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Given a split reductive Chevalley group scheme G over Z and a parabolicsubgroup scheme P in G, this paper constructs G-linear semiorthogonaldecompositions of the bounded derived category of noetherian representations ofP with each semiorthogonal component being equivalent to the bounded derivedcategory of noetherian representations of G. The G-linear semiorthogonaldecompositions in question are compatible with the Bruhat order on cosets ofthe Weyl group of P in the Weyl group of G. Their construction builds upon thefoundational results on B-modules from the works of Mathieu, Polo, and van derKallen, and upon properties of the Steinberg basis of the T-equivariantK-theory of G/B. As a corollary, we obtain full exceptional collections in thebounded derived category of coherent sheaves on generalized flag schemes G/Pover Z.

Highest weight category structures on $Rep(B)$ and full exceptional collections on generalized flag varieties over $\mathbb Z$