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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27
votes
2answers
2k views

Diophantine applications of Spec?

Let $f(\bar x)$ be a multivariable polynomial with integer coefficients. The zeros of that polynomial are in bijection with the homomorphisms $\mathbb Z[\bar x] \rightarrow \mathbb Z$ that factor ...
6
votes
2answers
859 views

on the adjointness of the global section functor and the Spec functor

In fact, it is an exercise on Hartshorne, Ex 2.4 to the second chapter of it (P79): let $A$ be a ring and $(X,\mathcal{O}_X)$ be a scheme. Given a morphism $f:X\longrightarrow \operatorname{Spec} A$,...
10
votes
3answers
678 views

Krull Dimension of a scheme

Can someone give a hint or a solution for showing that a scheme has Krull dimension $d$ if and only if there is an affine open cover of the scheme such that the Krull dimension of each affine scheme ...
3
votes
3answers
2k views

Definition of degree of finite morphism plus context

Let $f: X \rightarrow Y$ be a finite morphism of schemes, defined here, http://en.wikipedia.org/wiki/Finite_morphism I always assumed that the degree of $f$ was the degree of the induced field ...
5
votes
2answers
234 views

How is the notion of adjunction of two functors usefull?

Is there a secret or an intuitive idea behind the fact of creating the concept of adjunction of two functors ( Functor - Adjoint Functor ) ? How is this notion of adjunction of two functors usefull ? ...
15
votes
1answer
1k views

On limits, schemes and Spec functor

I have several related questions: Do there exist colimits in the category of schemes? If not, do there exist just direct limits? Do there exist limits? If not, do there exist just inverse limits? ...
6
votes
1answer
249 views

“This property is local on” : properties of morphisms of $S$-schemes

I am learning schemes theory at school and I have for now only lectures notes that I am taking during the course. The professor is quite often using following expressions, without having defined them :...
13
votes
1answer
297 views

When can stalks be glued to recover a sheaf?

Let $\mathcal{F}$ be a sheaf over some topological space. The stalks are $\mathcal{F}_x= \underset{{x\in U}}{ \underrightarrow{\lim}} \mathcal{F}(U)$. Is there a special name for a sheaf that ...
9
votes
1answer
341 views

Tangent space in a point and First Ext group

Let $X$ be an abelian variety over an algebraically closed field $k$. I have read that one has for an arbitrary closed point $x$ on $X$ a canonical identification $$T_x(X)\simeq \operatorname{Ext}^1(...
5
votes
2answers
226 views

An exercise in Liu regarding a sheaf of ideals (Chapter II 3.4)

I'm fairly certain there is both a typo and an omission in this exercise. It reads "Let $X$ be a scheme and $f \in \mathcal{O}_X(X)$. Show that $U \mapsto f|_U \mathcal{O}_X(U)$ for every affine open ...
8
votes
2answers
847 views

Why is every Noetherian zero-dimensional scheme finite discrete?

In the book The geometry of schemes by Eisenbud and Harris, at page 27 we find the exercise asserting that Exercise I.XXXVI. The underlying space of a zero-dimensional scheme is discrete; if ...
4
votes
2answers
584 views

Intersection of open affines can be covered by open sets distinguished in *both*affines

Suppose $X$ is an arbitrary scheme and $U \cong \operatorname{Spec} A$ and $V \cong \operatorname{Spec} B$ are affine upon subsets of $X$. It's not true in general that $U \cap V$ is affine, so if we ...
4
votes
1answer
266 views

Zariski Tangent Space and $k[\varepsilon]/\varepsilon^2$

Let $X$ be a scheme over a field $k$, and let $x\in X$ be a rational point, that is, we have $k(x):=\mathcal{O}_x/\mathfrak{m}_x\cong k$. Let $\alpha:\mathfrak{m}_x/\mathfrak{m}_x^2\rightarrow k$ be ...
2
votes
1answer
323 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 ...
1
vote
1answer
390 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 ...
6
votes
1answer
190 views

Is the global section ring of a Noetherian Scheme Noetherian as well?

