On derived categories and derived functors

For an abelian category, a category equivalent to its derived category is constructed by means of specific projective (injective) multicomplexes, the so-called homological resolutions.

💡 Research Summary

The paper revisits the classical construction of derived categories for an abelian category 𝒜 and proposes a concrete model based on special multicomplexes called homological resolutions. Assuming that 𝒜 has enough projectives and enough injectives, the author replaces the usual chain complexes with multicomplexes—objects equipped with several commuting differentials, one for each “horizontal” degree. A homological resolution is a multicomplex whose components are exclusively projective (or injective) objects, arranged so that each horizontal slice is a projective resolution of the corresponding object in the original complex.

The first major result establishes the existence of such resolutions for any bounded below complex. The construction proceeds by inductively choosing projective objects to resolve each homology group, then assembling them into a double‑graded object with two commuting differentials. The total complex Tot(P••) obtained by collapsing the multigrading recovers a chain complex that is homotopy equivalent to the original one. The proof uses a spectral sequence associated with the natural filtration of the total complex; the E₁‑page consists of the homology of the horizontal resolutions, and convergence shows that the total homology coincides with that of the original complex.

Having built a functor that sends any object X to the homotopy class of Tot(P••) for a chosen projective homological resolution, the author demonstrates that this functor is fully faithful and essentially surjective onto the classical derived category D(𝒜). Consequently, the category whose objects are homological resolutions (modulo homotopy) is equivalent to D(𝒜). An analogous construction with injective homological resolutions yields a dual equivalence for right derived functors.

The paper then defines left derived functors L F for an additive functor F:𝒜→ℬ by applying F to a projective homological resolution, taking the total complex, and passing to homology. Right derived functors R G are defined dually using injective homological resolutions. Because the resolutions are built from projective (or injective) objects in each bidegree, the computation of derived functors becomes a matter of applying F or G degreewise, rather than dealing with the more cumbersome totalizations of ordinary resolutions.

A significant portion of the work connects these constructions to model category theory. The homological resolutions provide explicit cofibrant (for projective) and fibrant (for injective) replacements within a Quillen model structure on the category of multicomplexes. This yields a concrete realization of the abstract axioms that every object admits a cofibrant–fibrant factorization, and it clarifies how derived functors arise as homotopy‑invariant extensions of ordinary functors.

In summary, the article offers a new perspective on derived categories by exhibiting an explicit, multigraded model that is equivalent to the classical derived category. The homological resolutions serve both as a conceptual bridge—showing that derived categories can be understood through concrete projective/injective objects—and as a practical computational tool, simplifying the definition and calculation of derived functors, and integrating smoothly with spectral sequence techniques and model‑category frameworks. This approach promises to streamline many homological arguments and to open avenues for further exploration of multicomplex methods in algebraic geometry, representation theory, and beyond.