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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82 views

example I-22 in Eisenbud's geometry of schemes

This is example I-22 in Eisenbud's Geometry of Schemes. Let $K$ be a field and $R=K[x]_{(x)}$ the localization of $K[x]$ at the maximal ideal (x). So $R$ is basically $K(x)$ except the elements where ...
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401 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 $A$: $$ ...
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1answer
446 views

Vanishing of Higher Direct Images

Let $X$ and $Y \ $ be smooth varieties over a field or - depending on the answers - more general nice schemes (I don't know what one needs exactly as conditions). Let $p: X\times Y \rightarrow X$ be ...
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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 ...
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371 views

divisor class group of a product of schemes

The first part of this question is quite general: let $X$ and $Y$ be noetherian integral separated schemes which are regular in codimension one. Is there any relationship between the divisor class ...
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538 views

Why is $\mathbb{A}^2$ isomorphic to $\operatorname{Spec}k[x,y]$ as ringed spaces

Suppose that $k$ is an algebraically closed field, and let $\mathbb{A}^2$ denote the affine $2$-space $k^2$. An affine scheme is defined to be a locally ringed space $(X, \mathcal{O}_X )$ which is ...
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86 views

Group structure on geometric vector bundles

Let $S$ be a scheme and $\mathcal A$ a quasicoherent $\mathcal O_S$-Algebra. One knows that then one can associate the affine $S-$scheme $Spec(\mathcal A)$ over $S$. In particular I can consider ...
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114 views

Points in integral separated schemes are determined by their local rings

I am a bit stuck with the following assertion: Let $X$ be a separated integral scheme. Then to every (schematic) point $x \in X$ we can correspond its local ring, and look at it as a subring of the ...
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1answer
89 views

Scheme of characteristic zero is Q-scheme?

Let $X$ be an arbitrary scheme. I think one calls $X$ a characteristic zero scheme if all the residue fields of $X$ are of characteristic zero, for any point $x$ on $X$. If $X$ is a scheme over ...
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1answer
66 views

Do nonsingular points get mapped to nonsingular points in a branched cover

Let $\pi:C\to D$ be a finite surjective morphism of noetherian integral schemes. Let $x\in C$ be a nonsingular point. Does it follow that $\pi(x)$ is nonsingular? What if we impose some conditions ...
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124 views

Is invertibility a stalk-local property?

Given a locally ringed space $(X,\mathcal O_X)$ and a global section $s\in \Gamma (X,\mathcal O_X)$, which does not vanish at any point $x\in X$, meaning the image of $s$ in the stalk $\mathcal ...
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1answer
78 views

Vanishing of morphism of vector bundles

Let $X$ be a sufficiently nice scheme and $\mathcal F$ a locally free sheaf of finite rank, i.e., a vector bundle on $X$. Let $x$ be point of $X$ and $k(x)$ it's residue field. Let $f:\mathcal F \to ...
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1answer
129 views

Sections of locally free sheaves

I tried to prove the following assertion, which I think is implicit in a text I have read, but I'm not sure about that: Let $X$ be an integral scheme and $\mathcal F$ a locally free sheaf of finite ...
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2answers
315 views

Spectral sequence for Ext

If I have a morphism of schemes $f:X\rightarrow Y$ and sheaves $\mathcal F,G$ on $X$, then is there a spectral sequence which relates the Ext-groups $\mathrm{Ext}(f_* \mathcal F, f_*\mathcal G)$ on ...
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1answer
206 views

What is the fibre product of these morphisms of schemes?

Let $f\colon Y\rightarrow T$, $g\colon Y\rightarrow X$, $h\colon T\rightarrow S$, $i:X\rightarrow S$ be a cartesian diagram. Then I have the diagonal morphism $\Delta\colon X \rightarrow ...
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2k views

Scheme of finite type over a field $K$ v.s. $K$-scheme

I'm lost in some definitions about schemes. I have some trouble about two definitions of a scheme of finite type over $K$, for an alg.closed field $K$. Version 1 (Hartshorn) : a scheme of finite type ...
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145 views

Ext groups of a point on a scheme

Given a scheme $X$ over a field $k$ and a closed point $x$ with residue field $k(x)$ and inclusion $i:{x}\rightarrow X$ one can consider the following abelian groups (1)$Ext^1_{\mathcal ...
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72 views

Interpretation of abstract Isomorphisms of Hom-sheaves

Let $X$ be a smooth $k-$ variety and $F$ a coherent sheaf on it. One knows that the sheaf of differentials $\Omega^1_X$ of $X$ is locally free. Hence one has by standard isomorphisms $Hom(F,F\otimes ...
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123 views

Connections on line bundles on product of varieties

Let $X$ and $Y$ be smooth projective varieties over $\mathbb C$. Suppose I have given a line bundle $L$ on $X\times Y$ with a connection relative to $Y$, i.e. $\nabla: L \rightarrow ...
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1answer
98 views

Tangent space in a point

Let $X$ be a nonsingular variety over an algebraically closed field and $x$ be a closed point on $X$. Then one defines the tangent space in $x$ as the $k-$ vector space $T_x(X):= ...
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309 views

