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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2
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1answer
43 views

Confused about short exact sequence involving $\mathcal{O}_{\mathbb{P}^n}$ and $\mathcal{O}_Z$ for $\pi:Z\rightarrow \mathbb{P}^n$ closed embedding

If $\pi: Z\rightarrow \mathbb{P}^n$ is a closed embedding where $Z$ is the zero set of some degree homogeneous polynomial, we have: $$0\rightarrow \mathcal{O}_{\mathbb{P}^n}(-d)\rightarrow ...
4
votes
1answer
68 views

Layman's Question on Schemes

I am reading Jordan Ellenberg's article on Arithmetic Geometry in the Princeton Companion to Mathematics. I have forgotten most of the algebra I learned since passing my qualifying exams more than 30 ...
0
votes
1answer
48 views

$(-1)$-curves and base change along a field automorphism.

Suppose that $E\subseteq S$ is a $(-1)$-curve inside a non-singular complex projective surface. By a $(-1)$-curve $E$, I mean that $E\cong\mathbb P^1_\mathbb C$ and $E^2=-1$. Now consider a field ...
2
votes
1answer
55 views

Varieties over a field $K$ are also varieties over any subfield of $K$.

Suppose that $f:X\longrightarrow\text{Spec} K$ is a variety over $K$, namely $X$ is an integral, separated $K$-scheme of finite type. Now if $L$ is a subfield of $K$, it is clear that there exists a ...
2
votes
1answer
101 views

Prove that two definitions are equivalent

Preliminary notion: Suppose that $X$ is an algebraic variety over a field $K$ ($K$-scheme, integral separated, of finite type). If $L$ is a subfield of $K$, we say that $X$ is defined over $L$ if ...
6
votes
1answer
85 views

What are the points of some schemes?

Let $X=\operatorname{Spec}\mathbb{C}[x,y,t]/(xy-t)$, $Y=\operatorname{Spec}K[x,y]/(xy-t)\rightarrow \operatorname{Spec}K$ and $Z=\operatorname{Spec}R[x,y]/(xy-t)\rightarrow \operatorname{Spec}R$, ...
0
votes
0answers
20 views

QR decomposition, borel groups and generalizations

Then every matrix $M$ in $M_{m\times m} (\mathbb{C})$ can be written in the form: $QR=M$, where $Q$ is unitary and $R$ is upper-triangular. My question is simple, does this generalize in the ...
0
votes
1answer
35 views

Existence of a rational function on a nonsingular curve with a simple zero at P and order 0 at Q

Let $X$ be a nonsingular curve over an algebraically closed field $k$ (by curve over $k$, I mean an integral separated scheme of finite type over $k$ which has dimension 1). Let $P$ and $Q$ be ...
1
vote
2answers
90 views

Eisenbud & Harris's The Geometry of Schemes proof of Prop I-18

There seems to be an error in the proof of Proposition I-18. The first inset equation (line 7) on page 20 only holds in $X_{f_a f_b}$. But it is used on line 15, where it needs to hold in $X_{f_b}$. ...
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 $G$ ...
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) $?
0
votes
1answer
50 views

Fibred product of schemes

I'm learning algebraic geometry, and I'm having some difficulties in developing intuition for the fibred product of schemes. I can take schemes to be over a field (but not necessarily separably ...
0
votes
1answer
65 views

short exact sequences of linear algebraic groups and $K$-forms

This is probably a stupid question, but I can't figure it out. Let $G, G', G''$ be linear algebraic groups over a characteristic $0$ field $K$. Say that $G' \in A$ and $G'' \in B$, where $A,B$ are ...
1
vote
1answer
70 views

degree of morphism of schemes

Let $\phi: Y \to X$ be a finite etale morphism of proper smooth connected schemes over a field $K$ and suppose that the induced morphism $\phi: \overline{Y} \to \overline{X}$ has degree $n$, where ...
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 ...
1
vote
1answer
88 views

Fibres of the map $Spec\mathbb{Z}[x] \rightarrow Spec\mathbb{Z}$

I am trying to understand what is a spectrum of the ring $\mathbb{Z}[x]$. I have read Spectrum of $\mathbb{Z}[x]$ but because of my very restricted knowledge of schemes I do not understand the ...
1
vote
1answer
78 views

etale neighborhoods

I've read that quasi-compact etale morphisms of schemes over a not necessarily algebraiclly closed field $F$ (I'm happy to take $F$ a field of char $0$) are the algebraic analogs of local ...
0
votes
1answer
43 views

