The homotopy category of flat modules, and Grothendieck local duality

Amnon Neeman – 26 July 2006

Let X be a projective variety. For many years now, we have known how to form an infinite completion for the derived category Db(Vect/X), whose objects are bounded complexes of vector bundles on X. This allows us to prove theorems about finite complexes of vector bundles, using infinite techniques. Only very recently have we understood how to handle the derived category Db(Coh/X), whose objects are bounded complexes of coherent sheaves on X. Krause published a paper, in 2005, showing that K(Inj/X) is an infinite completion for Db(Coh/X). Jorgensen, also in 2005, showed that, as long as X is affine, K(Proj/X) is an infinite completion for the dual category Db(Coh/X)op. Grothendieck's local duality studies conditions under which the categories Db(Coh/X) and Db(Coh/X)op are equivalent; it suffices for there to exist a "dualizing complex", and dualizing complexes exist for many interesting X. One can ask what happens to the infinite completions; it turns out that, when a dualizing complex exists, then K(Inj/X) and K(Proj/X) are also equivalent. We will discuss this, as well as related results.