t-structures and twisted complexes on derived injectives
Francesco Genovese (Universiteit Antwerpen)
Aula Beltrami – Venerdì 20 Dicembre 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 injectiveobjects 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.