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 ...

learn more… | top users | synonyms

24
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 ...
8
votes
0answers
125 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 ...
8
votes
0answers
281 views

Why do these two constructions of a sheaf associated to a module on a projective scheme agree?

Let $B$ be a graded ring an $M$ be a $\mathbb Z$-graded $B$-module. We can associate to $M$ a sheaf of modules on $\operatorname{Proj} B$ by defining the sheaf on the principal open sets $D_+(f)$ to ...
7
votes
0answers
180 views

Invertible rational functions

I am looking for references for the following facts. ...
6
votes
0answers
81 views

Vanishing of Tor sheaf on a union of subschemes with vanishing Tor.

Suppose $X$ is a scheme and $Y$ and $Z$ are closed subschemes such that $Y$ is a finite union of closed reduced irreducible subschemes $C$ that satisfy ...
6
votes
0answers
87 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 ...
5
votes
0answers
90 views

Eisenbud/Haris, Exercice I-53: morphism between global spectra

Let $X=\operatorname{Spec}(\mathscr{F})$ and $Y=\operatorname{Spec}(\mathscr{G})$ two global spectra over a scheme $S$ and let $f:X\to Y$ be a morphism. I want to show that for all prime ideal sheaf ...
4
votes
0answers
69 views

Splitting cotangent bundles over schemes

For smooth manifolds the followng is well known $T^*(M \times N) \cong p_1^*(T^*M)\oplus p_2^*(T^*N)$ as bundles over $M\times N$. Let $j: X \to M$ be an embedded submanifold and let $N^*X\subset ...
4
votes
0answers
97 views

Learning Galois theory geometrically?

Recently I started poking at algebraic geometry and commutative algebra. My background is basic category theory and basic algebraic topology. I don't know a lot of other mathematics. I noticed Galois ...
4
votes
0answers
44 views

Hartshorne problem III.5.2(a)

Consider problem III.5.2(a) in Hartshorne's Algebraic Geometry: Let $X$ be a projective scheme over a field $k$, let $\mathcal O_X(1)$ be a very ample invertible sheaf on $X$ over $k$, and let ...
4
votes
0answers
50 views

Birational map between reduced schemes.

I am reading Ravi Vakil's notes "Foundation of Algebraic Geometry" and in Proposition 6.5.5., it states the following: Suppose $X$ and $Y$ are reduced schemes. Then $X$ and $Y$ are birational if and ...
4
votes
0answers
62 views

Determining a scheme $X$ is affine from $Qcoh(X)$

My question is a subquestion of this question. I do not want to use full reconstruction theorems. The settings is the following. Let $X$ be a scheme. Assume that the adjunction in TAG01BH of the ...
4
votes
0answers
77 views

Utility and meaning of the relative setting in Scheme theory

I'm sorry if my question is rather trivial, but I'm starting to learn scheme theory and I have a very basic question. When talking about schemes I see that very often, instead of taking "a point of a ...
4
votes
0answers
29 views

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)
4
votes
0answers
125 views

Grothendieck group of a variety and Grothendieck group of its derived category

Let $X$ be a smooth projective variety over a field $K$. Let $K_0(X)$ denote the Grothendieck group of the abelian category of coherent sheaves on $X$. Let $\newcommand{\Pf}{\mathbf{Pf}}K_0(\Pf(X))$ ...
4
votes
0answers
148 views

Orbits of affine group scheme actions

Motivation Let $F$ and $K$ be distinct algebraically closed fields. Then $GL_n(F)$ acts on $M_n(F)$ by conjugation, and the nilpotent orbits of this action can be characterized by the Jordan normal ...
4
votes
0answers
100 views

Is there any equivalence between the category of schemes over $\mathbb R$ and the category real manifolds

The equivalence of the category of smooth projective curves over $\mathbb C$ and the category of compact Riemann surfaces is, I believe, well documented. For example, it is mentioned on the wiki page: ...
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
0answers
1k views

Proving that $\operatorname{Pic}^0(X) \times \operatorname{Pic}^0(Y) \cong \operatorname{Pic}^0(X \times Y)$

Let $k$ be a field of arbitrary characteristic and let $X$ and $Y$ be projective varieties over $k$. I have come across the formula $$\operatorname{Pic}^0(X) \times \operatorname{Pic}^0(Y) \cong ...
4
votes
0answers
130 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. $\mathscr{O}_X(U)$ ...
3
votes
0answers
30 views

Motivation of completions of schemes?

