Derived categories of Grassmannians over integers and modular representation theory (Q329481)

In the paper under review, the author studies the derived categories of coherent sheaves on Grassmannians \(\mathrm{Gr}(k, n)\) over the ring of integers. Then he proves that the category \(\text D^b{\text{Gr}(k, n)}\) has a semi-orthogonal decomposition, with components being full subcategories of the derived category of representations of \(\mathrm{GL}_k\). This implies existence of a full exceptional collection, which is a refinement of \textit{M. M. Kapranov}'s collection [Invent. Math. 92, No. 3, 479--508 (1988; Zbl 0651.18008)] over a field of characteristic zero.NEWLINENEWLINEThe author also describes the right dual semi-orthogonal decomposition which has a similar form, and its components are full subcategories of the derived category of representations of \(\mathrm{GL}_{n-k}\). The resulting equivalences between the components of the two decompositions are given by a version of Koszul duality for strict polynomial functors. Finally, he constructs a tilting vector bundle \(\mathcal {E}(k,n)\) on \(\mathrm{Gr}(k, n)\), shows that its endomorphism algebra has two natural structures of a split quasi-hereditary algebra over \(\mathbb{Z}\), and identifies the objects of \(\text D^b\mathrm{Gr}(k, n)\), which correspond to the standard and costandard modules in both structures.NEWLINENEWLINESimilar results were obtained by \textit{R.-O. Buchweitz} et al. [Compos. Math. 151, No. 7, 1242--1264 (2015; Zbl 1333.14017)] over the field of arbitrary characteristic, by different methods using in particular the Kempf vanishing theorem.