Questions tagged [affine-schemes]
The spectrum of a commutative ring with unit is the set of prime ideals endowed with the Zariski topology. One can define a sheaf of rings on this space : to each Zariski-open set is assigned a commutative ring, thought of as the ring of "polynomial functions" defined on that set. This topological space endowed with this sheaf is called the spectrum of the ring. Every locally ringed space isomorphic to such a spectrum is called an affine scheme.
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Geometric interpretation of Lefschetz number for local fields
I have the following problem.
Let $L/K$ be a finite galois extension of local fields with Galois group $G$. For nontrivial $g\in G$ define Lefschetz number $i_{L/K}(g):= \min\limits_{x\in \mathcal{O}...
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If $f:\operatorname{Spec} R \to X$ is a map with $R$ local and with closed point landing in an open $U\subset X$, then $im(f)\subset U$ too.
I’m taking an introductory course on Scheme theory. In one of the proofs of the course, we were considering a situation where we have a morphism of schemes $f: \operatorname{Spec} R \to X$, where $X$ ...
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Locus of the fibered product of two affine lines with doubled origin where projections coincide
This is an exercise (Exercise III-1) in the book Geometry of Schemes:
Exercise III-1. a) Let $Y$ be the line with doubled origin over a field $K$ and let $\varphi_1,\varphi_2:\mathbb{A}_K^1\to Y$ be ...
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Can we explicitly describe the derived pullback $\mathbf L\pi^* \widetilde M$ for a closed immersion $\pi$ of affine schemes?
Let $I$ be an ideal of a commutative Noetherian ring $R$. Let $M$ be a finitely generated $R$-module and $\widetilde M$ be its associated sheaf on $\text{Spec} (R)$. We have the closed immersion $\pi:...
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image of Segre embedding is cut out (scheme-theoretically) by vanishing minors
I am doing exercise 10.6.B of Vakil's FOAG (July 31, 2023 version).
Basically, I need to show that the image of the Segre embedding is cut out by equations so that the matrix
\begin{bmatrix}
a_{00}&...
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Morphism of schemes and closed points [duplicate]
I apologize for the vagueness of my question.
In the case of morphisms between schemes (that locally look like an $A$-algebra maybe), the textbook (Vakil's FOAG, July 31, 2023 version) often uses the &...
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Gluing in the sheaf of meromorphic functions on an affine scheme
Let $X = \operatorname{Spec}(R)$ be an affine Noetherian scheme, and $U_i = D(f_i)$ (with $f_i \in R$) be (finitely many) principal open sets with $X = \bigcup_i U_i$.
We have the sheaf of meromorphic ...
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If $\operatorname{Spec} f$ is a closed immersion, then $f$ is surjective?
Let $f: A\to B$ be a ring homomorphism. Then we can get a morphism of schemes $\operatorname{Spec} f:\operatorname{Spec} B\to \operatorname{Spec} A$.
In general, is this true? "If $\operatorname{...
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Open subschemes of open sets not contained in basic open sets
I'm just starting to learn algebraic geometry for the first time, and I had an elementary question concerning some of my intuitions about open subschemes (during this period where I haven't yet learnt ...
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global section of a B-scheme defines a morphism to projective $B$-scheme
This is problem 7.3.O in Vakil's AG notes (July 31, 2023 version). I am not sure how to define the map locally.
Problem 7.3.O. Let $B$ be a ring and $X$ a $B$-scheme. Suppose $f_0,\cdots,f_n$ are n+1 ...
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Rising Sea exercise 5.3.H
I wish to prove that if $(f_1, \ldots, f_n) = (1)$, and for each $i$, $B \to A_{f_i}$ is of finite type, then so is $B \to A$.
I am following Vakil's hints.
Let $r_{ij}$ be the finitely many elements ...