Sorry if this is a stupid question, but what are the uses of a completion of a scheme along a closed subscheme? Are there any nice universal properties it satisfies, or do certain morphisms factor ...
3
votes
0answers
49 views

Tricks to find $\dim H^0(O(D)(n))$

Let $X$ be a curve of genus $g\neq 0$ and $D$ a divisor on $X$. If $n \in \mathbb{Z}$ and $O(D)(n)=O(D)\otimes O(n)$ is the twist of $O(D)$ by $n$, there is a manageable proceedings to find the ...
3
votes
0answers
71 views

Sheaves with surjective map from the ring of global sections to stalks

Let $M$ be a smooth manifold, $\mathscr O_M$ the sheaf of vector fields. Then, by an argument involving partition of unity, we can show that the map $$\mathscr O_M(M)\to\mathscr O_{M,x}$$ is ...
3
votes
0answers
53 views

Confusion over common criterion for a line bundle on a scheme to be trivial?

Suppose you have a line bundle $L\to X$ on a scheme $X$. I'll denote by $\sigma\colon X\to L$ to be the zero section. Why is it that this bundle is trivial iff there is another section $s\colon X\to ...
3
votes
0answers
83 views

A computation related to Hironaka's Example

My questions are at the very end, first I'll describe the context. Let $f:\mathbb{P}^3\to \mathbb{P}^3$ be an involution whose fixed locus consists of two disjoint lines $L, L' \subset \mathbb{P}^3$. ...
3
votes
0answers
86 views

Hartshorne notation in section III.12

I am reading section III.12 in Hartshorne, the one about the Semicontinuity Theorem. For $f:X \rightarrow Y$, where $Y=\mathrm{Spec}A$ and $\mathcal{F}$ a coherent sheaf on $X$, he writes ...
3
votes
0answers
58 views

on what morphism of schemes look like locally

I am working on the following exercise (Exercise 6.3.C. on Ravi Vakil's online notes), which has two parts: Given a morphism of schemes $\pi: X \rightarrow Y$, show that if Spec $A$ is an affine open ...
3
votes
0answers
38 views

Definition of Formally Smooth from Stack Project

$T, T'$ are affine schemes. What is meant by $F\leftarrow T$ (or $G \leftarrow T')$
3
votes
0answers
45 views

trivialising cover for etale morphisms

Let $f:Y \to X$ be a finite etale morphism of smooth and proper schemes over a field $k$ (not necessarily sep closed). Is there a geometrically connected etale cover $\{U_i\}$ of $X$ which ...
3
votes
0answers
68 views

Why are morphisms of schemes locally given by ring homomorphisms?

I am trying to prove the statement: A morphism $f:Y \rightarrow X$ of schemes is determined locally by homomorphisms of rings. Here is my attempt: Given the morphism $f:Y \rightarrow X$ and any ...
3
votes
0answers
106 views

Weil restriction - from abstract nonsense to a practical procedure

Let $L/K$ be a finite field extension. Let $X$ be a $L$-scheme, that is $\exists$ a morphism of schemes $\pi : X \to \operatorname{Spec} L$. The Weil restriction of $X$ is a $K$-scheme $W_{L/K}X$ ...
3
votes
0answers
84 views

Hilbert Scheme and Chow variety in the case of Conics in $\mathbb{P}^{3}$

My question concerns the relationship between chow varieties and hilbert schemes in the case of conics in $\mathbb{P}^{3}_{k}$. More precisely, consider the Hilbert scheme $\mathsf{hilb}^{2t+1}_{3}$, ...
3
votes
0answers
64 views

Can Hom(X,Y) be given a natural scheme structure?

