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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18
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 ...
9
votes
0answers
195 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 ...
8
votes
0answers
231 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
148 views

Invertible rational functions

I am looking for references for the following facts. ...
6
votes
0answers
70 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
0answers
78 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
26 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
115 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
94 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
139 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
122 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
44 views
+50

Regular sequence of sections of line bundles over a coherent sheaf

I am reading the first chapter from the book by Huybrechts and Lehn, where I encountered the following definition. I have the following doubts regarding this definition : What is the map ...
3
votes
0answers
83 views

Equivalent definition of Schemes

I recall seeing that the category of schemes can be captured by a general construction as follows. Let $\mathbf{Spec}\colon \mathbf{CRing}^{op}\to \mathbf{LRS}$ be the usual functor from the ...
3
votes
0answers
37 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
33 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
79 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
61 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
58 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
56 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
125 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
48 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
58 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
67 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
85 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 $\newcommand{\Pf}{\mathbf{Pf}}K_0(\Pf(X))$ ...
3
votes
0answers
152 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
121 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
277 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 ...
2
votes
0answers
38 views

contracted products of torsors

I have a question about contracted products of torsors: Is $(A \times^B C) \times^D E \cong A \times^B (C \times^D E) $?
2
votes
0answers
40 views

When is the map “attaching irreducible components” an effective isomorphism?

Let $\mathcal C$ be a category with fiber products. We say that a morphism $X \to Y$ in $\mathcal C$ is an effective epimorphism if the sequence of sets $$ \text{Hom}(Y,S) \to \text{Hom}(X,S) ...
2
votes
0answers
41 views

Rational function on a noetherian scheme

Let $X$ be a noetherian scheme and $f$ a rational function on $X$, so by definition the domain of $X$ includes all associated points of $X$. I think the following is true: $f$ is regular on $X$ if and ...
2
votes
0answers
39 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 ...
2
votes
0answers
40 views

Unipotent Group Scheme

Can someone give me a reference of why this is true? Let A an associative k-algebra in wich every element is nilpotent, (non unital) and free of finite rank k-module, then for every commutative ...
2
votes
0answers
98 views

Closed points of a scheme (locally) of finite type over an algebraically closed field

If $X$ is an arbitrary scheme, I can prove that the set $X(k)$ of $k$-valued points is in bijective correspondence with the set $$\{(x,\iota) \, | \, x \in X, \, \iota:\kappa(x) \hookrightarrow k\},$$ ...
2
votes
0answers
120 views

Relative spectrum of a quasi-coherent algebra.

I'm working out the notion of spectrum of a quasi-coherent algebra over a scheme. $\require{AMScd}$ Let $(X,\mathcal O_X)$ be a scheme and $\mathcal A$ a quasi-coherent $\mathcal O_X$-algebra, i.e. a ...
2
votes
0answers
39 views

Do the cyclic or Hochschild homologies satisfy the addition axiom of Eilenberg Steenrod?

Do the cyclic or Hochschild homologies satisfy the addition axiom of ES? If so please provide a reference or proof (reference is preferable).
2
votes
0answers
135 views

Why does Fulton's Intersection Theory define $x \cdot_f y$ in this way?

His definition 8.1.1: Let $f:X\rightarrow Y$ be a morphism, with $Y$ non-singular of dimension $n.$ Let $p_X: X' \rightarrow X$, $p_Y: Y' \rightarrow Y$ be morphisms of schemes $X',Y'$ to $X$ and $Y$ ...
2
votes
0answers
96 views

Dimension of closure of set of closed points in a scheme

Let $X$ be a noetherian integral scheme. Let $Z\subset X$ be a set of closed points in $X$. What is the dimension of the closure of $Z$?
2
votes
0answers
69 views

Isomorphism of first infinitesimal neighborhoods

Take an abelian variety $X$ over a field and consider $Z$ to be the first infinitesimal neighborhood of the diagonal in $X\times X$. Let furthermore $Y$ be the first infinitesimal neighborhood of the ...
2
votes
0answers
96 views

Is there a $\mathbb{C}_g$, where $g$ is composite?

Analogous to the $p$-adic ring $\mathbb{Z}_p$, you can (at least formally), define the $g$-adic ring $\mathbb{Z}_g$, where $g$ is composite. Of course when completing to a field, you get in trouble ...
2
votes
0answers
516 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
42 views

Can we embed K(X_eta) canonically in K(X)

Let $f:X\longrightarrow S$ be a morphism of schemes. Assume $X$ and $S$ are integral. Let $\eta$ be the generic point of $S$ and let $X_\eta\longrightarrow \textrm{Spec} \ K(S)$ be the induced ...
1
vote
0answers
38 views

Closed sets flat morphism

Consider the sets $Y_{k}:=\{y\in Y\mid |f^{-1}(y)|\leq k\}$, where $f:X\to Y$ is a flat quasi-finite morphism between smooth irreducible affine varieties and $k$ a natural number. Are these sets ...
1
vote
0answers
13 views

Complexifying Lie group actions

In Atiyah-Pressley's paper Convexity and Loop Groups, it is claimed that the Bruhat cells $C_\lambda$ are invariant under the natural $S^1 \times T$-action and that the authors will prove this claim. ...
1
vote
0answers
40 views

Chow lemma + resolution of singularities

I am trying to understand the following simple reduction in Deligne's Theorie de Hodge II. He works in the category of schemes of finite type over $\mathrm{Spec}(\mathbf{C})$. Let $f : X \to S$ be a ...
1
vote
0answers
67 views

Some questions about Qing Liu's proof of Valuative Criterion of Properness

I'm studying algebraic geometry by following Mumford's Red Book, but recently I found Qing Liu's book "Algebraic Geometry and Arithmetic Curves", which seems to be more careful and detailed in its ...
1
vote
0answers
53 views

Point at infinity of the scheme $\mathbb{P}^1$

Let $\mathbb{P}^1_k$ be the scheme defined as either $\text{Proj }k[t]$ or obtained by gluing two affine lines appropriately. What is the point at infinity which in topology usually is given by ...
1
vote
0answers
27 views

Split extension of linear algebraic groups given by unipotent radicals and reductive groups

I've read that there is a split extension of any linear algebraic group $G$ over a perfect field given by $$ 1 \to R_u(G) \to G \to H \to 1$$ where $R_u(G)$ is the radical unipotent subgroup of $G$ ...
1
vote
0answers
55 views

Weil restriction for schemes

I'm trying to understand the Weil restriction of a scheme (since I'm reading a paper which uses it). I'm even having troubles trying understanding the following "toy" example. Toy example. Let $X$ be ...
1
vote
0answers
13 views

Group schemes decomposition

Given an abelian group scheme of finite type $(G,+)$ over $\mathbb{F}$ connected, and given two connected closed subgroup schemes of finite type $G$ over $\mathbb{F}$ connected $H$, $N$ of $G$. ...