The concept of scheme mimics the concept of manifold obtained by glueing pieces isomorphic to open balls, but with different "basic" glueing pieces. Properly, a scheme is a locally ringed space locally isomorphic (in the category of locally ringed spaces) to an affine scheme, spectrum of a ...

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13
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1answer
266 views

When can stalks be glued to recover a sheaf?

Let $\mathcal{F}$ be a sheaf over some topological space. The stalks are $\mathcal{F}_x= \underset{{x\in U}}{ \underrightarrow{\lim}} \mathcal{F}(U)$. Is there a special name for a sheaf that ...
1
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0answers
30 views

Construction of relative projective space via glueing

I would like to gain further practice on glueing schemes by constructing projective space over a ring. I am considering the following: I wonder how we get that $\mathcal{O}_{X_i} (X_{ij}) = ...
1
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0answers
23 views

Pulling back the sheaf of ideals under diagonal embedding

Let $X$ be a scheme (over a field $k$) and $\Delta:X\to X\times X$ be a diagonal embedding. Let $I$ be a sheaf of ideals of the diagonal $\Delta(X)$ in $X\times X$. For positive integer $n$ how could ...
1
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0answers
11 views

Classification of group schemes of order 2: Why is $sx=x\otimes 1+1\otimes x-cx\otimes x$?

In their paper Group Schemes of Prime Order, Tate and Oort state Suppose $$G=\operatorname{Spec}(A)\text,\quad S=\operatorname{Spec}(R) \text,$$ and suppose the augmentation ideal ...
0
votes
1answer
83 views

An algebraic variety as a scheme

If we take the definition of a variety as a reduced integral scheme of finite type over an algebraically closed field $k$, then a variety is in particular a scheme over $k$, so is a scheme $X$ with a ...
1
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0answers
30 views

Milne's definition of $G$-torsors

I am trying to understand the proof of proposition 4.1 of Milne's book Étale Cohomology (p.120) and I am getting really confused with some points of the reverse implication: if I understand correctly ...
2
votes
1answer
80 views

Inverse function theorem for etale morphisms

Looking around stackexchange, it seems there are many related questions, but I'm a beginner and I can't find a proof on the internet (without going through the more general results in stacks project). ...
1
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0answers
46 views

Pushforward of a quasicoherent sheaf on a noetherian scheme is quasicoherent

I'm reading through and trying to understand the proof of Proposition 5.8 in Chapter II of Algebraic Geometry by Hartshorne that the pushforward of quasicoherent sheaf $\mathcal{F}$ by a morphism ...
2
votes
1answer
58 views

Confusion Over Geometric Irreducibility

Let $X$ be a scheme over a field $k$. For an extension field $K$ of $k$, we can change the base to obtain the scheme $X_K$. Supposedly it is possible that $X$ is irreducible while $X_K$ is reducible, ...
4
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0answers
108 views

Learning Galois theory geometrically?

Recently I started poking at algebraic geometry and commutative algebra. My background is basic category theory and basic algebraic topology. I don't know a lot of other mathematics. I noticed Galois ...
3
votes
1answer
38 views

All open subsets of the spectrum of a number field are principal

Let $K$ be a number field, $\mathcal{O}_k$ its ring of integers, $\operatorname{Cl}(K)$ its class group and $h_K = \lvert \operatorname{Cl}(K)\rvert$ its class number. Let $X = ...
1
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0answers
31 views

Relative affine schemes as zero-locuses

Let $f : Y \to X$ be an affine morphism of schemes (say of finite type over a field $k$). I read in some notes that $Y$ can be written as $Y \cong E \times_{E'} X$ where $u : E \to E'$ is a morphism ...
1
vote
1answer
44 views

$f:X\rightarrow Y$ is a closed immersion iff $f:f^{-1}(U_i)\rightarrow U_i$ is a closed immersion.

I came across the following property of closed immersions on Wikipedia - A morphism $f:Z\rightarrow X$ is a closed immersion iff for some (equivalently every) open covering $X=\bigcup U_j$ the ...
0
votes
0answers
29 views

A finite morphism between affine schemes is closed. How can we generalize this to show any finite morphism of schemes is closed?

