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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if $Z\rightarrow X$ is closed immersion then $f:Y\rightarrow X$ factors through X iff $f^{\ast}\mathscr I\rightarrow f^{\ast}\mathscr O_X$ is zero
The following is Lemma 01HP at the Stacks Project:
Lemma. Let $X,Y$ be locally ringed space. Let $\mathscr I\subset \mathscr O_X$ be sheaf of ideals locally generated by sections. Let $i:Z\rightarrow ...
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Gortz, Wedhorn, Algebraic Geometry, Proof of Proposition 3.52 - why is there at most one reduced subscheme structure?
Proposition 3.52 in Görtz and Wedhorn's Algebraic Geometry reads as follows:
Let $X$ be a scheme and $Z \subseteq X$ a locally closed subset. Then there exists a unique reduced subscheme $Z_{\...
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Clarification on Hartshorne II.2.12: Glueing Lemma
Exercise II.2.12 from Hartshorne's Algebraic Geometry book (i.e. the Glueing Lemma) presents the following
II.2.12. Let $\left\{X_i\right\}$ be a family of schemes (possibly infinite).
For each $i\ne ...
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Global sections vs regular sections
I'm having problems with understanding the definition of a global section of a sheaf defined over a scheme $X$. In my head a global section can be thought as a function whose domain is all $X$. But, ...
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Value of sheaf on an open point?
$\newcommand{\O}{\mathscr{O}}\newcommand{\Spec}{\operatorname{Spec}}$
Let $X$ be a scheme, and $x\in X$ be an open point, that $\{x\}$ is open, I want to determine what $\O_X(x)$ is (up to isomorphism)...
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Sheafification of function presheaves
Let $X$ be a topological space, $Y$ a set, and let $\mathcal F$ be a presheaf on $X$ so that $\mathcal{F}(U)\subseteq Y^U$, for any open set $U$, and the restriction morphisms are function restriction....
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Example of a closed subscheme of $\operatorname{Proj} A$ which is not of the form $\operatorname{Proj}A/I$.
First, we know that in the case where $\operatorname{Proj} A$ is quasi-compact then no such example exists, so we need the irrelevant ideal to be finitely generated up to a radical.
Take $A=\mathbb C[...
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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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Showing that $\operatorname{Spec} k$ is the scheme quotient of $\mathbb{A}^1_k$ by an action of $\mathbb{Z}$
This is Exercise 2.3.22 in Liu's book.
The goal is to show that the quotient in the category of schemes of $\mathbb{A}^1_k$ under the action of $\mathbb{Z}$ given by $n : T \to T + n$ (on the ring $k[...
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Simple objects in category of quasi-coherent sheaves
Let $X$ be a scheme. Exercise 6.5L in Vakil's FOAG asks to show that the simple objects in $QCoh_X$ (i.e. quasi-coherent $\mathcal O_X$-modules that are not $0$ and have no non-trivial quasi-coherent ...
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Help with R. Pink complete lecture notes of "Finite group schemes". Theorem 12.2
I'm an undergrad and I'm doing an assignment on Richard Pink's lectures notes on finite group schemes.
I kinda understood anything until this theorem comes (I skipped the motivation because my "...
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$\operatorname{Spec}(R/I)\to \operatorname{Spec}(R)$ is closed immersion of schemes: why is the kernel locally generated by sections?
I know $Z:=\operatorname{Spec}(R/I)$ is homeomorphic to $V(I)$ which is closed in $X:=\operatorname{Spec}(R)$. Let $i:Z\rightarrow X$. Next we want show $\mathscr O_{X}\rightarrow i_*\mathscr O_Z$ is ...
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All closed subschemes of projective schemes (i.e. $X=\operatorname{Proj} A$) are projective (not necessarilly generated in degree 1)
$\newcommand{\Proj}{\operatorname{Proj}}\newcommand{\Spec}{\operatorname{Spec}}\newcommand{\O}{\mathscr{O}}$
So I asked a similar question here, but I have a different plan of attack this time, and I ...
