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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1answer
203 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 ...
8
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1answer
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 ...
3
votes
1answer
139 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 ...
2
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1answer
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 ...
4
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1answer
122 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 ...
2
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1answer
97 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):= ...
7
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1answer
303 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) ...
4
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1answer
195 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 ...
9
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1answer
291 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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vote
1answer
119 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 $$\operatorname{Hom}_X(f*G,F)\simeq \operatorname{Hom}_Y(G,f_{*}F).$$ ...
5
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1answer
231 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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0answers
75 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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0answers
93 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) ...
3
votes
1answer
177 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
votes
1answer
232 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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0answers
87 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 $X$ ...
2
votes
0answers
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 ...
11
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1answer
1k views

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: ...
1
vote
1answer
147 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
votes
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) ...
0
votes
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 ...
4
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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 ...
1
vote
1answer
277 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
votes
1answer
110 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
votes
2answers
189 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
votes
1answer
239 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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votes
3answers
331 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 ...
2
votes
0answers
585 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 ...
1
vote
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
votes
1answer
302 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} ...
6
votes
2answers
566 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 ...
4
votes
2answers
660 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} ...
5
votes
2answers
264 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, ...
2
votes
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
votes
1answer
290 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
votes
1answer
135 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 ...
2
votes
1answer
373 views

Reduction map from the generic to the special fibre

I have a few basic questions about Liu's [1, Section 10.1.3] description of the reduction map from the closed points of the generic fibre of a proper scheme over a complete DVR to its special fibre. ...
26
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 ...
2
votes
1answer
1k views

“Surjectivity is stable under base change” and field compositums

If $f:X\rightarrow Y$ is a surjective morphism of schemes and $g:X'\rightarrow Y$ is another morphism of schemes, one can show that $p_{2}:X\times_{Y}X'\rightarrow X'$ is also surjective. ...
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0answers
125 views

Isomorphism of schemes and invertible sheaves

I have a question more about terminology than anything else: If $f:X\rightarrow Y$ is an isomorphism of schemes, then what does it mean to say "the invertible sheaf $\mathscr{L}$ on $X$ corresponds ...
3
votes
1answer
327 views

Connected components of a fiber product of schemes

The underlying set of the product $X \times Y$ of two schemes is by no means the set-theoretic product of the underlying sets of $X$ and $Y$. Although I am happy with the abstract definition of fiber ...
12
votes
3answers
2k views

Learning schemes

Could someone suggest me how to learn some basic theory of schemes? I have two books from algebraic geometry, namely "Diophantine Geometry" from Hindry and Silverman and "Algebraic geometry and ...
17
votes
1answer
457 views

Can an integral scheme have closed points of both positive and zero characteristic?

Background Recall that an integral scheme $X$ is a scheme which is both irreducible and reduced; equivalently, its ring of functions is an integral domain on every open subset. Given any point $p$, ...
4
votes
2answers
325 views

Stalks on Projective Scheme

Let $k$ be an algebraic closed field. Let $x$ be a point in $X=P_k^1$. What is $O_{X,x}$? For example, if I have $x=(t-a)\in \text{Spec }k[t]$. Looking $x$ inside $P_k^1$, does ...
6
votes
1answer
185 views

Have I got the right definition of formal smoothness?

I'm trying to work out a basic example where formal smoothness should fail. I'm considering $\mathbb{R} \to \mathbb{R}[x,y]/(x^2-y^2)$. The idea is that not every $\mathbb{R}$-homomorphism ...
3
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1answer
294 views

What is a G-Galois Branched Cover

What is, in the language of Schemes, a G-galois branched cover?
4
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1answer
470 views

Fiber product of varieties vs schemes reference

Given two complex varieties over a common base, I can take their fiber product in the category of varieties, or I can take their fiber product in the category of schemes and then take the reduced ...
21
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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 ...
12
votes
1answer
465 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. ...