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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2
votes
1answer
124 views

$X_K$ normal imply $X$ normal

In Algebraic Geometry and Arithmetic Curves of Qing Liu, I have two problems with the lemma 4.1.18 (page 119). The lemma is so: let $\mathcal{O}_K$ a DVR (uniformizing parameter $t$) with residue ...
1
vote
3answers
156 views

The relation between been the quotient ring of a prime ideal and its localization

Let $A$ be a ring and $\mathfrak{p} \subset R$ be a prime ideal. Set $A_\mathfrak{p}=R[U^{-1}]$, where $U= A-\mathfrak{p}$. What is the relation between $A/\mathfrak{p}$ and $A_\mathfrak{p}$? My ...
1
vote
0answers
64 views

Morphisms from the group variety of $n$-th roots

In Milne - Lectures on étale cohomology, example 6.10 i came across the following. We fix a variety $X$ and work in the category $Var/X$ of varieties over $X$ (so with fixed morphisms to $X$!) and ...
3
votes
1answer
109 views

Tensor product of the structure sheaf with itsself

Let $\pi : X \to S$ be a morphism of schemes. Is the $\mathcal{O}_X$-module $\mathcal{O}_X \otimes_{\pi^{-1} \mathcal{O}_S} \mathcal{O}_X$ quasi-coherent? Here, $\mathcal{O}_X$ acts on the tensor ...
4
votes
1answer
584 views

Finite + surjective + projective implies flat?

Let $f: X \rightarrow Y$ be a morphism of irreducible projective varieties, that is both finite and surjective. Does this mean that it is flat? I have tried the following: By finiteness, the map is ...
2
votes
1answer
365 views

References for quasi finite and proper implies finite

Does anybody know a reference for a proof of: Let $f: X \rightarrow Y$ be a quasi-finite proper morphism of varieties. Then $f$ is finite. Is there one in Hartshorne? I could not find it. Thanks!
5
votes
1answer
270 views

Faithfully flat morphisms with all fibers complete

Prove or disprove: if $f: X \to Y$ is faithfully flat and each fiber is complete, then $f$ is proper. (I'd especially like to see a counterexample with a morphism of finite type between varieties over ...
4
votes
1answer
122 views

Irreducibility preserved under étale maps?

I remember hearing about this statement once, but cannot remember where or when. If it is true i could make good use of it. Let $\pi: X \rightarrow Y$ be an étale map of (irreducible) algebraic ...
5
votes
1answer
381 views

Rational/meromorphic functions on a scheme

In EGA (IV, §§ 20 – 21), the sheaf of meromorphic functions $\mathscr{M}_X$ on a ringed space $(X, \mathscr{O}_X)$ is defined as the sheaf associated to the presheaf that associates to an open $U ...
4
votes
1answer
448 views

Pullback of very ample sheaf again very ample? And other questions.

Let $S \subseteq \mathbb{P}^n$ be a smooth projective surface with given embedding in projective space. Moreover, let $X$ be another smooth surface and let there be a map $\pi: X \rightarrow S$ that ...
4
votes
1answer
237 views

Does the singular locus of a conical variety (or scheme) determine the singular locus of its projectivization?

Lets say $X$ is a conical affine algebraic variety (conical meaning $X$ is the zero set of homogeneous polynomials of positive degree, equivalenty $X \subsetneq k^n$ and $x \in X \Rightarrow ax \in X$ ...
2
votes
1answer
191 views

Quasi-Coherent modules over sheaves of rings (proof question)

Let $X$ be a scheme with structure sheaf $\mathcal O_X$ and let $\mathcal B$ be a quasi-coherent $\mathcal O_X$-algebra. Let $\mathcal F$ be any sheaf of $\mathcal B$-modules. In EGA I, § 9, Prop. ...
5
votes
1answer
164 views

$f(X)$ dense in $Spec A$

I have some problems solving the following exercise from Liu's book Algebraic Geometry and Arithmetic Curves, exercise 3.15 from chapter 2. Let $X$ be a quasi-compact scheme, $A=O_X(X)$. Let us ...
7
votes
1answer
151 views

Exercise on locally ringed spaces

I try to solve exercise 2.19 on page 65 in "Algebraic Geometry I" by U.Görtz/T.Wedhorn. The exercise reads: Let $(X,O_X)$ be a locally ringed space, and $f\in O_X(X)$. Define $X_f:=\{ x\in X; f(x) ...
1
vote
2answers
100 views

The image of a certain morphism in $\mathbb{A}^1_k$ is a closed point

Let $X$ be a nonempty, connected, proper scheme over a field $k$, and $f \in \mathcal{O}_X(X)$ (the global sections of $X$). Let $\varphi : X \to \mathbb{A}^1_k$ be the morphism induced by: ...
5
votes
1answer
114 views