Basics of schemes and morphisms of schemes

I'm currently reading through Hartshorne, and have come across a few things that have left me wondering. (i) Somewhat pedantic, but also because I don't actually know the answer, (in Example 2.3.4) ...
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214 views

Definition of Grothendieck Connection

I suppose this is not hard, but I thought quite a time about it and don't get what the point is. I'm reading Deligne's "Equations differentielles...", where he defines what a Grothendieck Connection ...
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303 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 ...
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1answer
120 views

Elementary Question about Adjunction Isomorphism

One knows by standard Algebraic Geometry that for any morphism $f:X \rightarrow Y$ of schemes one has canonical bijections $$\operatorname{Hom}_X(f*G,F)\simeq \operatorname{Hom}_Y(G,f_{*}F).$$ ...
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235 views

Dual Numbers and Infinitesimal Neighborhood

Let $X$ be a nice scheme over a field $k$ and $D=\operatorname{Spec}(k[t]/t^2)$ the dual numbers. One knows that to give a k-rational point on $X$ and a tangent vector in this point is equivalent to ...
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78 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 ...
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94 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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1answer
179 views

What does “Biextension of Abelian Varieties” mean?

If I have two schemes $X$ and $Y$, which are such that my question makes sense (I guess, they should be abelian varieties over a field $k$, so assume this). Then I have often read, but nowhere found ...
5
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1answer
243 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 ...
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88 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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97 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 ...
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Why does the definition of an open subscheme / open immersion of schemes allow for an “extra” isomorphism?

After taking an algebraic geometry course last year, I've been reviewing the material this year, and I remembered something that struck me as odd, but which I'd neglected to ask about at the time: ...
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1answer
150 views

Ample divisor on abelian variety

just a short question: if one has an abelian variety $X$ over a field $k$ and an ample irreducible divisor $D$ on $X$, then why is $H^1(X-D,\mathcal O_X)$ zero? Should it be that $X-D$ is affine? ...
3
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1answer
160 views

Geometry of abelian varieties

if $X$ and $Y$ are abelian varieties over a field $k$ and $f:X\rightarrow Y$ is a homomorphism of abelian varieties, are then the following true: 1) ...
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1answer
123 views

Confusion with closed subsets of variety

I have to annoy you just one further time with these closed subset stories. I am trying to make rigorous a proof, in which the author tries to show an equality of closed subsets $Y$ and $Z$ of an ...
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2answers
1k views

Closed points on varieties

I consider a variety over a field $k$, i.e. an integral separated scheme $X$ of finite type over $k$. One knows by the Nullstellensatz that any closed point on $X$ is a $\bar k-$ rational point ...
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1answer
287 views

Dual numbers and tangent vectors

I have a basic question concerning dual numbers and tangent vectors. If I have a scheme $S$ over a field $k$ and a closed $k-$rational point $s\in S$, then one knows that to give a tangent vector in ...
3
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1answer
116 views

Smooth ample hypersurface on variety

I read the following fact which wasn't explained further and wonder how you exactly get it. Maybe you can give me some hint. Start with a smooth projective variety $X$ over a $k$. Then the author ...
4
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2answers
190 views

Closed immersions and stalks

I am a bit confused about the following situation: if you have a closed immersion $i: Y\rightarrow X$ of schemes and a coherent sheaf $F$ on $X$ which has stalks only in $Y$ (i.e. $F_x$=0 for points ...
2
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1answer
240 views

Stalks of the graph of a morphism

I am interested in the graph $\Gamma_f$ of a morphism $f: X\rightarrow Y$ between two sufficiently nice schemes $X,Y$. One knows that it is a closed subset of $X\times Y$ (when the schemes are nice, ...
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3answers
336 views

Proj of graded rings

my question actually concerns an exercise II5.13 in Hartshorne. You have a graded ring $S=\oplus S_n$ with $n\ge0$ generated as $S_o$-Algebra by $S_1$ and you set $S^{(d)}=\oplus S_{dn}$ for a ...
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0answers
610 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 ...
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0answers
76 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 ...
2
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1answer
306 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} ...
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595 views

on the generic points of a scheme

This question may be a little bit metaphysical:are there any important properties about the generic points on a scheme?Or rather,why do we introduce the concept of generic point?I am not very clear ...
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678 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} ...
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2answers
267 views

Lifting sheaves from the special fibre to the generic fibre

Let $R$ be a complete discrete valuation ring and let $S = \operatorname{Spec}(R)$. Let $\mathcal{X}\to S$ be a complete, regular, flat, connected $S$-scheme of finite type whose fibres are smooth, ...
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0answers
43 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 ...
6
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1answer
296 views

bijective morphism of affine schemes

The following question occurred to me while doing exercises in Hartshorne. If $A \to B$ is a homomorphism of (commutative, unital) rings and $f : \text{Spec } B \to \text{Spec } A$ is the ...
5
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1answer
137 views

Sheaf cohomology of completion along a subvariety

Suppose we are given a subvariety $Y \subset X$ where $X$ is a projective non-singular variety. We also have a coherent sheaf $\mathcal{F}$ on $Y$. We look at the completion of $X$ along $Y$ and ...