Title of Talk: HFD objects and their application to the indecomposability of derived categories
Speaker: Gopinath Sahoo, TIFR, Mumbai
Date: October 17, 2023
Time: 16:00:00 Hours
Venue: AG-77
Abstract: Orlov introduced a useful finiteness property called homologically finite for objects of a triangulated category. The subcategory of homologically finite objects behaves well with respect to admissible semi-orthogonal decompositions. He proved that for Noetherian separated schemes of finite Krull dimension with enough locally free sheaves, the homologically finite objects of the bounded derived category of coherent sheaves are precisely the perfect complexes.
In the preprint: https://arxiv.org/abs/2211.09418v2, Kuznetsov, and Shinder suggest a modification of this notion. Given a small $dg$-enhanced triangulated category $T$, they introduce homologically finite-dimensional objects of the derived category of $dg$-modules over $T$. Unlike the notion of Orlov the new construction can be iterated, using which they have introduced reflexivity, and some other properties for triangulated categories. They have shown that the triangulated categories arising in geometric and algebraic contexts are reflexive. They have proved that the indecomposability of a reflexive triangulated category is equivalent to the indecomposability of its HFD partner, using which they have deduced the non-existence of semi-orthogonal decomposition for derived categories of various singular varieties.