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Questions about Corollary 5.23 of AG1: Schemes by Görtz & Wedhorn
In the book, a theorem is stated with a clear demonstration. I simply write the points of the theorem that interest me (in the following, $k$ is any field) :
Theorem 5.22. Let $X$ be an irreducible $...
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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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Cartier divisor from rational section of invertible sheaf?
This is continuing the answer in Each divisor arises from a rational section . Should the result be changed to a version like the following with some requirements of the scheme and the section?
Let $...
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Distinguished projective opens in degree $0$ aren't affine?
$\newcommand{\Proj}{\operatorname{Proj}}\newcommand{\Spec}{\operatorname{Spec}}$
In Vakhil's the rising sea, he defines the topology on $\Proj A$ as the Zarisk topology, where closed sets are of the ...
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Closed subscheme of projective scheme is projective (ring is not generated in degree $1$)
$\newcommand{\Proj}{\operatorname{Proj}}\newcommand{\Spec}{\operatorname{Spec}}$
Let $A$ be a commutative $\mathbb Z^+$ graded ring (so grading is $n \geq 0$) but which is not necessarilly generated ...
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Double point scheme and vanishing of function
In Example II-9 of Eisenbud's The Geometry of Schemes, he considers the scheme $X=\operatorname{Spec}K[x]/(x^2)$ and makes the following claim:
$\bullet$ A function $f\in K[x]$ on $\mathbb{A}^1_K$ ...
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Two definitions of a variety
$\newcommand{\Spec}{\operatorname{Spec}}$
In my algebraic geometry course we defined a variety to be $k$-scheme $X$ which is of finite type, reduced, and separated over $\Spec k$. This definition ...
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Definition of closed immersion of locally ringed spaces
I just read on math stack program the definition of surjectivity of morphisms of sheaves and presheaves. I found they are quite confusing. In the following definition $X$ denotes a topological space ...
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Intersection of zero schemes defined via sections of vector bundles
Let $X$ be a scheme, and let $\mathcal{E},\mathcal{F}$ be vector bundles over $X$.
Consider non-zero global sections $s \in \Gamma(X,\mathcal{E})$ and $t \in \Gamma(X,\mathcal{F})$. These define maps ...
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The beginning of deeper mathematical abstraction [closed]
In the past, Mathematicians first created a geometric space and then thought about functions on it. For example, we defined a vector space and then thought about a linear map, defined a topological ...
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Non-reduced loci of the flat limit of a family of projective varieties
When reading Hartshorne Example III.$9.8.4$, I have some trouble determining the non-reduced loci of the flat limit of a family of varieties. The construction of this limit is explained below.
Let $...
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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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Computing $K_{X'/\mathbb{A}^2}$
Let $k$ be an algebraically closed field of characteristic zero, $X=\mathbb{A}^2_k$ and $X'=\operatorname{Bl}_{(x,y)} X$. I'm looking to compute $K_{X'/X}$.
Note that $$X'=\operatorname{Proj} k[x,y,...
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Definition of dual abelian schemes
Let $A$ be an abelian scheme on $S$
Let $\mathcal{P}ic_{A/S}$ be the relative Picard functor.
For many authors, the dual abelian variety is the identity component of the relative Picard functor (which ...
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Equivalent ways to obtain the reduced closed subscheme associated to the whole space
Let $(X,O_X)$ be a scheme. Consider the sheaf of ideals $I$ defined by: $$I(U)=\{s\in O_X(U):s_x\in \mathrm{nil}(O_{X,x})$$ $$\forall x\in U\},$$ for any open set $U\subset X$.
I know that $(X,O_X/I)$ ...
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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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If the base change is a fiber product, is the original variety a fiber product too?
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\...
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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 $...
CommunityBot
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Clarify on Propositions 3.36 and 3.38 in Qing Liu's algebraic geometry book
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 ...
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Does pushforward commute with tensor for coherent sheaves under an open immersion?
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 ...
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