Let $X$ and $Y$ be $S$-schemes. Is there a "natural" scheme structure on $\operatorname{Hom}_S(X,Y)$, maybe subject to some conditions on the schemes in question? Interpret "natural" however you'd ...
3
votes
0answers
111 views

Noetherian schemes and varieties

What types of varieties (e.g. projective, affine,...) over a field $k$ (char = $0$) are Noetherian schemes?
3
votes
0answers
168 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 ...
3
votes
0answers
64 views

Valuative criteria with varieties

Let $X$ be an algebraic prevariety over an algebraically closed field $k$ (I.e. an integral scheme of finite type over $k$). In the valuative criteria for separatedness and properness of $X$ over ...
3
votes
0answers
204 views

Show that in a quasi-compact scheme every point has a closed point in its closure

Vakil 5.1 E Show that in a quasi-compact scheme every point has a closed point in its closure Solution: Let $X$ be a quasi-compact scheme so that it has a finite cover by open affines $U_i$. ...
3
votes
0answers
64 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 ...
3
votes
0answers
71 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 ...
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 ...
3
votes
0answers
141 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 ...
3
votes
0answers
450 views

“Hartog's Lemma” for locally free sheaves on a Noetherian normal scheme

Exercise 14.1.I of Ravi Vakil's notes is the following: Show that locally free sheaves on Noetherian normal schemes satisfy "Hartog's Lemma": sections defined away from a set of codimension at least ...
3
votes
0answers
650 views

Very ample line bundles and global sections

my question concerns a very ample line bundle $L$ on a projective k-scheme. It gives a closed immersion $i:X \rightarrow P^{N-1}_{k}$ to some projective space over k. It can be viewed as induced by ...
2
votes
0answers
23 views

A category of closed immersions

Fix a scheme $Z$, and consider a category whose objects are schemes $X$ equipped with a closed immersion $Z\to X$. Obviously, a morphism $f:X\to Y$ should commute with the respective closed ...
2
votes
0answers
39 views

Construction of Tate curve and formal schemes

In the notes websites.math.leidenuniv.nl/geom/tate.ps (and probably in other places), there is a construction of the Tate curve, where the steps are summarized below. 1) Take ...
2
votes
0answers
38 views

If $X$ is a proper scheme over $k$, is $X/G$ separated?

Let $X$ a proper scheme over a field $k$ and let $G$ a finite group of its automorphism (as $k$-scheme). Let suppose that the quotient $X/G$ exists, is it is separated? How to prove it? If the ...
2
votes
0answers
39 views

Can the theorem on semicontinuity of fiber dimension be explained as a semi-continuty of rank of a quasi-coherent sheaf?

Here are two theorems: If $F$ is a finite type quasi-coherent sheaf on a scheme $X$, then $ rank(F)(p) = dim_{k(p)} F_p \otimes_{O_p} k(p)$ is a upper semicontinuous function on $X$. If $\pi : X \to ...
2
votes
0answers
47 views

How should I think of schemes of the form $\operatorname{spec} k[f_1, \ldots, f_n]$, $f_i \in k[x_1, \ldots, x_m]$?

How should I think of schemes of the form $\operatorname{spec} k[f_1, \ldots, f_n]$, $f_i \in k[x_1, \ldots, x_m]$? What of the map on spectra induced by the inclusion $i : k[f_1, \ldots, f_n] \to ...
2
votes
0answers
37 views

Is there an example of a scheme $X$, irreducible, with a dense open subset $U$ so that $\dim U < \dim X$?

Is there an example of a scheme $X$, irreducible, with a dense open subset $U$ so that $\dim U < \dim X$? What about just for topological spaces? I know for varieties this doesn't happen (though ...
2
votes
0answers
23 views

Is this always quasi-projective?

Let $\phi:(X,x)\to (T,t)$ be a morphism of germs of algebraic schemes over $\mathbb{C}$. How can we always (without "quasi-projective") have an open embedding of the germ $(X,x)$ into a closed ...