Tagged Questions

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 ...

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 ...
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 ...
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 ...
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 ...
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 ...
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!
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 ...
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 ...
381 views

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: ...
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 $K$, ...
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 ...
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 ...
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 $f(Y) = U$, ...
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 ...
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 ...
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 ...
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 ...
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 ...
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?
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 ...
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 ...
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 $Z$ ...
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 ...
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 ...
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 ...
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 ...
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 ...
51 views

71 views

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 ...