Notation in the Semicontinuity Theorem

In Hartshorne's book Algebraic Geometry in Chapter III Section 12 we have the following situation: $f\colon X\to Y$ is a projective morphism of schemes, $Y$ is the affine spectrume of a ring $A$ , ...
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) ...
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 ...
2
votes
1answer
53 views

Morphism schemes after base extension

I have been reading Ravi Vakil notes on algebraic geometry, and one exercise asks if $phi:X\rightarrow Y$ and $\pi:X\rightarrow Y$ are morphisms of $k$-schemes and $\ell/k$ a field extension, if ...
0
votes
0answers
31 views

Invariant differentials on group schemes

I'm studying group schemes from http://www.math.ru.nl/~bmoonen/BookAV/BasGrSch.pdf and I have some trouble with the following proposition. (3.15)Proposition Let $\pi:G\to S$ be a group scheme. Then ...
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 ...
1
vote
1answer
53 views

Why is : $ \ \ V \times_U W = p^{-1} (V) \bigcap q^{-1} (W) $?

Let $ f: Y \to X $ and $ g : Z \to X $ be two morphisms of schemes. Suppose we know that $ Y \times_X Z $ exists, and let $ p $ dénote its projection on $ Y $, and $ q $ its projection on $ Z $. ...
0
votes
2answers
131 views

Associated points and reduced scheme

1) Let X is a locally Noetherian scheme without embedded point, show that X is reduced iff it is reduced at the generic points. 2) Let X is a locally Noetherian scheme (maybe has some embedded ...
1
vote
1answer
68 views

A question on smooth morphisms and 'pointwise' smooth morphisms

Let $X$ be a scheme, $x\in X$ a point and $f\colon \operatorname{Spec}(k(x))\to X$ the canonical morphism. Is $f$ always a smooth morphism? Now suppose $g\colon X\to Y$ is a scheme over some ...
2
votes
1answer
64 views

In $\Bbb Z[x,y]$ is $(x^2+1,y^2+1,-xy+1)$ prime?

This is a reality check for the following computations that I did: Consider the map $(\operatorname{id}, \iota): \Bbb A_\Bbb Z^1 \rightarrow \Bbb A_\Bbb Z^1\times \Bbb A_\Bbb Z^1$ from the definition ...
1
vote
1answer
82 views

Finite pushforward commute with taking cohomology

Let $f: X \to Y$ be a finite morphism of schemes. How one can show that $f_*H^i(G) \cong H^i(f_* G)$ for any $G \in D(X)$ and any $i \in \mathbb{Z}$? In english, $G$ is a complex of quasi-coherent ...
0
votes
1answer
38 views

The structure morphism of a projective variety induces a morphism of $k$-algeras

Suppose that $k$ is an algebraically closed field and that $X=\textrm{Proj}{\frac{k[T_1,\ldots,T_n]}{(f_1,\ldots,f_n)}}$ is a projective variety with a structural morphism $p:X\rightarrow\textrm{Spec} ...
14
votes
2answers
636 views

History of Algebraic Geometry: Motivation behind definition of schemes

I am trying to read an article by Jean Dieudonne which talks about development of Algebraic Geometry. The article was being published in the journal "Advances in Mathematics" Volume 3, Issue 3, Pages ...
3
votes
1answer
45 views

Surjective étale morphisms on points. [closed]

Let $X$ and $Y$ be schemes over a field $K$. We assume, moreover, $X$ and $Y$ to be of finite type, separated and geometrically integral. Let $f:X \rightarrow Y$ be a surjective étale morphism. Is it ...
2
votes
1answer
84 views

what are the “points” of the scheme $\mathbb{Z}_8[x] /(x^2 + 7)$

I noticed modulo 8 the quadratic $x^2 + 7$ is zero for four separate values $x = 1,3,5,7 \in \mathbb{Z}_8$. The number of zeros exceeds the degree. I would like to define the "variety" ...
3
votes
1answer
167 views