I am working on Exercise II.3.5 b) in Hartshorne. It asks to show that a finite morphism is closed. I am working on a way to show that this is true in the affine case, but I am having trouble showing ...
0
votes
0answers
90 views

Is this solution to Hartshorne Exercise III.10.2 correct?

Exercise III.10.2 in Hartshone's Algebraic Geometry is discussed in this question. The accepted answer proceeds by showing that the smooth locus is open on $X$, then transferring this to $Y$ using the ...
1
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0answers
34 views

Are $ \mathbb{A}^n (k ) = k^n $ and $ \mathbb{P}^{n} (k) = \mathrm{Proj} \ k[X_0 , \dots , X_n ]$ irreducibles?

Are $ \mathbb{A} (k ) = k^n $ and $ \mathbb{P}^{n} (k) = \mathrm{Proj} \ k[X_0 , \dots , X_n ]$ irreducibles when $ k $ is a domain ? Thanks in advance for your help.
2
votes
2answers
59 views

Sheafyness and relative chinese remainder theorem

The relative chinese remainder theorem says that for any ring $R$ with two ideals $I,J$ we have an iso $R/(I\cap J)\cong R/I\times_{R/(I+J)}R/J$. Let's take $R=\Bbbk [x_1,\dots ,x_n]$ for $\Bbbk $ ...
1
vote
1answer
30 views

Fiber product with diagonal morphism [duplicate]

Stacks tag 01KR states that the diagram of schemes is "by general category theory" "a fibre product diagram". I tried to show this using the universal property, but didn't obtain anything useful. ...
1
vote
1answer
40 views

Very ample divisors over non-algebraically complete field

For a projective scheme $V$ over an algebraically-closed field $k$ it is a well-known fact fact that a base-point-free linear system of divisors is very ample iff it separates $k$-points and tangent ...
5
votes
2answers
112 views

If $A=\Gamma(X,O_X)$ and $Spec(A)=X$ then $Spec(A)=X$ as schemes.

Let $X$ be a scheme and $A$ a commutative Ring. Assume that we have an isomorphism $A \rightarrow \Gamma (X, O_X)$ and the induced map $X \rightarrow Spec(A)$ is a homeomorphism. Question: Is it ...
1
vote
2answers
48 views

Why does $\operatorname{Proj}(B)$ not contain ideals containing the irrelevant ideal?

Let $A$ be a ring and let $B = \bigoplus_{d\geq 0} B_d$ be a graded $A$-algebra.In Liu's book Algebraic Geometry and Arithmetic Curves the set $\operatorname{Proj}B$ is defined as the set of all ...
6
votes
0answers
88 views

Vanishing of Tor sheaf on a union of subschemes with vanishing Tor.

Suppose $X$ is a scheme and $Y$ and $Z$ are closed subschemes such that $Y$ is a finite union of closed reduced irreducible subschemes $C$ that satisfy ...
5
votes
2answers
104 views

What does the case of $\operatorname{Spec}C^{\infty}(M)$ tell us about the relavance of scheme theory to general rings?

I guess a lot of people with previous exposure to differential geometry have had this naive question pop out in their mind when studying schemes for the first time. The category of compact smooth ...
2
votes
1answer
67 views

Coproduct decomposition of the quot scheme functor

I am reading FGA explained and stuck on this argument: The functor $\mathfrak{Q}uot_{E/X/S}$ naturally decomposes as a co-product $$\mathfrak{Q}uot_{E/X/S}=\coprod_{\Phi\in \mathbb{Q}[\lambda]} ...
2
votes
1answer
61 views

Hartshorne Exercise III.8.4(c)

Let $Y$ be a noetherian scheme, and let $\mathcal E$ be a locally free $\mathcal O_Y$-module of rank $n+1$, with $n\ge 1$. Let $X=\mathbb P(\mathcal E)$ [the projective bundle over $\mathcal E$], with ...
0
votes
0answers
27 views

Is the tensor product of a flasque sheaf with a locally free sheaf necessarily flasque?

