Derived categories were introduced by Grothendieck and Verdier as a
language to state various duality theorems. They are obtained from
categories of complexes by formally inverting "quasi-isomorphisms".
Unfortunately derived categories do not admit many functorial
constructions as they ignore higher homotopy information. In
particular one has little control over the functors between them.
Nonetheless a celebrated theorem by Orlov, proved almost 25 years ago,
provided some hope that, at least in algebraic geometry, derived
categories might be rich enough after all. It states that any (exact)
fully faitful functor between derived categories of coherent sheaves
D(X), D(Y) on smooth projective varieties is a so-called "Fourier-Mukai functor".
I.e. it is induced from a unique object living in D(X x Y).
Sadly, it appears that this is as far as it gets. We are now able to
routinely construct counter examples against Orlov's result when the
full faithfulness hypothesis is dropped. Ultimately the existence of
such counter examples hinges on precisely controlling the
(in)compatibility of functors with higher homotopy information.
Recently we developed some machinery to conveniently handle this and
this has also allowed us to construct new families of triangulated
categories without "enhancement". Previously, essentially only a
single example of this type, due to Muro, Schwede and Strickland, was
known.
This is joint work with Amnon Neeman, Theo Raedschelders and Alice
Rizzardo. |