The concept of scheme is from algebraic geometry: A locally ringed space that is glued together (much like a manifold) from affine schemes, that is spectra of rings with Zariski topology and structure sheaf.

learn more… | top users | synonyms

0
votes
1answer
65 views

Representability of transformations by morphisms

This is a very naive question. Suppose I'm given two functors $F,G\colon(Schemes/S)^{op}\to (Sets)$, over some base scheme $S$, represented by $S$-Schemes $X_F$ and $X_G$ respectively. There are ...
3
votes
2answers
104 views

Field automorphisms and varieties

Let $C$ be an algebraically closed field and consider a variety $X$ over $C$. In the language of schemes $X$ is a separated, integral scheme over $C$ with a morphism of finite type $f:X\longrightarrow ...
1
vote
1answer
36 views

An action of an automorphism of a field $C$ on a variety $X$ over $C$

My question comes from the reading of the first two pages of the article "Bernhard Köck - Belyi's Theorem Revisited". Consider a field $C$ and a variety $X$ over $C$. In particular $X$ is a ...
3
votes
2answers
82 views

Separated prevarieties and schemes

As it is shown in "Goertz,Whedorn - Algebraic geometry I" there is an equivalence of categories between the category of prevarieties over $k$ (field algebraically closed) and the category of integral ...
2
votes
1answer
162 views

Sum of Homogeneous Ideals vs Homogeneous Ideal of Intersection

Sorry for the recent question spam, but I've been striving furiously over the last few days to solve a problem in Hartshorne (Algebraic Geometry) discussed in this question. The problem (II.8.4a) ...
1
vote
1answer
41 views

Image of the diagonal map in a scheme

My question is similar to the question in The image of the diagonal map in scheme. I saw a hint to my question in the comments, but I was not able to prove it. Let $f:X\longrightarrow Y$ is a ...
6
votes
0answers
67 views

For this morphism of integral schemes $X\to Y$, if $X$ is geometrically reduced (irreducible) then is $Y$ also geometrically reduced (irreducible)?

Let $k$ be an arbitrary field. Suppose that $C/k$ is an integral curve which is birationally equivalent to a projective line. Is it true that $C$ is geometrically reduced and irreducible? This is of ...
4
votes
1answer
135 views

Hartshorne's definition of structure sheaf

Hartshorne at page $70$ defines the structure sheaf on Spec $A$. The elements of $\mathcal O_{\textrm{Spec}A}(U)$ are particular functions $s:U\longrightarrow\coprod_{p\in U}A_p$. With the symbol ...
3
votes
1answer
102 views

Hartshorne Lemma (II.4.5)

The statement of the lemma is - Let $f:X\longrightarrow Y$ be a quasi-compact morphism of schemes. Then the subset $f(X)$ of $Y$ is closed if and only if it is stable under specialization. One way is ...
2
votes
2answers
175 views

$A$-points of a fiber of a morphism of schemes over $k$

Suppose $f:X\to Y$ is a morphism of schemes over a field $k$. For any point $y\in Y$, we have the fiber of $f$ over $y$ defined as the fiber product $X_y=X\times_Y\mathrm{Spec }\;k(y)$, where ...
3
votes
0answers
56 views

Is the Abel-Jacobi map flat?

Let $X$ be a scheme corresponding to a smooth projective curve over $k=\bar{k}$. Let $Div_X$ denote the set of effective Cartier divisors of $X$ and $Pic_X$ the set of line bundles over $X$. We can ...
5
votes
1answer
156 views

Tangent sheaf of the Picard scheme

Let $X\overset{f}\to S$ be a flat morphism of schemes, and $S=\operatorname{Spec}(k)$ for an algebraically closed field $k=\bar{k}$. I'm just interested, for the moment, with schemes $X$ corresponding ...
1
vote
0answers
50 views

The localization exact sequence.

Let $X$ a scheme and $Y$ a close subscheme and $i:Y \to X$ the inclusion. Let $U:=(X-Y)$ and $j:U \to X$ the inclusion map. If I denote with $CH(-)$ the Chow group, the sequence $$ CH_r(Y) ...
0
votes
1answer
58 views

Possible to prove this via adjunction?

Let $X\overset{f'}\to S$ and $T\overset{g}\to S$ be schemes over $S=Spec(k)$ for a field $k$. Further, let $X_T = X \underset{S}\times T$ denote the fiber product over $S$, and $f$ the induced map ...
1
vote
1answer
30 views

$R^1$ and $H^1$ for a scheme over a field

Let $S=Spec(k)$ for a field $k$ and $\;X\overset{f}\longrightarrow S$ a scheme over $S$. Is it correct to say that $$ R^1 f_* \mathcal{O}_X \cong H^1(X, \mathcal{O}_X) \quad ? $$ Here $R^1f_*$ is the ...
2
votes
0answers
114 views

Relative spectrum of a quasi-coherent algebra.