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Zeros of a finite support function on elliptic curve are torsion
Let $X$ be a elliptic curve over a field $k$ and $f$ be a holomorphic function on $X$, if $f$ only has two zeros $P$ and $Q$, then I'd like to know how to prove see that $P$ is a torsion point of $X$.
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When can $\textbf{SpecMax}(R)$ be a scheme?
Let $R$ be a commutative ring, and let $(\text{Spec(R)},\mathcal{O}_R)$ be the affine scheme associated to $R$.
Let $\text{SpecMax}(R)$ the subspace the $\text{Spec(R)}$ of the maximal ideals.
My ...
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Proof criticize: restriction maps of integral schemes are injective
Claim. Let $X$ be an integral scheme, and $\emptyset\neq V\subseteq U$ for some open subsets $V,U$ of $X$. Then $
\rho_{UV}:\mathcal{O}_X(U)\to\mathcal{O}_X(V)$ must be injective.
Here is my proof of ...
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Intuition of projective line via glueing
I am reading the construction of projective line (4.4.6) following Prof. Vakil's Algebraic geometry notes FOAG. However, I fail to understand few lines during the discussion.
Let $X=\operatorname{Spec}...
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Relationship between affine lines with coordinate $z$ and coordinate $z^2$
Let $k$ be algebraically closed of characteristic zero; assuming $k=\mathbb{C}$ is fine.
Consider the subring inclusion $k[z^2]\to k[z]$ and the induced map
$$
f\colon\mathbb{A}^1_k=\mathrm{Spec}k[z]\...
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More examples of morphisms of ringed spaces that aren't local?
$\def\Spec{\operatorname{Spec}}$All questions and answers that I've found in MSE regarding a morphism of ringed spaces between affine schemes that isn't a morphism of locally ringed spaces are the ...
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Geometrical interpretation of $\operatorname{Spec}(\mathbb{R}[x,y]/(x^2+y^2))$
$\def\Spec{\operatorname{Spec}}$If am not mistaken, the prime spectrum of $\Spec(\mathbb{R}[x,y]/(x^2+y^2))$ consists of the points $(\overline{x},\overline{y})$, $(\overline{x}-a)$, $(\overline{y}-b)$...
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Scheme theoretic fiber vs. Degree of a point of a finite morphism of affine schemes
Let $f: R\to S$ be a finite morphism of Commutative Noetherian rings, so $S$ is module finite over $R$ via $f$. Let $Q \in \text{Spec}(S)$, and set $P=f^{-1}(Q)$, so we have an induced map $R_P \to ...
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Should a closed immersion of schemes be closed map of topological spaces by definition?
In Section 9.1 of his "Foundations of Algebraic Geometry" (version August 2022), Ravi Vakil defines a closed embedding (closed immersion) of schemes to be an affine morphism
$ \pi: X\...
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How is a vector space an affine algebraic variety?
Suppose $\mathfrak{g}$ is a Lie algebra, let $\mathfrak{g}^* $ be its dual. I need $\mathfrak{g}^* $ to be an affine algebraic variety. How is it one?
I suppose the only important thing is that $\...
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The equality $\operatorname{Spec}\operatorname{Fun}\mathfrak{g}^* = \mathfrak{g}^*$
I don't know how many of my hyphotesis will be needed, maybe this is a more general fact.
Suppose $\mathfrak{g}$ is a complex simple Lie algebra, and let $\mathfrak{g}^*$ be its dual (as a vector ...
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What is a germ of a presheaf?
I’m trying to figure out the notions of germs and stalks of a presheaf. I had understand the definition is there an example though? Also I was thinking why there was a reason to construct these two ...
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The category of affine schemes is anti-equivalent to the category of commutative rings
The category of affine schemes is anti-equivalent to the category of commutative rings.
Let $F$ be the contravariant functor from the category of commutative rings (with identity) to affine schemes ...
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Is every affine variety an affine scheme? [duplicate]
I have seen that the notion of affine scheme is a generalization of the notion of affine varieties where the coordinate ring is replaced by any commutative unit ring, and the variety with the Zariski ...