Is the tensor product of a flasque sheaf with a locally free sheaf necessarily flasque? I found myself asking this while working an exercise in Hartshorne. I suspect the answer is 'No' in general, ...
0
votes
1answer
73 views

Morphisms of schemes are $\mathcal{O}-$modules

The following might not be true and part of the purpose of this question is to verify my understanding so far. Let me start with a few statements: Since the sheafication of a presheaf only takes ...
2
votes
0answers
49 views

How should I think of schemes of the form $\operatorname{spec} k[f_1, \ldots, f_n]$, $f_i \in k[x_1, \ldots, x_m]$?

How should I think of schemes of the form $\operatorname{spec} k[f_1, \ldots, f_n]$, $f_i \in k[x_1, \ldots, x_m]$? What of the map on spectra induced by the inclusion $i : k[f_1, \ldots, f_n] \to ...
2
votes
1answer
41 views

Non-noetherian scheme with infinitely many irreducible components passing through a point? Point still a domain?

Is there a non-noetherian scheme with infinitely many irreducible components passing through a point? (I expect the answer to be yes, but I do not know of an example.) For extra internet points, I ...
1
vote
0answers
42 views

Is the set of regular points in a scheme open in general?

In the situation when smooth/k coincides with regularity (for a finite-type k scheme edit: k also perfect, thanks Remy), I think this should be true (?). But I am not sure about the situation for a ...
2
votes
0answers
37 views

Is there an example of a scheme $X$, irreducible, with a dense open subset $U$ so that $\dim U < \dim X$?

Is there an example of a scheme $X$, irreducible, with a dense open subset $U$ so that $\dim U < \dim X$? What about just for topological spaces? I know for varieties this doesn't happen (though ...
2
votes
1answer
61 views

Intuition for the valuative criterion for properness of morphisms?

I've always been told that the intuition for the valuative criterion for properness is something like this: a morphism $X\rightarrow Y$ is proper if, given a map of a small disk $D$ into $Y$ and a ...
5
votes
0answers
59 views

Hartshorne problem III.5.2(a)

Consider problem III.5.2(a) in Hartshorne's Algebraic Geometry: Let $X$ be a projective scheme over a field $k$, let $\mathcal O_X(1)$ be a very ample invertible sheaf on $X$ over $k$, and let ...
2
votes
1answer
63 views

Why is the rank of a locally free sheaf same everywhere if $X$ is connected?

Let $(X,\mathcal{O}_X)$ be a connected scheme.Let $\mathcal F$ be a locally free sheaf on $X$. This means that $X$ can be covered by open sets $U$ for which $\mathcal F|_U$ is a free $\mathcal O_X|_U$ ...
3
votes
1answer
68 views

Spectrum of infinite product of rings

$\def\Z{{\mathbb{Z}}\,} \def\Spec{{\rm Spec}\,}$ Suppose $R$ a ring and consider $\Spec(\prod_{i \in \mathbb{Z}} R)$. Now for the finite case, I know that holds $\Spec(R \times R) = \Spec(R) \coprod ...
0
votes
1answer
41 views

Image of not dominant morphism in Spec Z is finite

I have a very simple question that I seem not to be able to answer by myself. I want to understand the following: "If the structure morphism $f: X \to \operatorname{Spec}\mathbb{Z}$ is not dominant, ...
1
vote
1answer
74 views

Very basic question on projective bundles: Why is the fiber a vector space?

Consider an integral scheme $(X,\mathcal O_X)$ of dimension $n$ and a "good" locally free sheaf of $\mathcal O_X$-modules $\mathcal E$ of rank $n+1$. Then, thanks to the "global proj construction" we ...
0
votes
1answer
37 views

Maps induced by divisors and degree

I feel very stupid to ask this question but I've serached everywhere and I've not found a clear (to me) answer. A well known result in Curve theory (over $k=\bar{k}$) states that a divisor $D$ on a ...
1
vote
2answers
56 views

The Hirzebruch surface $\mathbb F_n$ contains a unique irreducible curve with negative self-intersection.