Proof of $X(\mathcal{O}_K)\simeq X_K(K)$

I have problems to understand a proof of the following theorem (Algebraic Geometry and Arithmetic Curves, Qing Liu, Theo 3.3.25, page 107). Theorem: let $\mathcal{O}_K$ be a valuation ring over $K$, ...
4
votes
2answers
348 views

Intuition on base change of schemes

Let $S=Spec(A)$ and $S'=Spec(B)$ be two affine schemes for some rings $A$ and $B$ such that there is a morphism of schemes $f:S'\rightarrow S$. For any $S$-scheme $X$, one can consider the fiber ...
1
vote
1answer
349 views

The image of a proper scheme is closed

Let $X$ be a scheme, $Y$ a proper $X$-scheme and $Z$ a separated $X$-scheme. Let $f : Y \to Z$ be a morphism of $X$-schemes. I would like to prove that the image of $f$ is closed in $Z$. Here is what ...
3
votes
2answers
312 views

Intersection of quasi-compact open subsets of an affine scheme

Let $X = \mathrm{Spec}(A)$ be an affine scheme. Let $U$ be a quasi-compact open subset of $X$. Then there exist an affine scheme $Y$ and a morphism $f\colon Y \rightarrow X$ such that $f(Y) = U$, ...
4
votes
0answers
145 views

If $X\times C$ is isomorphic to $Y\times C$, does it follow that $X$ is isomorphic to $Y$

Let $k$ be a field. Let $X$, $Y$ and $C$ be varieties over $k$ such that $X\times_k C $ is isomorphic to $Y\times_k C$. Assume that $C$ is a curve. (The varieties $X$, $Y$ and $C$ are assumed to be ...
4
votes
1answer
489 views

Why surjectivity stable under base change?

I want to prove that surjectivity is stable under base change: if $f:X\to S$ a surjective morphism of scheme and $\varphi:T\to S$ then $f_T:X\times_S T\to T$ is surjective. Idea 1: I know that for ...
1
vote
0answers
102 views

places of function field and closed point of a scheme

Given an integral scheme $X$, let $K(X)=\mathrm{Frac}(R)$ be its function field, where $\mathrm{Spec}(R)$ is some non-empty open affine subscheme of $X$. Take the maximal ideal $P$ of some DVR of ...
5
votes
2answers
235 views

Basic definition of Inverse Limit in sheaf theory / schemes

I read the book "Algebraic Geometry" by U. Görtz and whenever limits are involved I struggle for an understanding. The application of limits is mostly very basic, though; but I'm new to the concept of ...
3
votes
0answers
174 views

Which local ringed spaces are schemes?

This paper gives a proof that the underlying topological spaces of affine schemes are precisely the spectral spaces (compact, T0, sober, and the compact open subsets are closed under finite ...
0
votes
1answer
216 views

Separated schemes

How is it connected that scheme $X$ is separated over $Y$ and over $Spec \; \mathbb{Z}$, where $X \rightarrow Y$ is some scheme morphism?
6
votes
1answer
148 views

Set of points where ring of germs is reduced is open

I want to solve an exercise from Liu's book Algebraic Geometry and Arithmetic Curves, namely exercise 4.9 in chapter 2: Let $X$ be a Noetherian scheme. Show that the set of points $x\in X$ such that ...
8
votes
1answer
257 views

Hom between 2 schemes

Why is the set $Hom(X,Y)$ between 2 schemes $X$ and $Y$ a scheme as well? Where can I read the construction? For example, $Hom(\mathbb{A}^1,\mathbb{A}^1)$ is the set of all polynomial, and what is the ...
4
votes
1answer
114 views

Irreducible trivialization of a finite etale morphism

Let $X$ be an irreducible scheme and $Y \to X$ a finite étale morphism. Is there some finite étale cover $Z \to X$ which trivializes $Y$ (i.e. $Y \times_X Z$ is a union of copies of $Z$) such that $Z$ ...
4
votes
1answer
355 views

The canonical divisor of the projective line

Let $A$ be a ring. Assume it has some nice properties if necessary, e.g., $A$ is a Dedekind domain. Let $\mathbf{P}^1_A$ be the projective line over $A$. I want to show that $-2 [\infty]$ is a ...
9
votes
2answers
668 views

Closed points of a scheme correspond to maximal ideals in the affines?

Let $X$ be a scheme. If $x\in X$ is a closed point, then it corresponds to a maximal ideal in $\mathcal{O}_X(U)$ for some affine open subset $U\subseteq X$. If I take an arbitrary open affine ...
3
votes
1answer
236 views

Does the functor of points commutes with inverse limits?

