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.

Filter by
Sorted by
Tagged with
1
vote
1answer
27 views

Two points in projective space: $(\mathbf{P}^2)^{[2]} \simeq \text{Bl}_\Delta (\mathbf{P}^2 \times \mathbf{P}^2)/S_2$

Wikipedia says that Hilbert scheme of points on projective space is equivalent to the Blowup of two copies of projective space: $$(\mathbf{P}^2)^{[2]} \simeq \text{Bl}_\Delta (\mathbf{P}^2 \times \...
0
votes
0answers
20 views

A question on the subscheme of a $k$-scheme of finite type

I am reading Wedhorn's Algebraic Geometry. On the page 88, it says: Example 3.45. If $k$ is a field, and $X$ is a $k$ -scheme of finite type, then all subschemes of $X$ are of finite type over $k$. ...
0
votes
1answer
29 views

Two different morphism of $K$-schemes have different image points?

If $K$ is a field and $X\rightarrow \operatorname{Spec}K$ a $K$-scheme, and I have two morphisms of $K$-schemes $f_1,f_2:\operatorname{Spec} K\rightarrow X$ that are different, does it follow that $...
0
votes
0answers
31 views

Typo in Qing Liu? Ramified extension of number fields example with all finite fibres

I encountered the following lemma in Qing Liu's Algebraic Geometry and Arithmetic Curves: Let $f:X\to Y$ be a morphism of finite type between locally Noetherian schemes. Then $f$ is unramified if and ...
0
votes
1answer
56 views

Could we deal with smooth variety as manifold?

Let $X$ be a smooth variety (or scheme) over an algebraically closed field $k$. Then we always regard it as a manifold. I'm wondering if this analog is literally? For example: Is there an open ...
0
votes
0answers
50 views

Hartshorne's Algebraic Geometry, Ex 9.5 (a) in Chapter III construct a flat family $X_t$ such that it's cone is not flat

I have some problems to understand a solution I found of following exercise from Hartshorne's Algebraic Geometry, Ex 9.5 in Chapter III, part (a): Very Flat Families. For any closed subscheme $X \...
3
votes
1answer
55 views

An étale morphism that restricts to an isomorphism on a closed subvariety.

If an étale morphism $f:X\rightarrow Y$ induces an isomorphism $f:f^{-1}(Z)\rightarrow Z$ for some closed subvariety $Z$ of $Y$. Doesn't it imply that $f$ is an isomorphism (I believe the answer ...
1
vote
0answers
33 views

How to construct $S(q^* E) \rightarrow T(L) $ for morphism to projective bundle?

This is 4.2.2 in EGA II. Let $q : X \rightarrow Y$ be a morphism of schemes. Let $L$ be an invertible sheaf on $X$ and let $E$ be a finite type quasicoherent sheaf on $Y$. Let $\phi : q^*(E) \...
1
vote
1answer
58 views

Definition of locally closed subscheme

I'm wondering how to define a locally closed subscheme formally. My attempt is to define it as a morphism $f:X\rightarrow Y$ which can factor as $f = g\circ i$ where $g$ is an open immersion and $i$ ...
1
vote
0answers
19 views

When the pullback of etale sheaves commutes with the forgetful functor to the Zariski site?

Let $f:X\rightarrow Y$ be a morphism of schemes. Let $f^*$ be the pullback functor from the sheaves on the big etale site (or small etale/Zariski site depending on the context!) of $Y$ to $X$. Let $L$ ...
0
votes
0answers
32 views

When is $Hom(F,G)$ quasicoherent?

If $\mathcal{F}$ is coherent, and $\mathcal{G}$ is quasicoherent, $Hom(\mathcal{F}, \mathcal{G})$ is quasicoherent. In the standard proof of this, I am not sure where coherence is applied (it seems ...
1
vote
2answers
58 views

How does a morphism of k-varieties induce a morphism of their k-rational points?

Let $k$ be an infinite field, not necessarily algebraically closed. By a k-variety I mean a k-scheme which is separated and of finite type. By a $k$-rational point of a $k$ variety I mean a point $x$ ...
2
votes
2answers
89 views

Why should we study morphism of $k$-schemes instead of morphisms of schemes?

I have been self-studying algebraic geometry through youtube video lecture of Prof. Richard E. BORCHERDS. This is the link of the lecture that I have question about: https://www.youtube.com/watch?v=...
3
votes
1answer
44 views

Ideal sheaf is quasi-coherent if and only if its generated by local sections.

My confusion is lies in Schemes Lemma 10.1 of the Stacks project. First, Modules Definition 8.1 states that a sheaf $\mathcal{F}$ of $\mathcal{O}_X$-modules is locally generated by sections if for all ...
0
votes
0answers
40 views

Dimension of a scheme $X$ at a closed point $x$ and dimension of its local ring.

The dimension of an irreducible scheme $X$ at $x$, dim$_x(X)$ is defined as the smallest dimension among its open neighbourhoods and the dimension of its local ring dim$(\mathcal{O}_{X,x})$ is just ...
3
votes
0answers
16 views

Relation between first sheaf cohomology of scheme X and the global section of sheaf on diagonal.

