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

Abstract

We consider analogues of the Bernstein-Gelfand-Gelfand resolution in a highest weight category $\mathcal{P}$. 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 $D^b(\mathcal{P})$. In particular, an object admits a Bernstein-Gelfand-Gelfand resolution if and only if a certain $\mathrm{Ext}$ vanishing with costandard objects holds. These results appear to be new even for Category $\mathcal{O}$.

24 pages