**R**³. Then we got to K-froms in

**R**ª (wish I knew how to put 'n' instead of 'a' cause that's the convention...) Then, we got to the part where were taking the exterior derivatives as an almost exact sequence from 0-forms -> 1-forms->..->k-forms->(k+1)-forms->... See, it's not exact so the kernel of one doesn't equal the image of the other, right? (oh, sequences and kernels and images are all stuff from linear). Well then you take the part that's in the kernel that

*didn't*come from the image, and that's the Kth deRham cohomology. See that set is made up of basically all the singularities and stuff in any one particular k. And we've been doing stuff with singularities in complex analysis! All three of my classes came together today! It was cool...

Right? well that's just the start! cause we're doing tensor products in linear, which is what the GUT seminar is this year (which really isn't apart of the linear course, but is ver cool nonetheless), and we finally got enough information to do the homework. But that's not the cool part, the cool part is like well, a tensor product is basically defined by its properties. You have two R-Modules, say M and N, and you take an R-bilinear map (say f) from MxN to C, some abelian group, right? Well you then have a unique homomorphism from M®N to C (say g) where g(m®n)=f(mxn). Oh, that ®? I'm using that to represent tensor product.

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