Quasi-projective dimension for complexes via filtrations

We extend quasi-projective dimension and quasi-projective length from finitely generated modules to homologically finite complexes by using finite filtrations in the derived category. Our definitions recover the original invariants of Gheibi–Jorgensen–Takahashi for modules and behave well under exact functors, which simplifies the proofs of several results. We establish the Auslander–Buchsbaum formula, the derived depth and width formulas, and the dependency formula for complexes of finite quasi-projective dimension. Extending a result of Gheibi–Jorgensen–Takahashi, we show that every homologically finite complex has finite quasi-projective dimension over a suitable complete intersection ring. We also prove a new intersection theorem and a descent theorem for Serre’s conditions. Finally, we obtain vanishing results for Tor, Ext, and Tate (co)homology, and deduce symmetry of eventual Ext vanishing over Gorenstein local rings.