I'm trying to read an arXiv paper using the terms projective and locally free modules interchangeably within the context of finitely generated number fields. To even attempt to understand the results of the paper, I need to understand these terms and the relation between them in this context. However, I'm only familiar with free modules and not projective modules or localisation.
I know I need to understand projective modules, torsionfree, localisation and the results/implications for such in PIDs but I can't wrap my head around it.
I've across 3 different definitions of projective modules, which I assume are equivalent; a direct summand of a free module, a definition using lifting and commutative diagrams and exact sequences. I'm not comfortable with category theory so the only accessible definition is the first.
Could someone point me towards some literature that deals with these topics from very elementary principles? I've tried to understand it through Knapp's Advanced Algebra textbook but I can't.
If someone could explain it on here that would be great, but unfortunately, due to my lack of knowledge I think it would take a lot of explanation for me to understand.