Let $X$ be a scheme and $\mathfrak{U} = \{U_i\}$ be an open cover of $X$. Let $\mathscr{F}$ be a sheaf on $X$. Denote $C^1$ for a first component of Cech complex. What I observe is : $C^1 = \Pi_{i &...
7
votes
2answers
165 views

When considering a finite-type scheme as a ringed space, is it enough to look at its $k$-points?

I am reading a set of notes by Michel Brion about automorphism groups of projective varieties. The following claim appears in the proof of a theorem stating that if G is a connected group scheme, $X$ ...
4
votes
1answer
121 views

$k$-rational points of the automorphism functor of a scheme

Let $X$ be a scheme and let $\operatorname{Aut}_X$ denote the functor sending a scheme $T$ to the set of $T$-automorphisms of $X \times T$. Assume that $\operatorname{Aut}_X$ is representable by a ...
1
vote
1answer
32 views

Projective space as the glue of affine schemes: checking the cocycle condition

One construction of projective space over a ring $A$ is to take $n+1$ affine opens given by $$ U_i = \operatorname{Spec} \frac{A\left[\frac{x_0}{x_i}, \dots, \frac{x_n}{x_i}\right]}{x_i/x_i-1},$$ and ...
0
votes
0answers
9 views

Showing that quasicoherence is affine-local

Let $\operatorname{Spec}R$ be an affine scheme. For an ideal $I$ of $R$, we obtain a ring homomorphism $R \to R/I$, and a corresponding scheme morphism $\alpha:\operatorname{Spec}(R/I) \to \...
2
votes
3answers
110 views

Is a subfunctor of a representable functor also representable?

I'm trying to learn about representable functors but I'm very new to this; even to categories... Suppose $G:C^{op}\rightarrow Set$ is a representable functor - as I understand this means that there ...
-1
votes
0answers
21 views

Definition of separable map between curves and separated morphism of schemes

Let $C1$ and $C2$ be smooth curve and $Φ$ be map between them. We usually define that $Φ$ is separable: field extension $K(C1)/K(Φ^*(C1))$ is separable. My question is : Is this definition of ...
3
votes
1answer
79 views

Blowup extends a regular map to $\mathbb{P}^{N+1}$

Let $(X_0,X_1,...,X_n)$ homogeneous coordinates of $\mathbb{P}^n$ and let assume that $X^r \subset \mathbb{P}^n$ is a complex variety where $x:= (1,0,...,0) \in X$ and $X$ isn't a cone with vertex $x$....
1
vote
1answer
128 views

Localization of sheaves: 'particular' proof wanted

The question is closely related to this one. Although the OP accepted already an answer there, I think that a subtely detail is still not solved and would like to ask about it here explicitely. The OP ...
4
votes
3answers
152 views

Localization of sheaves

Let $(X,\mathscr{O}_X)$ a ringed space and $\mathscr{F}$ an $\mathscr{O}_X$-module. Let $U\subseteq X$ be an open subset, and suppose we have $\mathscr{F}|_U=\mathscr{O}_U$, then for $x\in U$, how ...
0
votes
1answer
83 views

Understanding Hartshorne's example II 3.2.2

Example 3.2.2. If $P$ is a point of a variety $V$, with local ring $\mathcal{O}_P$, then $X:=\operatorname{Spec} \mathcal{O}_P$ is an integral noetherian scheme, which is not in general of finite type ...
0
votes
0answers
34 views

Functional topology for affine schemes

In the book "Categorical foundations", chapter III ("A functional approach to general topology"), an interpretation of topology is given in terms of factorization systems. It ...
0
votes
2answers
118 views

Question on Smooth Completion of curves

Let $X$ be a smooth irreducible curve of finite type $C$ over a separably closed field $k$. For such curves is known that they always have a smooth compactification, that is there exist a smooth ...
1
vote
1answer
43 views

Morphisms between affine schemes

Suppose we have two affine schemes $X=\operatorname{Spec} A$ and $Y=\operatorname{Spec} B$ for commutative rings $A,B$. I encountered this statement in my homework that $\operatorname{Mor}(X,Y)=\...
0
votes
1answer
27 views

Can a closed subset of a scheme $X$ be a support?

Suppose $X$ is a scheme, and $T$ its closet subset. I want to ask that is there a canonical way to obtain a quasi-coherent $\mathscr O_X$-ideal $I$ such that $T=supp \mathscr O_X/I$? This is the ...
0
votes
0answers
36 views

Basics about action on a Scheme by finite Group

Let $Y$ be a scheme and a finite group $G$ acts on $Y$ in the sense that $G$ embeds in $Aut_{Sch}(Y)$ in category of schemes. I have two question concerning general properties dealing with action by ...
-1
votes
1answer
55 views

Diagonal action by $\mathbb{Z}/2$ on $\mathbb{P}^1 \times \mathbb{P}^1$

I have a basic question about a certain action on product of projective lines $\mathbb{P}^1 \times \mathbb{P}^1$ by group $\mathbb{Z}/2$ which Sasha introduced in this MO discussion. How does it here ...
0
votes
0answers
40 views