As the title suggests, I am asked to prove that, given a Noetherian scheme $(X,\ \mathcal{O}_{X})$ and any open subset $U\subseteq X$, $\Gamma(U,\ \mathcal{O}_{X}):=\mathcal{O}_{X}(U)$ is a Noetherian ...
4
votes
1answer
303 views

The image of the diagonal map in scheme

Let $X\rightarrow S$ be a separated morphism of schemes, that is, the diagonal map $\Delta:X\rightarrow X\times_S X$ is a closed inmersion. In general (see Closure of image of diagonal morphism of S-...
4
votes
2answers
179 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 ...
31
votes
0answers
2k views

Pullback and Pushforward Isomorphism of Sheaves

Suppose we have two schemes $X, Y$ and a map $f: X\to Y$. Then we know that $\operatorname{Hom}_X(f^*\mathcal{G}, \mathcal{F})\simeq \operatorname{Hom}_Y(\mathcal{G}, f_*\mathcal{F})$, where $\mathcal{...
6
votes
2answers
838 views

Geometric interpretation and computation of the Normal bundle

My question concern the definition, geometric meaning, and usage of the normal bundle in algebraic geometry. Let $X$ be a nonsingular variety over an algebraically closed field $k$, and $Y\subseteq X$...
27
votes
1answer
824 views

Quasi-coherent sheaves, schemes, and the Gabriel-Rosenberg theorem

In the context of commutative rings, a ring is completely determined by its category of modules. That is, two commutative rings $R$ and $S$ are isomorphic if and only if the category of $R$-modules is ...
11
votes
2answers
1k views

How to think of the Zariski tangent space

The Zariski tangent space at a point $\mathfrak m$ is defined as the dual of $\mathfrak m/\mathfrak m ^2$. While I do appreciate this definition, I find it hard to work with, because we are not given ...
10
votes
3answers
651 views

When is the pushforward of a quasi-coherent sheaf quasi-coherent? Hartshorne proof

In the following we have $f:X \rightarrow Y$ morphism of schemes, $\mathcal{F}$ a quasi-coherent sheaf on $X$ and I am referring to proposition 5.8 page 115 in Hartshorne. To prove that the ...
9
votes
1answer
307 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 $\textbf P^1\times\textbf A^1$...
11
votes
1answer
453 views

Problem about Complete Intersection in $\textbf P^n$ (from Hartshorne).

I am in trouble with Exercise 8.4 in Hartshorne's Chapter II; I am doing part (a). It is about (global) complete intersection in $\textbf P^n$. For those without Hartshorne' book at hand, I describe ...
5
votes
3answers
654 views

Open properties of quasi-compact schemes

I am following Ravi Vakil's Math 216: Foundations of Algebraic geometry notes, and there is a remark following an exercise that I don't understand at all, and if anyone could enlighten me then that ...
12
votes
1answer
562 views

Why do we need noetherianness (or something like it) for Serre's criterion for affineness?

Serre's criterion for affineness (Hartshorne III.3.7) states that: Let $X$ be a noetherian scheme. Suppose $H^1(X, \mathcal{F})= 0$ for every quasi-coherent sheaf on $X$. Then $X$ is affine. ...
10
votes
1answer
909 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, ...
10
votes
2answers
625 views

Inverse of open affine subscheme is affine

This seems ridiculously simple, but it's eluding me. Suppose $f:X\rightarrow Y$ is a morphism of affine schemes. Let $V$ be an open affine subscheme of $Y$. Why is $f^{-1}(V)$ affine? I noted that $...
7
votes
2answers
353 views

Scheme over S and morphisms

Quoting from Hartshorne Let $S$ be a fixed scheme. A scheme over $S$ is a scheme $X$, together with a morphism $X \to S$. If $X$ and $Y$ are schemes over $S$, a morphism of $X$ to $Y$ as schemes ...
3
votes
1answer
209 views

Closure of image of diagonal morphism of S-scheme

Let $X$ be an $S$-scheme with structural morphism given by $f : X \to S$. The image of the diagonal morphism $\Delta : X \to X \times_S X$ is contained in the subset $Z := \{ z \in X \times_S X : p(z)...
2
votes
1answer
355 views

Proj construction and ample dualizing sheaf

my question concerns a smooth projective variety $X$ with dualizing sheaf $\omega_X$: if I have that this dualizing sheaf is ample, then I have read you can conclude that $X\simeq Proj(\oplus_{k} H^{...
11
votes
1answer
285 views

Reality check: $\mathcal{O}_{\mathbb{P}_\mathbb{R}^1}(1)$ and $\mathcal{O}_{\mathbb{P}_\mathbb{R}^1}(-1)$ are both Möbius strips?

