Here are some notes on vector bundles in scheme theory.
Some of the terminology I decided to use is nonstandard. There are also more accurate definitions when it comes to determining the essential images of some of the functors I describe in the notes. For example, the stacks projects’ definition of a vector bundle can be summarized “an object in the essential image of .”
When it came to the more technical notions which require a description of some costructures on the symmetric algebra, for instance, I didn’t go into too much detail. These points are more subtle and more technically demanding. People use the structure all the time without explicitly being able to describe it though, so don’t take this as a deterrent.