Comment by nxobject

My bad – that was a misleading thing to say! Thanks for pointing that out. I figured out what I said wrong. (Caveat emptor, I do biostatistics now.)

The context IIRC was this: one of the key results of the class was generalized Stokes' theorem, but this case (since was a 200-level class) we mostly just looked at differential forms on open spaces in R^n, and then said a few quick things about differentiable manifolds.

At this more concrete level, then, I remember that we constructed de Rham cohomology (fixing an open subset of R^n) beginning with the cochain complex given by vector spaces of k-differential forms and exterior derivatives, instead of working more generally with a cochain complex on modules.

But think I said something wrong here, which I why you were (rightly) confused. I'm not sure that the above distinction matters anyway since IIRC, you can get Mayer-Vietoris by showing that de Rham cohomology satisfies the Eilenberg-Steenrod axioms (stated for cohomology), and the Eilenberg-Steenrod axioms only need abelian groups anyway.

But I'm also 90% sure that TFA did something more direct to get to Mayer-Vietoris that I've forgotten, since we didn't use that much homological algebra.