Bernstein-Gelfand-Gelfand resolutions and the constructible $t$-structure

TL;DR

This paper establishes a connection between Bernstein-Gelfand-Gelfand resolutions and the constructible t-structure in highest weight categories, providing new insights even for Category O.

Contribution

It introduces analogues of BGG resolutions in highest weight categories and shows they form a chain-level lift of the constructible t-structure's heart.

Findings

Resolutions characterize objects via Ext vanishing with costandard objects.

The category of complexes forms a chain-level lift of the t-structure.

Results are new even for Category O.

Abstract

We consider analogues of the Bernstein-Gelfand-Gelfand resolution in a highest weight category $\mathscr{P}$. We prove the resulting category of complexes is a chain-level lift of the heart of the constructible $t$-structure on its bounded derived category $D^b(\mathscr{P})$. In particular, an object admits a Bernstein-Gelfand-Gelfand resolution if and only if a certain $\operatorname{Ext}$ vanishing with costandard objects holds. These results appear to be new even for Category $\mathscr{O}$.