What are the relationship between various definitions for quasi coherent sheaves?

It seems that there are many definitions of quasi coherent sheaves(modules). There is a nice page on nLab quasi coherent sheaves

My questions are:

1. Are there any other definitions of quasi coherent sheaves? For what?(I mean in which situation, they should be redefined to satisfy the working environment)

2. How are they related?

For question 2, nLab says these definitions are equivalent. There are four definitions in nLab. It seems except first definitions, the other three are equivalent, just restate in different language: i.e presheaves <---> pseudo functors <---> fibered categories.

But are these definitions exactly the same as the classical definitions? (as locally presentable modules)

• Good questions. I hope that answerers can also say something about the different definitions of coherent sheaves. – Kevin H. Lin Feb 14 '10 at 5:55
• ooh ooh big flashing red warning sign. There is an independent issue regarding coherent sheaves, and that is that when Hartshorne was writing his book he went for the easy "f.g. module" definition, which he knew wasn't Grothendieck's definition (and he says as much). His definition coincides with Grothendieck's (the "correct" definition) when the base is Noetherian, but not otherwise. It's a theorem (well, a lemma) in EGA that O_X (structure sheaf of a scheme) is Noetherian! This issue is independent of all this pseudo functor twaddle. – Kevin Buzzard Feb 14 '10 at 8:29
• By the way, the defintion of coherent sheaves given in nlabs is not correct. See EGA, chapter I, 0.5.3.1. – Qing Liu Feb 14 '10 at 22:38
• Typo! It's a theorem that O_X is coherent! Apologies. The point was supposed to be that this is self-evident using Hartshorne's definition. – Kevin Buzzard Feb 15 '10 at 8:30
• Kevin, it isn't true that the structure sheaf of a general scheme is coherent (using the "right" definition, as in Serre's FAC). Not sure if you were claiming to the contrary. For affine $X$, it amounts to saying that there are non-coherent rings (i.e., rings with finitely generated ideals that are not finitely presented as modules). An example: let $R$ be the valuation ring of an algebraically closed complete non-archimedean field, and a non-coherent ring is $R/I$ where $I$ is the ideal of elements $x \in R$ such that $|x| < c$ for any $c \in (0,1)$. Proof of non-coherent is an exercise. – BCnrd Mar 12 '10 at 6:14