Question

Derived geometry generalization of Lefschetz fixed-point theorem?

Answer 1

The proof of the Lefschetz fixed point theorem relates the alternating trace sum to the intersection number so if your intersection is not transverse, nothing bad happens except that the intersection product is harder to compute. For instance, if you take f = id so you are intersecting the diagonal in MxM with itself, this is maximally non transverse. However, the self intersection of the diagonal is well known to be the euler characteristic and this is what the LFT computes in this case.

Answer 2

What class or classes would I have to take to learn about this kind of stuff?

Answer 3

So, I'm far from an expert, but I think you might want to look into perverse sheaves.  
I've been reading up on them lately, they're objects in the derived category of sheaves on a space, and I have seen a lot of stuff about Lefschetz theorems and their derived analogues pop up.  Pramod Achar recently published a great book on the topic.

The "perversity" one chooses with which to defined perverse sheaves is related to tranversality, and allowing it to fail.

Answer 4

This is not a sub for ur homework bro.