Colloquium

Orateur(s) Michel Van den Bergh - Bruxelles et Hasselt,
Titre Counter examples in the theory of derived and triangulated categories
Date06/02/2020
Horaire17:00 à 18:00
Résume

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.

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