I'm working out the notion of spectrum of a quasi-coherent algebra over a scheme. $\require{AMScd}$ Let $(X,\mathcal O_X)$ be a scheme and $\mathcal A$ a quasi-coherent $\mathcal O_X$-algebra, i.e. a ...
3
votes
1answer
73 views

Line bundles pullback trick

I'm trying to prove a statement about line bundles, and the following question is crucial to complete a possible proof that I have in mind. Could you give me any hint please? Let $A$ and $B$ two ...
1
vote
1answer
74 views

Scheme over a DVR

Let $\mathcal{O}$ be a discrete valuation ring with finite residue field $k$ of characteristic $p$. Let $S=\mathrm{Spec}(R)$ be a noetherian scheme over $\mathrm{Spec}(\mathcal{O})$. Are there ...
4
votes
2answers
135 views

Inverse image of a closed subscheme

Let $f:X\to Y$ be a surjective morphism of schemes, and $Z\subset Y$ a closed subscheme with short exact sequence $$ 0\to I_Z \to \mathcal{O}_Y \to \mathcal{O}_Z \to 0. $$ What are sufficient ...
1
vote
1answer
58 views

Computing Sheaf of differentials of a scheme

Can some one please explain how to see that sheaf of differentials of the scheme $\mathbb A^n_Y =Spec(\mathbb A^n_\mathbb Z)\times _Z Y$ is $\mathcal O ^n_X$
7
votes
1answer
74 views

Irreducibility of the space of divisors on a curve

Let $X$ be a smooth projective and irreducible curve over a field $k$. Further, define $$ X_d = \{ \text{ Effective Cartier divisors of degree } d \text{ on } X \;\} $$ and $$ W_d = \{ \text{ Line ...
6
votes
1answer
97 views

How to use flatness here?

Let $X\to S$ be a scheme. Definition: A relative effective Cartier divisor on $X/S$ is a closed subscheme $D\subset X$ such that the ideal sheaf $I$ of $D$ is invertible and $D\to S$ is flat. Let ...
2
votes
1answer
70 views

Retrieve affine schemes by adjunction

In an introductory course about schemes, I've seen the adjunction $$ {\mathbf{LocRngSpace}}^{\mathrm{op}} \overset{\mathrm{Spec}}{\underset{\Gamma}{\leftrightarrows}} \mathbf{Ring}, $$ where ...
3
votes
1answer
213 views

Action of a Galois Group on an Algebraic Variety

I've to solve the following exercise. Let's be $X$ a connected algebraic variety over $k$ and let $K$ be a finite Galois Extension of $k$ with Gaolis Group $G$. Now I have to prove that $G$ acts ...
3
votes
0answers
66 views

Divisors as a scheme

Let $S$ be a scheme and $X$ be a scheme over $S$ corresponding to a smooth projective curve. Define $X_d := X^d / \mathfrak{S}_d$, where $X^d$ denotes the usual $d$-fold cartesian product of $X$ and ...
5
votes
1answer
75 views

Relative effective Cartier divisors

I have two different definitions of a relative effective Cartier divisor. The first one is a bit outdated and defines the notion over analytic spaces, in the following way: Definition 1: Let $X$ be ...
5
votes
1answer
106 views

Why is Weil restriction right adjoint to base change?

Let $k'/k$ be a finite field extension, and let $X'$ be an affine group scheme over $k'$. We can define the Weil restriction of $X'$ to be the affine group scheme $\mathrm{Res}_{k'/k}(X')$ over $k$ ...
3
votes
1answer
103 views

Theorem 4.2.21 in Liu: Regular closed point in a geometrically reduced algebraic variety

I dont't understant the beggining of the proof of theorem 4.2.21 in Liu. Theorem: A geometricaly reduced algebraic variety X contain always a regular closed point. The proof begin in saying that we ...
2
votes
1answer
30 views

About closed map beween schemes

Probabily it's trivial but I've no idea for a proof. Let $f: X \rightarrow Y $ a continuous map between Topological Spaces, with $Im(f)$ closed in $Y$. I know there exist a covering $\{Y_i\}$ of $Y$ ...
4
votes
1answer
63 views

Exact sequences in the Mumford example

This question concerns the proof for the Mumford example of an Hilbert Scheme that has a non-reduced component. I am studying the proof given on R. Hartshone, "Deformation Theory", pp. 91-94. I do not ...
2
votes
0answers
38 views

Do the cyclic or Hochschild homologies satisfy the addition axiom of Eilenberg Steenrod?

Do the cyclic or Hochschild homologies satisfy the addition axiom of ES? If so please provide a reference or proof (reference is preferable).
2
votes
1answer
59 views

Question on geometrically reduced, geometrically connected.

