Scheme theoretic definition of a vector bundle

Question

I have started reading the book "Lectures on Vector Bundles" by J. Le Potier, where at the very beginning he gives the following two definitions, which are very familiar in smooth or holomorphic settings:

$X=$variety over $\mathbb{C}$ - separated finite type scheme over $\mathbb{C}$. "Points" in this book mean closed points.

Definition 1. Let $X$ be an algebraic variety. A (complex) linear fibration over X is given by a surjective morphism of algebraic varieties $p: E \to X$, where for each point $x\in X$, $p^{-1}(x)$ has the structure of complex vector space.

Definition 2. An algebraic vector bundle of rank $r$ is a linear fibration $E \to X$ which is locally trivial, that is for each point $x\in X$ there exists an open neighbourhood $U$ with an isomorphism of fibrations $E|_U \to U \times \mathbb{C}^r$.

1. However, in this algebraic case I have the following confusion.:

What exactly does the structure of vector space mean in this case? Say if we take $\mathbb{C}^n$, scheme-theoretically this should be taken as $Spec \ \mathbb{C} [x_1,...,x_n]$, and on closed points we indeed have the vector space structure. But what about non-closed points, do we care about them at all?

So is it correct to understand the first definition in the sense that each (scheme-theoretic) fiber over closed point $x$ is isomorphic to $Spec \ \mathbb{C}[x_1,...,x_n]$? Also what about fibers over non-closed points, do I require something from them as well?

2. Another problem arises when the author starts treating associated bundles.

Say for the Whitney sum, he takes $E \oplus F = \coprod_{x\in X} (E_x \oplus F_x)$ as sets, and then locally induces the structure of algebraic variety by considering bijection $(E \oplus F)|_{U_i} \to U_i \times (\mathbb{C}^r \oplus \mathbb{C}^s)$, given by $\phi_i \oplus \psi_i$ where the latter are corresponding trivializations for $E$ and $F$.

But what does $\mathbb{C}^r \oplus \mathbb{C}^s$ mean rigorously? Is it just $\mathbb{C}^{r+s} = Spec \ \mathbb{C} [x_1,...,x_r, y_1,...,y_s]$?

But then it is a well known fact, that $Spec \ \mathbb{C} [x_1,...,x_r, y_1,...,y_s]$ has more points than direct product as sets $Spec \ \mathbb{C}[x_1,..,x_r] \times Spec \ \mathbb{C}[y_1,...,y_s]$ (thus more points than $E_x \oplus F_x$ for that matter). So $\phi_i \oplus \psi_i$ can only be a bijection if one restricts to closed points. So if I wanted to keep track of non-closed points, how should I modify the definition of $E \oplus F$ as sets?

The only rigorous solution which comes to my mind is to directly glue this scheme using the "direct sum" cocycle, starting from $U_i \times \mathbb{C}^{r+s}$ as constituent schemes.

3. I am aware that this approach of describing vector bundles should ultimately be the same as using locally free sheaves. The proof given in the book isn't rigorous enough for my tastes because of the aforementioned complications. Is there a source describing this approach to vector bundles and related constructions more rigorously?

Answer 1

It seems that non-closed point are giving you a lot of troubles. So before talking about vector bundles, I would like to say :

• Yes there are non-closed points. Either forget about them completely (you are working over $\mathbb{C}$, so this is in fact legitimate), or read some general scheme theory to see what are their use, what do they represent, why they are more technical than geometric features, and why they should not be seen as an additional difficulty.
• As you said, additional non closed point appears in a product $X\times Y$ (and in fact, over a non-algebraically closed field, additional closed points appears too). Now this is not a problem at all and in fact just something you will need to forget : never ever talk about the set-theoretic product $X\times Y$. This is never something interesting. When you have a product $X\times Y$, this will always be the scheme-theoretic one, with maybe a bunch of new non-closed points, even a bunch of new closed points (over other fields), but it won't matter.
• There is a notion of group schemes (and vector-spaces scheme). This is the same as the notion of group (and vector spaces) but written in the language of scheme theory. For example $\mathbb{C}^n$ is a scheme, you want to put a group structure on it. Thus you need an addition which will be morphism : $\mathbb{C}^n\times\mathbb{C}^n\rightarrow\mathbb{C}^n$ satisfying some properties. This is a morphism of schemes so $\mathbb{C}^n\times\mathbb{C}^n$ needs to be a scheme and to be endowed with the scheme theoretic product (and not the set-theoretic one). Yes there will be additional non-closed points, but don't worry.

I should probably say at this point what is this morphism. Either you agree that there is one (after all $(x,y)\mapsto x+y$ is obviously polynomial), or you want to go deeper in scheme theory. In that cas this is : $$\operatorname{Spec}\mathbb{C}[x_1,...,x_n]\times\operatorname{Spec}\mathbb{C}[x_1,...,x_n]=\operatorname{Spec}\mathbb{C}[x_1,y_1,...,x_n,y_n]\rightarrow\operatorname{Spec}\mathbb{C}[z_1,...,z_n]$$ the map induced by the ring morphism $\mathbb{C}[z_1,...,z_n]\rightarrow\mathbb{C}[x_1,y_1,...,x_n,y_n]$ such that $z_i\mapsto x_i+y_i$. (The coordinates at the target is the sum of the coordinates of the source)

See : I don't even need to understand what the morphism are on non-closed point. I just need to see that this morphism give the right thing on closed point.

Now your specific questions :

1. Yes we care about non-closed point in the scheme theoretic language. With times you will understand that they behave much like any other points. And often when you say a statement specific to closed point, you will understand that the same (or with just a small change) statement holds for non-closed point. In this case, if $p:E\rightarrow X$ is a linear fibration and $x\in X$ is not necessarily closed, let $\kappa(x)$ be its residue field, then $p^{-1}(x)$ needs to be $\kappa(x)$ vector space.

2. I think I already answered this question : $E_x\oplus F_x=E_x\times F_x$ ($\oplus$ is used because we are talking about vector spaces) is the scheme theoretic product. (Never the set theoretic one). So I don't see any issue here.

3. I did not read "Lectures on Vector Bundles" by J. Le Potier so I don't know how rigorous it is. Be careful though : do not mix lack of rigor with you having a difficulty. I have met a lot of student complaining about some books or teacher being not rigorous whereas they just did not understand a new concept (I was one of them once, even twice or more). And of course it is normal to have difficulties at this point : these are hard math. That said, changing point of view with another book is not a bad idea either.