A scheme $X$ defines a (covariant) functor from commutative rings to sets by $A\mapsto X(A)=Mor(SpecA,X)$ Does this functor commutes with inverse limits? It does when $X$ is affine, but I can't see ...
3
votes
0answers
140 views

smooth morphism on schemes

Imagine a projective, noetherian and flat family of curves $C \rightarrow S$ (i.e. every geometric fiber is a curve, this is a integral, non singular, proper scheme of dimension 1 over an ...
1
vote
0answers
131 views

Fine moduli solutions VS objects with automorphisms

In more than one place I read that, given a moduli problem, the existence of an object with nontrivial automorphisms prevents the existence of a fine solution. I'd like to understand in which sense ...
1
vote
1answer
51 views

One construction about sheafs

Let $(X,O_X)$ be a ringed space, $E$ - finite locally free $O_X$-module. Let $E^*=Hom_{O_X}(E, O_X)$. How to show, that $E^{**} = E$? It's clear, that locally $E|_U = O_X^n|_U$, and then $E^*|_U = ...
0
votes
2answers
55 views

Union of immersions

I'm working to understand the Grothendieck topology version of the Zariski topology of schemes. Explained simply, it replaces the notion of "open subschemes" with "open immersions". So instead of ...
1
vote
1answer
192 views

Proj construction and fibered products

How to show, that $Proj \, A[x_0,...,x_n] = Proj \, \mathbb{Z}[x_0,...,x_n] \times_\mathbb{Z} Spec \, A$? It is used in Hartshorne, Algebraic geometry, section 2.7.
2
votes
1answer
59 views

Smoothness of invertible elements

consider a proper, flat family of schemes $X\rightarrow S$, with $S$ affine. I would like to know under which condition on the family the functor $Sch/S \rightarrow Ab$ $ T \rightarrow ...
3
votes
1answer
71 views

Limits of subrings and surjectivity

Let $A$ be a ring and let $\mathcal{F}$ be the inductive system of subrings of $A$ which are of finite type over $\mathbb{Z}$: $$ \mathcal{F} = \{ \mathbb{Z}[a_1,\dots,a_n] \subseteq A \mid n \geq 0, ...
1
vote
1answer
282 views

Separated scheme

How to show, that the affine line with a split point is not a separated scheme? Hartshorne writes something about this point in product, but it is not product in topological spaces category! Give the ...
4
votes
1answer
144 views

Ring homomorphism and affine scheme

How to describe all ring homomorphisms $f: A \rightarrow B$, such that corresponding affine scheme morphism $f: Spec \, B \rightarrow Spec \, A$ is open immersion?
1
vote
1answer
125 views

Functional sheaf (Hartshorne, Cartier divisors)

In Hartshorne there is the following description of the sheaf $K$ on the scheme. For each open $U = Spec \, A$ we define $K(U) = S^{-1} A$, where $S$ is the set of non-zero-divisors. Why is it a ...
1
vote
1answer
132 views

Are smooth relative curves over an arbitrary base normal?

Let $X/S$ be a smooth, projective scheme of relative dimension one over a scheme $S$ (which we may assume is affine Noetherian, but need not be reduced nor irreducible nor even connected). For $s \in ...
1
vote
1answer
63 views

When is the cokernel of a homomorphism of flat sheaves flat?

Let $X\to S$ be a scheme and let $D'$ and $D$ be relative effective Cartier divisors on $X$ satisfying $D' \subset D$ and let $D''$ satisfy $D = D' + D''$. Let $$0 \to \mathscr{O}_X(D'') \to ...
4
votes
1answer
237 views

Regular in codim one scheme and DVR

Let $X$ be a noetherian integral (separated) scheme which is regular in codimension one. Let $Y$ be a prime divisor and let $\eta$ be the generic point of $Y.$ It seems I am missing something easy but ...
8
votes
1answer
294 views

Möbius strip and $\mathscr O(-1)$. Or $\mathscr O(1)$?

On the real $\textbf P^1$ we have these algebraic line bundles: $\mathscr O(1)$ and $\mathscr O(-1)$. Which one corresponds to the Möbius strip? (Both are $1$-twists of $\textbf P^1\times\textbf ...
1
vote
1answer
119 views

Torsion free monoids and different definitions

I just read the following question Groupification and perfection of a commutative monoid and being also interested in the same topic, I found out the definition of torsion-free monoid I am using is ...
0
votes
1answer
113 views

If $f$ an isomorphism of ringed spaces, is $f$ necessarily an isomorphism of locally ringed spaces?

I'm not sure that this is generally true, but Harthorne p73 seems to suggest it. If it is true could someone give me a hint for the proof?
14
votes
2answers
1k views

Where do Chern classes live? $c_1(L)\in \textrm{?}$

If $X$ is a complex manifold, one can define the first Chern class of $L\in \textrm{Pic}\,X$ to be its image in $H^2(X,\textbf Z)$, by using the exponential sequence. So one can write something like ...
3
votes
1answer
168 views

etale fundamental group of a product

Let $X,Y$ be noetherian connected schemes over an algebraically closed field $k$ and let $\overline{x},\overline{y}$ be geometric points on them. There is a canonical homomorphism $\pi_1(X \times ...
18
votes
2answers
2k views

What are normal schemes intuitively?

A ring is called integrally closed if it is an integral domain and is equal to its integral closure in its field of fractions. A scheme is called normal if every stalk is integrally closed. Some ...