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What ring's spectrum corresponds to the affine scheme that is the inverse image of a morphism between two affine schemes?
Giving morphism of schemes $\pi$ : $\operatorname{Spec}A \rightarrow \operatorname{Spec}B$, by definition we have $\pi^{-1}\mathscr{O}_{\operatorname{Spec}B}$ is a sheaf of rings on $\operatorname{...
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Spectrum as functor
I have just started studying about affine schemes and I am searching for books about Spec as a functor from the category CRings to the category AffineSchemes can you suggest me some? Also I am ...
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$\mathbb{Z}$-scheme from $\mathbb{Q}$-scheme
Let $I$ be an ideal of $\mathbb{Q}[T_1,\dotsc,T_n]$, and let $J=I\cap\mathbb{Z}[T_1,\dotsc,T_n]$.
Consider $X=\operatorname{Spec}(\mathbb{Q}[T_1,\dotsc,T_n]/I)$ and $Y=\operatorname{Spec}(\mathbb{Z}[...
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Noetherian condition of Serre's theorem on affineness
I am currently reading the proof of Hartshorne Theorem 3.7, which is a famous result of Serre. It stated the following:
Let $X$ be a noetherian scheme, then the following are equivalent:
$X$ is ...
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Description of $\operatorname{Spec}(A\times_C B)$ in terms of the spectra for $A,B,C$
I assume rings are commutative. Given surjective ring morphisms $f:A\to C$, $g:B\to C$, I wonder if it's possible to determine the prime ideals of the pullback $A\times_C B$ in terms of the prime ...
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Problem $I-24$ from Eisenbud and Harris's The Geometry of Schemes
The following is a statement from Eisenbud & Harris geometry of schemes
Let $(X, \mathcal{O})$ be any ringed space, and let $R = \mathcal{O}(X)$. For any $f \in R$ we can define a set $U_f \...
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Intuitive reason for why $\operatorname{Spec}(k[x,y,z]/(x-yz))$ is smooth at $O$ while $\operatorname{Spec}(k[x,y,z]/(x^2-yz))$ isn't?
Let $k$ be an algebraically closed field and consider the closed subschemes $X=\operatorname{Spec}(k[x,y,z]/(x-yz))$ and $Y=\operatorname{Spec}(k[x,y,z]/(x^2-yz))$ of $\mathbb{A}^3$. Then the gradient ...
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Smoothness of affine variety consisting of finitely many points
Let $R$ be a commutative ring such that $X:=\mathrm{Spec}(R)=\{p\}$, where $p$ is a prime ideal. Then, obviously, the affine variety $X$ consists of one point only. However, can we decide if this ...
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Curves homeomorphic under Zariski topology
Prove any two curves over some field $k$ are homeomorphic, where $k$ might not be algebraically closed. Curves are defined to be varieties (integral separated scheme of finite type) of dimension $1$.
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Similarities and Differences between Spec and Proj
I've been learning algebraic geometry, and I'm working on some problems about projective space and projective schemes. One struggle I've come across is working with morphisms between projective ...
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$R[x]$-algebra structure that on $R[y]$ seen as quotient of $R[x,y]$
In a broader exercise about the fibered product of schemes, I'm given a ring (commutative and unitary) $R$ and the following ring homomorphism:
$\begin{align*}
R[x]&\to R[y]\\
x&\mapsto y^2\\
\...
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Proof Proposition 4.1 Hartshorne
In this proposition, Hartshorne says and proves that any morphism of schemes between two affine schemes $f:X=\text{Spec }A \to Y=\text{Spec }B$ is separated.
To prove this he says that the diagonal ...
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Unicity in Hartshorne Corollary II.7.15
What am I missing in the following:
Let $X=\mathbb{A}^2_k$ be the affine plane over an algebraically closed field $k$, and let $O$ be the origin. Let $\tilde{X}$ be the blow-up of $O$. If $\mathcal{I}$...
