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 ...

learn more… | top users | synonyms

3
votes
1answer
42 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 = \operatorname{Spec}(\...
1
vote
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
45 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
33 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
101 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
vote
0answers
35 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
61 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
42 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
115 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
50 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
98 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 $$\mathscr{Tor}_i^{\mathcal{O}_X}(\mathcal{O}_C,\...
5
votes
2answers
108 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
71 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
69 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
30 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
80 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 k[...
2
votes
1answer
49 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
59 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
42 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 ...
3
votes
1answer
90 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
65 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
77 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
85 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
78 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
38 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
59 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
37 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
71 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
68 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
52 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
47 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 (say)$\mathrm{Bl}_{Z}(X)=\mathrm{...
3
votes
2answers
196 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
57 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
29 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 ...
6
votes
1answer
200 views

What does Hartshorne do wrong?

I'm currently trying to learn algebraic geometry from Hartshorne's Algebraic Geometry. I've often heard it said, both on MathOverflow and in my department, that Hartshorne's treatment of certain ...
0
votes
1answer
92 views

Question on the proof of Hartshorne IV.3.7

Proposition 3.7 in Hartshorne's Algebraic Geometry is the following. Let $X$ be a curve in $\mathbb P^3$, let $O$ be a point not on $X$, and let $\phi: X \rightarrow \mathbb P^2$ be the morphism ...
1
vote
0answers
38 views

Fibers of the Blow up of $\mathbb{A}^6$ over the center defined by ideal defined by vanishing of rank two minors.

Let us consider set of all matrices \begin{pmatrix}{} x & u & v \\ u & y & w \\ v & w & z \end{pmatrix} where $x,y,z,u,v,w\in \mathbb{R}$ which is equivalent to $\mathbb{A}^...
2
votes
1answer
69 views

if the canonical divisor is nef, then a multiple if effective

Let $S$ be a complex projective non-singular surface. Couldd you explain the following implication: If the canonical divisor $K_S$ is nef then there exists a number $m>>0$ such that $mK_S$ ...
1
vote
0answers
62 views

Why is $ \mathrm{Spec} \ A [T_1 , \dots , T_n ] \to \mathrm{Proj} \ A [T_0 , \dots , T_n ] $ an embedding?

Let $ A $ be a commutative ring with identity. I would like to know how to establish that : $ \mathrm{Spec} \ A [T_1 , \dots , T_n ] \to \mathrm{Proj} \ A [T_0 , \dots , T_n ] $ is an embedding ? ...
1
vote
1answer
80 views

Understand the corrispondence between rational maps and linear systems.

A key fact in "birational geometry" (on $\mathbb C$) is the following theorem: Let S be a surface. Then there is a bijection between the following sets: (i) {rational maps $\phi:S-\...
0
votes
0answers
47 views

$k$-structure on $K$-schemes

I'm reading A. Borel's "Linear Algebraic Groups". At an early point in the book, the author establishes the following concepts: (Let $K$ be an algebraically closed field, and $k$ a subfield of $K$) $...
2
votes
2answers
89 views

Corollary of Riemann Roch theorem

I'm studying about Riemann Roch theorem for curves and its consequences (in algebaic gemoetry). In Hartshorne I've found an exercise equivalent to this assert: Let $D$ be an effective divisor of ...
1
vote
2answers
71 views

What exactly does the Hilbert scheme of points parametrize?

The Hilbert scheme of points is defined as $$ \text{Hilb}^n(X) = \{ I \subset \mathbb{C}[x,y] \text{ such that } \text{dim}_{\mathbb{C}}/I = n \} $$ or, in words, the Hilbert scheme of points ...
1
vote
0answers
56 views

$\Omega_X$ locally free $\implies$ $X$ smooth

Let $X$ be $n$-dimensional scheme of finite type over an algebraically closed field. In the proof of Proposition 7.4.11 in Gathmann, the first paragraph reads If $\Omega_X$ is locally free of rank ...