of Görtz-Wedhorn's "AG1 : Schemes". Here is the statement with the previous theorem ($k$ is any field) : Theorem 5.32. Let ...
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Show that a geometrically normal scheme of finite type has singular locus of codimension at least 2.

I am trying to prove that a geometrically normal scheme of finite type over a field $k$ has singular locus of codimension at least 2. So far, I written the following argument to prove the result in ...
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How calculate $\chi(O_P)$, a skyscrapper sheaf?

I'm beggining with the calculation of cohomology groups in sheaf theory and I have a doubt about how to calculate some things. So I would like to ask for help if some step in this calculation are ...
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Fibered product of schemes locally of finite type over a field is smooth if and only if its factors are smooth.

Let $X$ and $Y$ be $k$-schemes locally of finite type. Suppose their product $X \times_k Y$ is smooth at a point $z$ which projects to $x \in X$. I want to show that $X$ is smooth at $x$. So far, I ...
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Quasicompactness of $\operatorname{Proj}A$

$\newcommand{\Proj}{\operatorname{Proj}}$ Let $A$ be a graded ring of the form: \begin{align} A=\bigoplus_{n=0}^\infty A_n \end{align} So in particular this ring is positively graded, but it is not ...
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Two morphisms are equal if they are equal fiberwise.

Suppose $S$ is a reduced scheme and $X$, $Y$ are finite type $S$-schemes. When can we say that two morphisms $f$, $g$ $: X\longrightarrow Y$ are equal iff they are equal fiberwise, i.e. $f(s)=g(s)$, ...
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Checking Ampleness by Faithfully Flat Descent on the Base

Let $f:X \to S$ be a morphism of schemes and $\mathcal{L}$ an invertible sheaf on $X$. Question: Can relative $f$-ampleness be checked by faithfully flat descent on the base? Namely, if $h: S' \to S$ ...
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Local Sections of Dualizing Sheaf

Let $f : C\rightarrow \operatorname{Spec} k$ be a proper connected nodal curve over algebraically closed field $k$ and $\omega_C$ its dualizing sheaf. Let $\nu : C'\rightarrow C$ be the normalization ...
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Is the cartier duality of a group scheme the Hom_{R-Mod}(-,R) or Hom_{R-Mod}(-,R)?

I'm reading and reassuming R. Pink "finite group schemes" lecture course notes (in particular sections 2, 3, 4, 5, 11, 12, 13, 14) as an assignment. At section \s4 there's a definition of ...
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how $\mathrm{Gal}(\overline{K}/K)$ acts on automorphisms of a $\overline{K}$-scheme

Let $K$ be a number field and $\overline{K}$ its algebraic closure. Let $(X,\mathcal{O}_{X},s)$ be a scheme over $\overline{K}$ where $s\colon X\to\mathrm{Spec}\overline{K}$ is the structural morphism....
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Qing Liu's Algebraic Geometry Exercise 3.1.6

The exercise states: Let $\pi : T\rightarrow S$ be a morphism of schemes. Supposing that $\pi : T\rightarrow S$ is an open or closed immersion, or that $S=$Spec$A$, and $\pi$ is induced by a ...
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Smooth morphisms of schemes

I am trying to understand smooth morphisms of schemes as in https://stacks.math.columbia.edu/tag/01V4 All the definition seems to only take in considerations the source of the morphism, as the ...
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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 ...
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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 $X$, $\{U_i\}$. ...
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Hartshorne theorem II.8.19

There is a step in this proof I don't understand and maybe someone who does can help clarify it to me. The theorem is the following: Let $X$ and $X'$ be birationally equivalent non-singular ...
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Schemes to the rescue?

I am reading chapter 1 of Gille and Szamuely's book Central Simple Algebras and Galois Cohomology, on quaternion algebras. In it they prove (remark 1.3.1 on p. 18) that the conic associated to a ...
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two definitions of degree of invertible sheaf on projective curve

Hi I saw two defs from Vakil's FOAG and Görtz&Wedhorn’s AG for the degree of invertible sheaves on projective curve: (Vakil's) 18.4.2. Important definition: degree of a line bundle on a ...
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Assume we have an extension of algebraically closed fields $L/K$ and varieties $V_1$ and $V_2$ over $K$. Let $W_1\subseteq (V_1)_L$ and $W_2\subseteq (V_2)_L$ subvarieties, so $W_1\times_L W_2\... 4 votes 1 answer 1k views Local ring of a generic point on an integral scheme is a field [duplicate] Let$X$be an integral scheme and let$\eta \in X$be its generic point. Then the local ring$K(X) := \mathcal{O}_{X, \eta}$is a field. Moreover, if$U = \text{Spec} A$is any open affine subset of$...
I have two questions, that I put together as they are related to the same topic, the projective scheme of a graded $A$-algebra $B$. Question 1. Do you confirm that points (a), (b) and (c) of ...
One may define the higher direct image $R^if_*(E)$ of a quasi coherent sheaf $E$ on $X$, where $f:X \rightarrow Y$ is a morphism of schemes. How does this functor behave with respect to tensor ...