遇到一个问题，请问谁能说说class number, picard group, line bundle之间的联系，谢谢。

1.固定一个scheme X, 则有一个一一对应。(see Hartshorne)

line bundles over X <--> invertible sheaves on X

在代数几何中，很多人更喜欢用sheaf这个概念，因为vector bundle
的范畴不是Abel范畴，但是加上coherent sheaf以后，就可以定义任何映射的kernel和cokernel了。

2. X 上所有line bundles的同构类形成一个群Picard group，乘法定义为张量积。

3. 如果X是仿射的（affine）, 假设X=Spec A, A是一个环（ring）。则X上的locally free sheaf 和A上的projective module之间有一个一一对应。在这个对应之下，invertible sheaves
给出A上的秩为1的projective module M. 如果A是整环，令K为分式域。则M与K的张量积（over A）是K上一维向量空间，即同构于K. 好久没有碰代数了，如果我没记错的话，这样的模就是分式理想（fractional idea）。如果是对的，证明其实很容易。

4.关于Class number的现代解释，强烈推荐Milnor的代数K理论前面的章节。他可以通过projective module以及K_0群来定义。他和line bundle的关系就是通过上面几条建立的。一般来说，Picard group 是几何的观点，对一般的scheme可以定义。Class group 是代数的观点，它仅仅对数域有定义（或Dedekind domain）。对函数域的情形，一般用几何的观点更方便。 或者说如果环A是一个域k上的代数，则考虑的是几何学。

Actually, when we say "line bundles", do we mean holomorphic line bundles in ag?

vector bundle的范畴不是Abel范畴
==============================
Could you name some non-abelian operations defined on vector bundles?

他可以通过projective module以及K_0群来定义。
==============================
Please express this with more details, and in your favourite way. Dont worry if I dont understand it.

一般来说，Picard group 是几何的观点，对一般的scheme可以定义。Class group 是代数的观点，它仅仅对数域有定义（或Dedekind domain）。对函数域的情形，一般用几何的观点更方便。 或者说如果环A是一个域k上的代数，则考虑的是几何学。
==============================
Yes, I think working with geometry is better, as it is more "flexible".

Many constructions in commutative algebra are so mysterious, I can't "really" understand them.

Actually, when we say "line bundles", do we mean holomorphic line bundles in ag?

--------Yes, in AG, the main interest is holomorphic object, however, in complex geometry, we should start with continous one and apply D bar operator to get holomorphic object. But in "intrinsic" algebraic geometry, every thing is holomorphic(to be precise,algebraic).

vector bundle的范畴不是Abel范畴
==============================
Could you name some non-abelian operations defined on vector bundles?

--------Let f: E --> F be a homomorphism between two vector bundles E and F. The kernel and cokernel is in general not a vector bundle. So we can not define kernel and cokernel for a general homomorphism in the category of vector bundles. They exist only when the fiber map of f_s: E_s --> F_s have same rank. However, the kernel and cokernel always exist as coherent sheaves.

他可以通过projective module以及K_0群来定义。
==============================
Please express this with more details, and in your favourite way. Dont worry if I dont understand it.

--------I read the book of milnor a long time ago, and I didn't work in that area. So I almost forgot all the details. The thing I am sure is, you need to understand the following concepts:
1.Dedekind domain 2.ideal 3.ideal class group 4.projective module
5.Grothedieck group K_0
I recommend to pick up the book of milnor and read the first chapter or maybe the second too.

Let f: E --> F be a homomorphism between two vector bundles E and F. The kernel and cokernel is in general not a vector bundle.
===============================
Good point. Does this have something to do with the failure of exactness in some sequence?

1.Dedekind domain 2.ideal 3.ideal class group 4.projective module
5.Grothedieck group K_0
===============================
En, I know a little on all of these, although I think asking other ppl is always much quicker than learning by books [for my purpose]. Maybe I will consider someone else. Thank you!