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I Derived All Night

20 December, 2007
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A couple of years ago I started trying to read up on some of the basic terminology of derived categories, as they are often espoused as a natural home for homological invariants. (More specifically, I was trying to generalise a Künneth theorem for Banach modules.) After a while I felt (and I suspect I’m not alone) that I was in danger of replacing my hoped-for theorem with a tautology at the derived level.

I was reminded of this mixture of confusion, frustration and enthusiasm by this comment on a thread at the n-category café:

…maybe I have become a bit disaffected in the past few years with derived stuff—after we derive everything in sight up to the moon, then what? The Riemann hypothesis is still open, the Weil conjectures still have a ridiculous proof, and so on. My own interests are now mostly in the then-what part. Of course, I’m still interested in the deriving, and I think it would be good for mathematics if lots of people think about how to do the deriving and I would encourage other people to hire them (but for the sake of everyone they shouldn’t forget completely about the then-what part).

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