Derived categories of Grassmannians over integers and modular representation theory

TL;DR

This paper investigates the derived categories of Grassmannians over integers, establishing semi-orthogonal decompositions, exceptional collections, and tilting bundles, with implications for modular representation theory and Koszul duality.

Contribution

It constructs semi-orthogonal decompositions and exceptional collections for Grassmannians over integers, extending known results from characteristic zero to arbitrary base rings.

Findings

Existence of semi-orthogonal decompositions with components related to $GL_k$ and $GL_{n-k}$ representations.

Construction of a tilting bundle with split quasi-hereditary endomorphism algebra over $Z$.

Extension of results to arbitrary commutative rings and perfect complexes.

Abstract

In this paper we study the derived categories of coherent sheaves on Grassmannians $\operatorname{Gr}(k,n),$ defined over the ring of integers. We prove that the category $D^b(\operatorname{Gr}(k,n))$ has a semi-orthogonal decomposition, with components being full subcategories of the derived category of representations of $GL_k.$ This in particular implies existence of a full exceptional collection, which is a refinement of Kapranov's collection \cite{Kap}, which was constructed over a field of characteristic zero. We also describe the right dual semi-orthogonal decomposition which has a similar form, and its components are full subcategories of the derived category of representations of $GL_{n-k}.$ The resulting equivalences between the components of the two decompositions are given by a version of Koszul duality for strict polynomial functors. We also construct a tilting vector bundle on $\operatorname{Gr}(k,n).$ We show that its endomorphism algebra has two natural structures of a split quasi-hereditary algebra over $\mathbb{Z},$ and we identify the objects of $D^b(\operatorname{Gr}(k,n)),$ which correspond to the standard and costandard modules in both structures. All the results automatically extend to the case of arbitrary commutative base ring and the category of perfect complexes on the Grassmannian, by extension of scalars (base change). Similar results over fields of arbitrary characteristic were obtained independently in \cite{BLVdB}, by different methods.