I have a question from a book which I am trying to attempt. Let $k$ be a field not of characteristic 2 and let $a\in k$ be not a square (i.e. for all $b\in k$, $b^{2}\neq a$). I want to show that ...
2
votes
1answer
47 views

Dimension of schemes over fields

In my reading I recently encountered the following problem: Let $A$ be a DVR and let $Y=\mathrm{Spec}\,A$. Let $K$ be the fraction field of $A$ and let $k$ be the residue field. The canonical ...
2
votes
1answer
64 views

Interpreting results concerning the global sections ring being finitely generated

Let $A$ be a ring and $X$ be an $A$-scheme. (Hartshorne exercise II.2.17) Suppose there exist $f_0,\dots,f_n\in \mathcal{O}_X(X)$ such that a) $X_{f_i}:=\{x\in X: f_i(x)\not=0\}\subset X$ is ...
1
vote
1answer
60 views

Properties of Quasi-Coherent Modules

Let $X=\mathrm{Spec}\,A$ be an affine scheme and $M$ an $A$-module. Show that the following two conditions are equivalent: (a) $\tilde{M}$ is a locally free $\mathcal{O}_{X}$-module of finite type. ...
2
votes
1answer
65 views

Scheme-Theoretic Nakayama's Lemma

Let $X$ be a noetherian scheme and $\mathscr{F}$ a coherent $\mathscr{O}_{X}$-module. For a point $x \in X$, let $k(x)=\mathscr{O}_{X,x}/\mathfrak{m}_{x}$ be the residue field at $x$. (a) Suppose $x ...
1
vote
1answer
66 views

Open immersion from a proper scheme to a separated, irreducible scheme.

Fix a scheme $S$ and let $X$ and $Y$ be $S$-schemes. Assume that $X$ is proper over $S$ and $Y$ is separated over $S$. Let $f: X \rightarrow Y$ be an open immersion of $S$-schemes. If $Y$ is ...
2
votes
2answers
203 views

Morphism from a proper irreducible scheme into an affine scheme of finite type

Let $K$ be a field and $X$ be a proper irreducible $K$-scheme. Show that the image of any $K$-morphism $X \rightarrow Y$ into an affine $K$-scheme $Y$ of finite type consists of a single point. I ...
7
votes
1answer
114 views

How to tell whether a scheme is reduced from its functor of points?

Suppose I have a scheme $X$ and I want to know if $X$ is reduced, but all I have access to is the functor $$ R\mapsto X(R)=Mor(\operatorname{Spec}(R),X) $$ from commutative rings to sets (rather ...
3
votes
1answer
183 views

Questions on Gluing Schemes

I am trying to do an exercise in a book and here is the question. I have attempted it but I am not sure if my answer is correct. I would appreciate if someone corrects my attempt. Note here that the ...
9
votes
2answers
380 views

Some basic examples of étale fundamental groups

I'm trying to get a better understanding of étale fundamental groups, and I think that the overall idea -- the big picture -- is beginning to become clear, but my computational ability seems to be ...
5
votes
1answer
170 views

Explicit description of discrete valuations corresponding to prime divisors

I'm reading Hartshorne p. 130 on Weil divisors. Let $X$ be a noetherian integral separated scheme regular in codimension 1, $Y$ a prime divisor on $X$, and $\eta \in Y$ its generic point. ...
4
votes
1answer
147 views

function field of an integral scheme

Suppose $X$ is an integral scheme, and let $\eta \in X$ be its generic point. Then the local ring $\mathcal{O}_{X,\eta}$ is a field, called the function field of $X$ and denoted $K(X)$. Why is $K(X)$ ...
2
votes
1answer
132 views

Graph morphism for a separated morphism of schemes

I want to prove the following: Let $f: X \rightarrow S$ be a separated morphism of schemes. Show that any section $g: S \rightarrow X$ of $f$ i.e. a morphism such that $f \circ g=\textrm{id}_{S}$ is ...
2
votes
1answer
145 views

Separated Morphisms of Schemes

(a) Let $f:X \rightarrow S$ be a separated morphism of schemes. Show that for any subscheme $U \subset X$, the restriction $f\mid_{U}:U \rightarrow S$ is separated. (b) Let $R$ be a commutative ring ...
3
votes
2answers
153 views

Why are non-separated schemes schemes?

In "the old days", e.g. in the famous texts by Grothendieck and Mumford, a scheme was defined as what we now call a separated scheme. (i.e. a scheme where the image of the morphism $\Delta:X \to X ...
3
votes
1answer
78 views

Morphisms from spectra to schemes

Let $X$ be a scheme. Show that for any $x \in X$ there exists a canonical morphism $\textrm{Spec}\, \mathcal{O}_{X,x} \rightarrow X$. If $k(x)=\mathcal{O}_{X,x}/\frak{m}_{x}$ is the residue field at ...
1
vote
1answer
39 views

Mappings for an integral scheme with generic point

Let $X$ be an integral scheme with generic point $\eta$. Show that for any $x \in X$, there is a canonical ring homomorphism $\mathcal{O}_{X,x}\rightarrow \mathcal{O}_{X,\eta}$. For any open ...
2
votes
1answer
58 views

Frobenius twist commutes with fiber product

Let $k$ be a field, and let $X$ be a scheme over $k$. Let $f:k\to k$ be the embedding of fields defined by $f(\lambda)=\lambda^p$. Define the first Frobenius twist of $X$ to be the scheme $X\times_f ...
1
vote
1answer
105 views

The Frobenius twist of a scheme

Let $k$ be a field of characteristic $p$, and define an embedding of fields $f:k\to k$ by $f(\lambda)=\lambda^p$. Then, given any scheme over $k$, $X$, define the Frobenius twist of $X$ as ...