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Unitary group modulo the Jacobson radical
Let $F$ be any field, and let $A$ be an associative unital finite dimensional $F$-algebra, with an $F$-linear involution $*$ (possibly trivial), that is an isomorphism of the $F$-vector space $A$ of ...
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For $f : X \rightarrow Y$ a morphism of schemes, for $U \subset X$ a nonempty affine open, must $f(U)$ be contained in some affine open of $Y$?
Is there a morphism of schemes $f : X \rightarrow Y$ admitting a nonempty affine open $U \subset X$
such that $f(U)$ is not contained in any affine open of $Y$?
Clearly if such $f$ and $U$ exist then $...
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Intersection and preimage of standard affine open sets give another standard affine open set.
I have trouble understanding a key step of Lemma 01ST on the Stack Project. The problem is showing that $U_u'=f^{-1}(V_u) \cap U_u$ is standard affine open in $U_u$, where $U_u\subset U$ itself is ...
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Is the stalk of an irreducible scheme at the generic point always a field?
Let $X$ be an irreducible scheme. It can be proved that it has a unique generic point, i.e. there is a unique point $\xi \in X$ such that $\overline{\{ \xi \}} = X$. One can identify $\xi$ as the ...
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Intersection of affine opens is affine for separated schemes of an affine scheme
Let $X$ be a scheme over $S=Spec A$ and let $U, V$ be affine opens in $X$. Let $X$ be separated over $A$ and let $\delta$ be the diagonal morphism $X \rightarrow X \times_S X$. Then the $U \cap V$ is ...
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What is the connection between sheaf and schemes?
I have some problems in understanding the notion of a presheaf $\mathcal{O}_X$ on $\text{Spec}(A)=:X$. So is it true that $\mathcal{O}_X$ is the presheaf $$\mathcal{O}_X: \text{Op}\left(\text{Spec}(A)\...
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Cokernel of canonical skyscraper map not generated by global sections.
I am trying to solve the following exercise (Görtz and Wedhorn Exercise 7.7):
Let $A$ be an integral domain with field of fractions $K$, set $X=\operatorname{Spec} A$ and let $\eta \in X$ be the ...
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On the Picard group of the affine n-space over a normal affine base scheme [duplicate]
Question:
Let $S=Spec(\Lambda)$ be a normal affine scheme. Consider the morphism of schemes $\mathbb{A}^{n}_{S}=Spec(\Lambda[x_{1}, \ldots, x_{n}]) \rightarrow S$. Is it true that the induced map $Pic(...
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Is $\mathbb{A}\setminus0$ an affine scheme for non algebraically closed field [duplicate]
If $k$ is algebraically closed, it's known that $\mathbb{A}^2\setminus0$ is not an affine scheme (see, e.g. this). But what about the scenario when $k$ is not algebraically closed?
My thought: in the ...
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the meaning of $[x_0, x_1, \ldots, x_n]$ formed by projective coordinates on projective space $\Bbb{P}^n_A$ in Vakil's FOAG 7.3.F
In Vakil's FOAG, the projective space $\Bbb{P}^n_A$ is defined to be $\operatorname{Proj} A[x_0, x_1, \ldots, x_n]$. (there is another definition in the book too, just glueing $n+1$ affine space). ...
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Post regarding Hartshorne II Ex. 3.1.
3.1. Show that a morphism $f : X \to Y$ is locally of finite type if and only if for every open affine subset $V = \textrm{Spec} \, B$ of $Y$, $f^{-1}(V)$ can be covered by open affine subsets $U_j = \...
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does natural structure morphism from Proj $S$ to Spec $A$ require $S$ a finitely generated graded ring?
The question is already asked here natural structure morphism from Proj $S$ to Spec $A$ . And It’s exercise 7.3.I in Vakil’s FOAG:
7.3.I. EASY EXERCISE.
If $S$ is a finitely generated graded A-...
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