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WHAT I KNOW ABOUT GROTHENDIECK RINGS FOR TORSION FREE CATEGORIES (AND ONE OR TWO THINGS I'D LIKE TO KNOW)

It seems to me that understanding the ring structure induced by the tensor product on the Grothendieck group of quasi-isomorphism classes of finite rank torsion free groups is a worthwhile and intriguing goal. Apparently no one else feels the same way, since nobody else seems to be working on the problem, a situation which certainly has its advantages for me but which also seems just a little bit sad. So I thought it might help inspire and/or encourage a few others if I presented a summary of what I know at this point, including the main results from my two papers on the subject along with some theorems that have been sitting down in the basement gathering dust because I haven’t yet managed to find a suitable family for them to belong to.