t-structures and twisted complexes on derived injectives
Francesco Genovese (Universiteit Antwerpen)
Aula Beltrami - Friday, December 20, 2019 h.12:00
Abstract. I will report on joint work with Wendy Lowen and Michel Van
den Bergh. An abelian category with enough injectives can be
reconstructed as the category of finitely presented modules over the
category of its injective objects; this allows to understand
deformations of abelian categories in terms of deformations of their
injective objects. In the derived setting, abelian categories are
replaced with (enhanced) triangulated categories together with a
t-structure - which specifies an abelian subcategory - and injective
objects can also be enhanced to so-called “derived injectives”. These
derived injectives can be used to make resolutions of objects of the
given triangulated category, and in the end achieve a derived version
of the aforementioned reconstruction theorem: more precisely, we can
reconstruct a triangulated category with a "nice enough" and left
bounded t-structure as a category of left bounded 'twisted complexes'
of derived injectives. This can also be viewed as a generalisation of
the well-known result that the left bounded derived category of an
abelian category is isomorphic to the left bounded homotopy category
of complexes of injective objects, and it is the foundational step
towards the development of a deformation theory of triangulated
categories with a t-structure.
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