This question follows up this previous question, which has an accepted answer that I am having trouble believing. (Edit: the answer has now been fixed; thanks Georges!) I have been working on giving ...
9
votes
1answer
202 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 ...
8
votes
1answer
289 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 ...
6
votes
1answer
194 views

Separated schemes and unicity of extension

In point set topology, we have the following result, which is easily proved. Theorem. Let $Y$ be Hausdorff space and $f,g:X \to Y$ be continuous functions. If there exists a set $A\subset X$ such ...
5
votes
2answers
174 views

Isomorphism of Proj schemes of graded rings, Hartshorne 2.14

This question is based on exercise $2.14$ of chapter $2$ of Hartshorne. Suppose $\varphi:S\rightarrow T$ is a graded homomorphism of graded (commutative, unital) rings such that $\varphi_d := \...
5
votes
1answer
594 views

When are complete intersections also local complete intersections?

First, let us recall some definitions. Let $P=\mathbb P^d_k$ be a projective space over a field $k$. Let $X$ be a closed subvariety of $P$ of dimension $r$. We say that $X$ is a complete ...
3
votes
2answers
225 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 ...
3
votes
1answer
283 views

“Push-Pull” Morphisms of Higher Direct Image Sheaves

(This is 20.7.B in Ravi Vakil's notes) Suppose $f:Y \to Z$ is any morphism, and $\pi: X\to Y$ is quasicompact and quasiseparated. Suppose $\mathcal{F}$ is a quasicoherent sheaf on $X$. Let $$\...
2
votes
1answer
79 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 $X=\...
2
votes
1answer
131 views

Scheme glued out of finitely many spectra of local rings

Let $X$ be a noetherian separated scheme over the spectrum of a field $K$. By definition, I can find an open covering of $X$ by finitely many affine schemes $Spec(R_i)$ where each $R_i$ is a $K$-...
8
votes
1answer
607 views

Basic understanding of Spec$(\mathbb Z)$

So, I'm looking into schemes, and found that I have no intuition in the field, so I decided to look into some simple (as in affine and well-known) examples. As I like to dwell on the basics for a ...
8
votes
1answer
416 views

Toric Varieties: gluing of affine varieties (blow-up example)

Let $\Delta$ be a fan, consisting of cones $\sigma_0=conv(e_1,e_1+e_2)$ and $\sigma_2=conv(e_1+e_2,e_2)$ and $\tau=\sigma_0\cap\sigma_1=conv(e_1+e_2)$. The dual cones are $\sigma_0^\vee= conv(e_2,e_1\!...
7
votes
1answer
230 views

Why are projective spaces over a ring of different dimensions non-isomorphic?

Let $A$ be a nonzero commutative ring with unit. Define $\mathbb P_A^n$ to be the scheme $\operatorname {Proj} A[T_0,\dots,T_n]$, where the grading on the polynomial ring is by degree. Why is it true ...
5
votes
1answer
317 views

Hartshorne ex III.10.2 on smooth morphisms

I need some help with the following exercise: Let $f:X\rightarrow Y$ be a flat proper morphism between varieties over $k$, where variety means separated, finite type, integral, and $k$ not ...
5
votes
1answer
1k views

Finite morphisms of schemes are closed

I want to prove that finite morphisms of schemes are closed, but I cannot prove the affine case, namely: Given a finite morphism of rings $\varphi :B \to A$ prove that the induced morphism of ...
5
votes
1answer
291 views

First Ext group of a sheaf

Let $F$ be a quasicoherent sheaf on a scheme $X$, which is supposed to be sufficiently nice. Does one then have a canonical isomorphism $Ext^1(F,F) \simeq H^1(X, \underline{End}(F))$, where with $\...
4
votes
1answer
116 views

Question on morphism locally of finite type

The exercise 3.1 in GTM 52 by Hartshorne require to prove that $f:X \longrightarrow Y$ is locally of finite type iff for every open affine subset $V=\text{Spec}B$, $f^{-1}(V)$ can be covered by open ...
4
votes
1answer
271 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 ...