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In this paper we study the derived categories of coherent sheaves onGrassmannians $\operatorname{Gr}(k,n),$ defined over the ring of integers. Weprove that the category $D^b(\operatorname{Gr}(k,n))$ has a semi-orthogonaldecomposition, with components being full subcategories of the derived categoryof representations of $GL_k.$ This in particular implies existence of a fullexceptional 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 asimilar form, and its components are full subcategories of the derived categoryof representations of $GL_{n-k}.$ The resulting equivalences between thecomponents of the two decompositions are given by a version of Koszul dualityfor strict polynomial functors.  We also construct a tilting vector bundle on $\operatorname{Gr}(k,n).$ Weshow that its endomorphism algebra has two natural structures of a splitquasi-hereditary algebra over $\mathbb{Z},$ and we identify the objects of$D^b(\operatorname{Gr}(k,n)),$ which correspond to the standard and costandardmodules in both structures.  All the results automatically extend to the case of arbitrary commutativebase ring and the category of perfect complexes on the Grassmannian, byextension of scalars (base change).  Similar results over fields of arbitrary characteristic were obtainedindependently in \cite{BLVdB}, by different methods.

Derived categories of Grassmannians over integers and modular representation theory