The concept of scheme is from algebraic geometry: A locally ringed space that is glued together (much like a manifold) from affine schemes, that is spectra of rings with Zariski topology and structure sheaf.

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16
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1k 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
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197 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
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158 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 ...
6
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63 views

Schemes to the rescue?

I am reading chapter 1 of Gille and Szamuely's book Central Simple Algebras and Galois Cohomology, on quaternion algebras. In it they prove (remark 1.3.1 on p. 18) that the conic associated to a ...
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126 views

Invertible rational functions

I am looking for references for the following facts. ...
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62 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
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69 views

A morphism from $\mathbb P^1_\mathbb C$ to $\mathbb P^1_\mathbb C$

Consider the projective scheme $\mathbb P^1_\mathbb C$ that is different from the projective line $\mathbb P^1(\mathbb C)$. Now look at the following lemma: I don't understand what is a ...
4
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69 views

Global generation of $\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)$ and $\mathcal{E}$

I'm trying to prove that $\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)$ is generated by global sections on $\mathbb{P}(\mathcal{E})$ if and only if $\mathcal{E}$ is generated by global sections ($\pi: ...
4
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23 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
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97 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 ...
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84 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: ...
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132 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
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794 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 ...
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101 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
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34 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
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48 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
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42 views

Noetherian schemes and varieties

What types of varieties (e.g. projective, affine,...) over a field $k$ (char = $0$) are Noetherian schemes?
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80 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
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44 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
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45 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 ...
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61 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
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65 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
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99 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
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213 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
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37 views

Does $\operatorname{Proj}(\sigma)$ fix some points?

Consider a subfield $K\subseteq\mathbb C$, then by some properties of the fibered product of schemes we have that: $$\mathbb P^1_\mathbb C\cong\mathbb ...
2
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22 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
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33 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).
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135 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 ...
2
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121 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 $Y$ ...
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91 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
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65 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
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94 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
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414 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
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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 ...
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36 views

Check a non-projective morphism

The following is from the wiki: http://en.wikipedia.org/wiki/Proper_morphism Projective morphisms are proper, but not all proper morphisms are projective. For example, it can be shown that the ...
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19 views

The Chow variety of conics in $\mathbb{P}^{3}_{k}$

Consider the family of all conics in $\mathbb{P}^{3}_{k}$, with $k$ an algebraically closed field. Such curves all have degree $2$ and genus $0$, and they can be uniquely defined to be all curves in ...
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35 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 $$\{(x,\iota) \, | \, x \in X, \, \iota:\kappa(x) \hookrightarrow k\},$$ ...
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39 views

The localization exact sequence.

Let $X$ a scheme and $Y$ a close subscheme and $i:Y \to X$ the inclusion. Let $U:=(X-Y)$ and $j:U \to X$ the inclusion map. If I denote with $CH(-)$ the Chow group, the sequence $$ CH_r(Y) ...
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70 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 ...
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70 views

Projective scheme

How can I prove that the two different construction of $\mathbb{P}_k^n$ (as $Proj K[x_0,x_1,...,x_n]$ and by gluing copies of $\mathbb{A}_k^n$) agree? And how can I prove that if $A$ is reduced also ...
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70 views

Closed sets of $Spec(R)$.

I have to dounbs: Let $R$ be the ring $\mathbb{C}[x]_{(x)}$. So we have that $Spec(R)=\{(0),(x)\}$. I have to descrive closed sets of $Spec(R)$. I think that they are $\{\emptyset, Spec(R), (x) \}$ ...
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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 ...
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25 views

von Neumann Stability help

Using the forward time centered space scheme, I transformed the equation: $u_t-2u_{xx}-u_{yy}=0$ to ...
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50 views

Algebraic analogue of maximum modulus principle applied to Riemann surface.

Let $X$ be an abstract curve in the following sense: $X$ is a scheme, proper over $k$ which is noetherian, integral, dimension 1, and normal. The important thing to point out is that I am not assuming ...
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51 views

Morphisms from the group variety of $n$-th roots

In Milne - Lectures on ├ętale 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 ...
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89 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 ...
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120 views

Why topological stratification is useful?

My main focus is on the applications of stratification in complex/abstract Algebraic geometry especially, from the scheme-theoretic viewpoint and (Added) Moduli spaces. I have a vague feeling that ...
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81 views

Morphisms of abelian variety and torus in additive group

let $A$ be an abelian variety and $T$ be an algebraic torus over a field $k$; furthermore denote with $\mathbb G_a$ the additive group over $k$, i.e. just the affine space. Why does then hold (i) ...
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60 views

Segre classes of singular projective varieties

Corollary 4.2.4 from Fulton's Intersection Theory gives a method for computing the Segre class of varieties, but in particular it allows computation of the Segre class for singular varieties. Let $X$ ...
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62 views

Tangent vectors and length two subschemes

can someone explain what the datum of a point $x$ on a scheme $X$ together with a tangent direction has to do with subschemes of length two concentrated in that point? It seems as if there would be ...