2
votes
1answer
57 views

In $\Bbb Z[x,y]$ is $(x^2+1,y^2+1,-xy+1)$ prime?

This is a reality check for the following computations that I did: Consider the map $(\operatorname{id}, \iota): \Bbb A_\Bbb Z^1 \rightarrow \Bbb A_\Bbb Z^1\times \Bbb A_\Bbb Z^1$ from the definition ...
2
votes
1answer
77 views

what are the “points” of the scheme $\mathbb{Z}_8[x] /(x^2 + 7)$

I noticed modulo 8 the quadratic $x^2 + 7$ is zero for four separate values $x = 1,3,5,7 \in \mathbb{Z}_8$. The number of zeros exceeds the degree. I would like to define the "variety" ...
0
votes
1answer
30 views

Non-Noetherian ring $R$ with Spec($R$) a Noetherian Scheme

In looking at the examples of Non-Noetherian rings I knew/found I wasn't able to find one where I could conclude that Spec($R$) was a Noetherian scheme (not just merely a Noetherian topological ...
3
votes
2answers
198 views

Scheme: Countable union of affine lines

Let $X$ be a countable union of $A_n$ ($n \in \Bbb{N}$), where $A_i$ are affine lines, i.e., $A_i=\operatorname{Spec}k[x]$, with $k$ algebraically closed field, such that $A_i$ meets $A_{i+1}$ in the ...
2
votes
1answer
66 views

Geometric reducedness (integral) versus reducedness (integral)

All the schemes here are over $\mathbb{C}$. Suppose $X \to Y$ is a morphism of varieties, then the geometric reducedness (integral) of the generic fibre implies the geometric reducedness (integral) ...
3
votes
0answers
111 views

Infinitesimal thickening of a smooth closed subscheme

Let $A$ be a noetherian ring (if it is useful I can assume that $A$ is an algebra of essentially finite type over a field) and $I \subset A$ is an ideal s.t. $A/I$ is smooth. Is it true that extension ...
4
votes
1answer
97 views

Definition of algebraic variety

In general, the definition of an algebraic variety differs from one reference to other. The definition that I was used to is to consider an algebraic variety as an integral scheme of finite type. ...
5
votes
0answers
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 ...
3
votes
1answer
48 views

Relationship between maps and maps of rings

I take this exercise from Ravi Vakil's book. Consider the map of complex manifolds sending $\mathbb{C} \rightarrow \mathbb{C}$ via $x \mapsto y=x^2$. We interpret $\mathbb{C}$ as the $x$-line and ...
1
vote
0answers
65 views

How can I calculate $\mathrm{Spec}(\mathbb{Z}_{(3)})$? And $\mathrm{Spec}(\mathbb{Z}_3)$?

Let $\mathbb{Z}_{(3)}$ be the localization (in $\mathbb{Z}$) of the ideal generated by $3$. So I have to put in $\mathbb{Z}$ all the inverses of the complement of $(3)$. How can I calculate ...
3
votes
1answer
76 views

Generic point of a curve in affine plane

Consider the affine plane over $k$, i.e. Spec $k[x,y]$. There are three kinds of prime ideals: $(0)$, $(x-a,y-b)$, and $(f(x,y))$, for $f$ irreducible. Let the ideal $(f(x,y))$ correspond to the point ...
3
votes
1answer
92 views

Does this morphism necessarily give rise to a finite extension of residue fields?

Let $f:X\rightarrow Y$ be a morphism of finite type of locally Notherian schemes. Let $x\in X$ and $y=f(x)$. Recall that $f$ is said to be unramified if the map of stalks $g:\mathcal O_{Y,y} ...
4
votes
1answer
78 views

Flat closed immersion into a Noetherian scheme is open

Let $X$ be an irreducible Noetherian scheme. Consider some flat closed immersion into it. I want to show that it is also open, so that the morphism is surjective. I have a few thoughts, but I can't ...
3
votes
2answers
111 views

Irreducibility is preserved under base extension

I want to prove that if $A$ is a finitely generated $k$-algebra ($k$ is a field) with prime nilradical then for any field extension $k\rightarrow K$, the $K$-algebra $A\otimes_kK$ has also prime ...
2
votes
1answer
118 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
131 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 ...
3
votes
1answer
320 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 ...
1
vote
1answer
49 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 = ...
1
vote
1answer
142 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
66 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, ...
4
votes
1answer
123 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
106 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 ...
0
votes
1answer
102 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?
1
vote
1answer
82 views

Relation between DVR's of a local domain and localizations of its integral closure.

$\textbf{1.}\,\,\,\,\,\,\,\,$ Let $(A,\mathfrak m_A)$ be a one dimensional local domain and let $B$ be its integral closure in the fraction field $L=\textrm{Frac}\,A$. Assume that $B$ is finitely ...
4
votes
0answers
107 views

Is reducedness an open condition?

If $X$ is a (general) scheme and $X$ is reduced at $p$, i.e. $\mathscr{O}_{X,p}$ is reduced, does there necessarily exist an open neighborhood of $p$ on which $X$ is reduced, i.e. $\mathscr{O}_X(U)$ ...
8
votes
1answer
549 views

Affine scheme $X$ with $\dim(X)=0$ but infinitely many points

As the title says, I'm looking for an affine scheme of dimension zero, but with infinitely many points. At first I doubted that something like this could exist, and I still can't think of an example, ...
6
votes
1answer
359 views

Prop. 2.3 Hartshorne: $\varphi:A\to B$ induces a morphism $\operatorname{Spec}(B)\to\operatorname{Spec}(A)$

I don't fully understand a step in the proof of the above-mentioned Proposition; more precisely, in part (b): If $\varphi:A\to B$ is a homomorphism of rings, $X=\operatorname{Spec}(A)$, ...
8
votes
1answer
291 views

The spectrum of a product of rings

Let $A$ be the product of a family $(A_i)_{i\in I}$ of commutative rings, and $c$ the canonical continuous map from the disjoint union $U$ of the spectra of the $A_i$ to the spectrum of $A$: $$ ...
0
votes
1answer
79 views

Exterior product of Modules, problem wih tensor product

Let $X$ and $Y$ be schemes over a field $k$ and $p,q$ the projections of $X \times Y$ on $X$ and $Y$. Let $M$ and $N$ be modules on $X$ and $Y$. Then the exterior product $M \boxtimes N $ is defined ...