Hall algebras in the derived category and higher-rank DT invariants

We remark that the combination of the works of Ben-Bassat-Brav-Bussi-Joyce and Alper-Hall-Rydh imply the conjectured local description of the moduli stacks of semiSchur objects in the derived category of coherent sheaves on projective Calabi-Yau 3-folds. This result was assumed in the author's previous papers to apply wall-crossing formulas of DT-type invariants in the derived category, for example DT/PT correspondence, rationality, etc. We also show that the above result can be applied to prove the higher-rank version of the DT/PT correspondence and rationality.