Surjectivity of proper morphism

I am trying to prove that, given a proper morphism of schemes $f \colon X \rightarrow Y$, if $Y$ and all fibers of $f$ are connected then $X$ is connected. It seems to me that it would help knowing if ...
0
votes
0answers
20 views

open subscheme and étale fundamental group

Let $X$ be an irreducible scheme, $U\subset X$ be a nonempty open subscheme. Is $\pi^{ét}_1(U)\to \pi^{ét}_1(X)$ surjective? For example, if $X$ is normal integral scheme, then this follows from ...
0
votes
0answers
12 views

Universal cover with respect to étale topology of scheme

Let $X$ be a connected quasi-compact quasi-separated scheme. I tried to define a universal cover as follows (thanks to the help of one friend). Consider $I$ the directed set of all open normal ...
1
vote
1answer
46 views

Subset of a scheme where a section vanishes

Suppose $(X,\mathscr{O})$ is a scheme, $s\in \mathscr{O}(U)$. I am trying to prove that $\{x\in U: s_x=0\in \mathscr{O}_x(\text{the stalk})\}$ is open in $U$ but not necessarily closed. I can see why ...
2
votes
1answer
50 views

Smoothness is a geometric property

I read that "smoothness is a geometric property" meaning that if $k\subset K$ is a field extension and $X_k$ is a scheme over $k$ then $X_k$ is smooth over $k$ if and only if $X_K = X_k \...
0
votes
0answers
31 views

Regularity of quotient of polynomial ring

I am trying to solve Exercise IV.2.3 from Liu's book. I want to prove that $K[x,y]/(x^2+y^3+t^n)$ is regular, where $K$ is the fraction field of a discrete valuation ring and $t$ is a uniformizing ...
3
votes
2answers
79 views

$\mathbb{P}^1$ isomorphic to conic in $\operatorname{Proj}(K[x,y,z])$ context

I have recently started to study schemes and I found my self on the follow situation: I want to prove that $\mathbb{P}^1$ is isomorphic to a conic. I have used the morphisim $\mathbb{P}^1\...
0
votes
1answer
70 views

Presheaf of $\mathcal{O}_X$-modules with restriction given by localization

Let $X$ be a scheme, and consider the distinguished affine open base of topology on $X$. That is, the data of all affine opens $\mathrm{Spec}(A)\subset X$ and inclusions only of the form $\mathrm{Spec}...
3
votes
0answers
65 views

Understanding Hartshorne's theorem II.3.1: a scheme is integral iff is both reduced and irreducible

I am trying to understand the last part of theorem 3.1. of Hartshorne's Algebraic Geometry, which says: A scheme is integral iff is both reduced and irreducible. In the $(\Longleftarrow)$ part: ...
1
vote
2answers
85 views

Factorization as morphism between schemes vs. as morphism between underlying sets

I'm confused about an argument used by Mohan in his answer of this question. We have two algebraic curves $C $ and $C'$ (algebraic curve = complete and nonsingular over an algebraically closed field) ...
1
vote
1answer
30 views

Irreducible components of the closure of a locally closed subset

I would like to ask for a reference of the following fact: the irreducible components of the closure of a locally closed subset are the closures of the irreducible components of the subset. Thank you ...
1
vote
0answers
48 views

Understanding geometric visualisation of a double point (fat point)

This question comes from Example 12.21 (a) of Gathmann's 2019 notes, here. In the example, take $R = K[x]/(x^2)$ for some field $K$, so that $\operatorname{Spec} R$ is a single point $\mathfrak{p} = (...
2
votes
0answers
35 views

Blowup of $\Bbb{A}^2$ at the intersection of a cubic with a triple line

I'm familiar with blowups in classical algebraic geometry but I'm still learning about blowup of schemes. For now I'm trying to be as concrete as possible, because the formal definition is ...
0
votes
1answer
39 views

Proof that a set is open using closed points on a $k$-scheme

Let $S \subset X$, where $X$ is an affine scheme of finite type over $k$ and $S(k)$ is not empty. Let us suppose that the set of points of $S(k)$ is inside of the interior $\text{int } S$, that is, ...
1
vote
1answer
64 views

Short Exact Sequence of Tangent Space of smooth morphism

I'm reading the proof of Proposition 4.3.39 in Qing Liu's book. Proposition 4.3.39 Let $f:X\rightarrow Y$ be a morphism of finite type to a locally Noetherian scheme. Let us suppose $f$ is smooth at $...
1
vote
1answer
72 views

Global sections and Fiber products $X \times_S \operatorname{Spec} k$

Let $X $ a $S$-scheme ($S$ another scheme). Suppose $k$ is a field and $S$ has a $k$-valued point, that is a map $p: \operatorname{Spec} k \to S$. If $X \times_S \operatorname{Spec} k$ denotes the ...
0
votes
0answers
25 views

Why formal power series is in a object of the category of formal scheme?

What is an object and an morphism of the category of formal scheme? I heard formal group is an group object in a category of formal scheme. I wonder why formal power series is in a object of formal ...
1
vote
1answer
36 views

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 $X$, $\{U_i\}$. ...

1
2 3 4 5
44