Consider the $n$-th Hirzebruch surface $\mathbb F_n:=\mathbb P_{\mathbb P^1}(\mathcal O_{\mathbb P^1}\otimes\mathcal O_{\mathbb P^1} (n))$, and let $C_0\subseteq \mathbb F_n$ a section of the ruling ...
2
votes
1answer
34 views

Morphism between affine scheme corresponding with ring homomorphism

Let $X = \text{Spec} R$ and $K = \text{Spec} S$ be two affine schemes with $f,g: K \rightarrow X$. I know that a morphism between affine schemes correspond with a ring homomrphism, so denote $\varphi: ...
1
vote
1answer
55 views

$\mathbb P^1\times \mathbb P^1$ is minimal

I'm trying to prove that the complex surface $\mathbb P^1\times \mathbb P^1$ is minimal. I'd like to prove it directly, namely by showing that there are not exceptional curves. Suppose that $E$ is a ...
5
votes
1answer
69 views

Degree of $f:\mathbb{P}^1_k\rightarrow \mathbb{P}^1_k$

Let $k$ be an algebrically closed field and consider $\mathbb{P}^1_k$ the $1$-dimensional projective space over $k$. My question is the following: Let consider $f:\mathbb{P}^1_k\rightarrow ...
0
votes
0answers
29 views

What exactly does the phrase “the quadratic cuts out a quadratic cone”?

Let $u,w,v \in \mathbb{C}$. Then I read in a review on affine schemes that The quadratic $uw-v^2$ cuts out a quadratic cone $X \in \mathbb{A}^3$ with coordinate ring $\mathbb{C}[u,v,w]/(uw-v^2)$. ...
2
votes
1answer
64 views

Geometrically ruled surface, sections and intersection numbers.

Consider a geometrically ruled surface $\pi: S\longrightarrow C$ where both $S$ and $C$ are projective, non-singular and complex ($C$ is obviously a curve). By using Tsen theorem one can show that ...
3
votes
0answers
50 views

Tricks to find $\dim H^0(O(D)(n))$

Let $X$ be a curve of genus $g\neq 0$ and $D$ a divisor on $X$. If $n \in \mathbb{Z}$ and $O(D)(n)=O(D)\otimes O(n)$ is the twist of $O(D)$ by $n$, there is a manageable proceedings to find the ...
0
votes
0answers
25 views

Zariski tangent space and subschemes of length 2

Let $X$ be a scheme, $x\in X$.The Zariski tangent space of $X$ at $x$ is the dual of the vector space $\mathfrak{m}_{x}/\mathfrak{m}_{x}^2$ over $k(x)$. It is a general fact that there is a ...
1
vote
0answers
40 views

Description of Blow up via Ress Algebra

Let $X=\mathrm{Spec}(R)$ be an affine scheme and $Z=\mathrm{Spec}(R/I)$ be a closed subscheme of $X$ defined by an ideal $I$ of $R$. The blow up of $X$ along $Z$ is ...
3
votes
2answers
186 views

Why is the “smallest non-affine scheme” not affine?

Exercise I-25 of Eisenbud and Harris' "Geometry of Schemes" defines the following scheme, which is then claimed to be the smallest non-affine scheme: Let $X=\{p,q_1,q_2\}$ with the open subsets ...
1
vote
1answer
49 views

Projective line is not isomorphic to the affine space with a doubled origin (schemes)

In the book 'Geometry of Schemes' (Eisenbud-Harris), there is a question that says the following (page 34, I-44): put $Y = \rm{Spec\,} K[s]$ and $Z = \rm{Spec\,} K[t]$. Let $U \subset Y$ be the open ...
2
votes
1answer
28 views

If $f : X \to Z$ is a morphism of schemes, which factors through an open $i : U \to Z$ on the level of topological spaces…

If $f : X \to Z$ is a morphism of schemes, which factors through an open $i : U \to Z$ on the level of topological spaces... then is there a (unique) morphism of schemes $g : X \to U$ which makes the ...