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.
8
votes
1answer
214 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 ...
2
votes
3answers
107 views
Definition of degree of finite morphism plus context
Let $f: X \rightarrow Y$ be a finite morphism of schemes, defined here,
http://en.wikipedia.org/wiki/Finite_morphism
I always assumed that the degree of $f$ was the degree of the induced field ...
22
votes
2answers
963 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
2answers
261 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} ...
11
votes
3answers
834 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 ...
0
votes
1answer
67 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 ...
2
votes
1answer
108 views
Closure of image of diagonal morphism of S-scheme
Let $X$ be an $S$-scheme with structural morphism given by $f : X \to S$. The image of the diagonal morphism $\Delta : X \to X \times_S X$ is contained in the subset $Z := \{ z \in X \times_S X : ...
5
votes
1answer
78 views
Separated schemes and unicity of extension
In point set topology, we have the following result, which is easily proved.
Theorem. Let $Y$ be Hausdorff space and $f,g:X \to Y$ be continuous functions. If there exists a set $A\subset X$ such ...
4
votes
1answer
146 views
Möbius strip and $\mathscr O(-1)$. Or $\mathscr O(1)$?
On the real $\textbf P^1$ we have these algebraic line bundles: $\mathscr O(1)$ and $\mathscr O(-1)$.
Which one corresponds to the Möbius strip? (Both are $1$-twists of $\textbf P^1\times\textbf ...
2
votes
1answer
92 views
Scheme glued out of finitely many spectra of local rings
Let $X$ be a noetherian separated scheme over the spectrum of a field $K$. By definition, I can find an open covering of $X$ by finitely many affine schemes $Spec(R_i)$ where each $R_i$ is a ...
2
votes
1answer
124 views
“Push-Pull” Morphisms of Higher Direct Image Sheaves
(This is 20.7.B in Ravi Vakil's notes)
Suppose $f:Y \to Z$ is any morphism, and $\pi: X\to Y$ is quasicompact and quasiseparated. Suppose $\mathcal{F}$ is a quasicoherent sheaf on $X$. Let
...
1
vote
1answer
61 views
Checking flat- and smoothness: enough to check on closed points?
I am currently studying varieties over $\mathbb{C}$, i know some scheme theory.
Let $f: X \rightarrow Y$ be a morphism of varieties. If we want to show flatness, is it enough to check the condition ...
5
votes
0answers
92 views
Reality check: $\mathcal{O}_{\mathbb{P}_\mathbb{R}^1}(1)$ and $\mathcal{O}_{\mathbb{P}_\mathbb{R}^1}(-1)$ are both mobius 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 ...
3
votes
1answer
74 views
The image of a proper scheme is closed
Let $X$ be a scheme, $Y$ a proper $X$-scheme and $Z$ a separated $X$-scheme. Let $f : Y \to Z$ be a morphism of $X$-schemes. I would like to prove that the image of $f$ is closed in $Z$.
Here is what ...
2
votes
1answer
77 views
Nice proof for étale of degree 1 implies isomorphism.
Let $f: X \rightarrow Y$ be a finite étale covering of degree 1 of varieties over some field $k$, so
$$
[K(Y):K(X)] = 1.
$$
If $k=\mathbb{C}$ and the varieties are smooth, one can apply complex ...
2
votes
1answer
65 views
Regular in codim one scheme and DVR
Let $X$ be a noetherian integral (separated) scheme which is regular in codimension one. Let $Y$ be a prime divisor and let $\eta$ be the generic point of $Y.$ It seems I am missing something easy but ...
1
vote
1answer
124 views
(Continued:) finiteness of étale morphisms
I was writing a question, it became too long, and i decided to split it into two parts. I hope posting two questions at the same time is not a problem.
First question: Checking flat- and smoothness: ...
0
votes
1answer
94 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 ...
