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.

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Is a nonzero divisor locally nonzero divisor?

Suppose $X$ is a scheme (or locally ringed space), $a\in \Gamma (X,O_X)$, not a zero divisor. Can we show $a_x$is also a nonzero divisor in $O_{X,x}$? (It is true if $X$ is an affine scheme)
2
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1answer
75 views

Finite fiber- unramified morphisms

I'm in trouble understandig the proof of Proposition 3.2 Chapter 1 of Milne's Book "Étale Cohomology". Let $f:Y\rightarrow X$ be locally of finite-type. The following are equivalent. $(a)$ f is ...
4
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1answer
59 views

Two definitions of Scheme theoretic image

Suppose $f:X \to Y $ is a morphism. I saw two definitions of scheme theoretic image. The first one requires $f$ to be quasi-compact and quasi-separated, or quasi-compact, which ensures the kernel ...
6
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1answer
104 views

Morphisms from a proper scheme to the affine line over a field must be constant.

Let $k$ be a field. Let $X$ be a (non empty) connected proper $k$-scheme. I would like to prove the maximum principle, that is for any $k$-morphism $\varphi \colon X \to \mathbb A_k^1$, the image of ...
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73 views

Normal cone to a closed subscheme

Suppose $S$ is a closed subscheme of a smooth variety $M$ and that its ideal sheaf $\mathscr{I}_S$ factors as $\mathscr{I}_S=\mathscr{I}_{S_1}\cdot \mathscr{I}_{S_2}$, where $S_1$ and $S_2$ are closed ...
2
votes
1answer
28 views

Universally closed morphism and closed morphism

If $f:X\longrightarrow Y$ is a universally closed morphism of schemes (that is the map obtained by base extension is closed), then does it imply $f$ is closed? Or, is the assumption of $f$ being ...
0
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1answer
58 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
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2answers
103 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 ...
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1answer
35 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
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2answers
79 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
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1answer
147 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) ...
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1answer
35 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 ...
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66 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
122 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 ...
2
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1answer
80 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
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2answers
151 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 ...
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52 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
149 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 ...
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22 views

Functions on $K(X)$ and DVR.

In our definition a variety is an integral and separated scheme $X$ and we denote with $K(X)$ the fild of rational functions on $X$. Let $X$ be a normal variety. Let $D$ be an integral codimension-one ...
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47 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) ...
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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 ...
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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 ...
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85 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
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1answer
64 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 ...
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vote
1answer
65 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
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2answers
107 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 ...
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1answer
56 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
72 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 ...
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1answer
96 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 ...
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1answer
63 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
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1answer
168 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
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64 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
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1answer
60 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
77 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
98 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
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1answer
29 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
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0answers
37 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
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1answer
54 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
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1answer
45 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
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1answer
59 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 ...
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1answer
53 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. ...
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1answer
63 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 ...
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vote
1answer
58 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
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2answers
189 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 ...
6
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1answer
87 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
163 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 ...
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244 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
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1answer
144 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
126 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 $K(X)$ ...