The chain complex of the total space of a fibration can be enriched with filtration by the basic degree which gives rise to the Serre spectral sequence. This is a wonderful tool for the knowledge of the fibration and perturbations of the \(E_{r}\)-stage is a process widely used. For instance, with it, S. Halperin and the reviewer creates various models of the total space of a fibration [S. Halperin and D. Tanré, Ill. J. Math. 34, No. 2, 284–324 (1990; Zbl 0679.55011)] and G. Laumon studies modules over the ring of differential operators on complex algebraic varieties [G. Laumon, Lect. Notes Math. 1016, 151–237 (1983; Zbl 0551.14006)]. In the paper under review, the authors raise this approach to a systematic study of Quillen closed model structures. They consider categories of filtered and bigraded chain complexes of modules over a commutative ring with unit and define cofibrantly generated closed model structures on them, by varying the notions of fibrations and weak-equivalences. Mostly, the weak-equivalences are the \(E_{r}\)-quasi-isomorphisms of bigraded complexes at the \(r\)-stage of the associated spectral sequence.
In the filtered case, the authors show that the Deligne functors, shift and décalage, connect by Quillen equivalences the model categories corresponding to \(E_{r}\)-quasi-isomorphisms and fibrations defined by the surjectivity on the \(r\)-cycles, when varying \(r\geq 0\). The bicomplex situation is more rigid and the shift-décalage adjunction cannot be directly transferred. Two different cofibrantly generated model structures are presented, with weak-equivalences the \(E_{r}\)-quasi-isomorphisms. This part extends previous works of F. Muro and C. Roitzheim [J. Pure Appl. Algebra 223, No. 5, 1913–1939 (2019; Zbl 1409.18020)]
Despite the technical nature of the subject, this paper is pleasant to read. Several remarks help to understand the choices made in the definition of the closed model structures.