Hartshorne ex III.10.2 on smooth morphisms

I need some help with the following exercise: Let $f:X\rightarrow Y$ be a flat proper morphism between varieties over $k$, where variety means separated, finite type, integral, and $k$ not ...
3
votes
0answers
81 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 $W_{L/K}X$ ...
0
votes
1answer
45 views

Non-Noetherian ring $R$ with Spec($R$) a Noetherian Scheme

In looking at the examples of Non-Noetherian rings I knew/found I wasn't able to find one where I could conclude that Spec($R$) was a Noetherian scheme (not just merely a Noetherian topological ...
0
votes
0answers
65 views

Grothendieck topology

Hello : Here is a small parapgraph that i try to understand : If $ M $ is a smooth manifold, then we can recover the underlying set of $ M $ by considering the set $ \mathrm{Hom} ( \{ \star \} , ...
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 $G$. ...
4
votes
1answer
91 views

Is the global section ring of a Noetherian Scheme Noetherian as well?

As the title suggests, I am asked to prove that, given a Noetherian scheme $(X,\ \mathcal{O}_{X})$ and any open subset $U\subseteq X$, $\Gamma(U,\ \mathcal{O}_{X}):=\mathcal{O}_{X}(U)$ is a Noetherian ...
1
vote
0answers
52 views

What is the opposite of category of schemes

The category of schemes sits between $Aff=CRing^\text{op}$ and its free cocompletion $\hom(Aff^\text{op},Sets)$, as the locally representable functors. Dualizing, we have $CRing\subseteq ...
0
votes
0answers
99 views

Compare étale morphisms in Top and Schemes

Perhaps because I find the algebraic definition of étale morphisms (of schemes) hard to approach, I'm trying to bootstrap from my understanding of étale morphisms of topological spaces. In topological ...
5
votes
1answer
60 views

quasicompact scheme are finite union of affine scheme

This is a problem from Liu`s book. Show that a scheme $X$ is quasi-compact if and only if it is a finite union of affine schemes. If the scheme is quasicompact then it is obviously a finite union of ...
1
vote
1answer
58 views

Connected scheme but not quasicompact

I am searching for a connected scheme which is not quasi compact. My try: Suppose $A_i$=Spec$k[x]$, where $k$ is algebraically closed field are affine schmes . I am planning to glue $0$ of the ...
0
votes
0answers
48 views

Sheaf of ideals [duplicate]

Let $(X, \mathcal{O}_X)$ be a scheme and and $f \in \mathcal{O}_X(X)$. We define the sheaf of ideals by the assignment $$ U \longmapsto f|_U \mathcal{O}_X(U) $$ and denote this sheaf by ...
0
votes
0answers
16 views

Equivariant Sheaves, Local system

Let $(G,m)$ be a group scheme unipotent, and $L$ a local system of rank 1 on $G$ such that: $m^*(L) \simeq L \boxtimes L $. Then why is $L$ an equivariant sheaf on $G$ with the action the ...
3
votes
2answers
206 views

Scheme: Countable union of affine lines

Let $X$ be a countable union of $A_n$ ($n \in \Bbb{N}$), where $A_i$ are affine lines, i.e., $A_i=\operatorname{Spec}k[x]$, with $k$ algebraically closed field, such that $A_i$ meets $A_{i+1}$ in the ...
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
1answer
36 views

complex automorphisms acting on a projective variety

Consider a complex projective variety $X=\operatorname{Proj}\frac{\mathbb C[T_1,\ldots,T_n]}{(f_1,\ldots,f_n)}$ with $f_1,\ldots,f_n$ homogeneous polynomials. If $\sigma\in\operatorname{Aut}(\mathbb ...
0
votes
0answers
37 views

Generic points and base change

Let $X$ be a scheme over a field $k$. Let $K$ be a field extending $k$. What can we say about the generic points of $X_K$ with respect to the generic points of $X$? In particular I suspect that doing ...
1
vote
1answer
33 views

Graph of a morphism between two $K$-schemes: an open covering

Consider a separated morpshism $f:X\longrightarrow Y$ between two $K$-schemes ($K$ is a field). The graph of $f$ is the image of the morphism $(Id_X,f):X\longrightarrow X\times_{\operatorname {spec}K} ...