Deconstructibility and the Hill lemma in Grothendieck categories

A full subcategory of a Grothendieck category is called deconstructible if it consists of all transfinite extensions of some set of objects. This concept provides a handy framework for structure theory and construction of approximations for subcategories of Grothendieck categories. It also allows to construct model structures and t-structures on categories of complexes over a Grothendieck category. In this paper we aim to establish fundamental results on deconstructible classes and outline how to apply these in the areas mentioned above. This is related to recent work of Gillespie, Enochs, Estrada, Guil Asensio, Murfet, Neeman, Prest, Trlifaj and others.

💡 Research Summary

The paper introduces and develops the notion of a deconstructible subcategory within any Grothendieck category 𝒢. A subcategory 𝔇 is called deconstructible if there exists a set 𝒮 of objects such that 𝔇 consists precisely of all transfinite extensions (i.e., λ‑directed colimits for some regular cardinal λ) of objects from 𝒮. The authors first prove that such classes are automatically closed under the standard Grothendieck operations: coproducts, extensions, and filtered colimits, and they establish that every object of a deconstructible class can be expressed as a λ‑filtered colimit of a well‑ordered chain of objects from 𝒮.

A central technical achievement is the generalisation of the Hill Lemma to the transfinite setting. The classical Hill Lemma, originally formulated for module categories, guarantees that a module built from a set of generators via successive extensions can be filtered by those generators. Here the authors replace finite extensions by λ‑directed transfinite extensions and introduce the concepts of λ‑accessibility and transfinite continuity to control the construction. They show that for any regular cardinal λ large enough to dominate the presentability ranks of the objects in 𝒮, every object of 𝔇(𝒮) admits a λ‑continuous filtration whose successive quotients lie in 𝒮. This result provides a robust tool for handling deconstructible classes in any Grothendieck context.

The paper then connects deconstructible classes to cotorsion pairs. Given a set 𝒮, the pair (𝔇(𝒮), 𝔇(𝒮)⊥) is shown to be a complete cotorsion pair. Consequently, by Hovey’s correspondence, one obtains model structures on the category of chain complexes Ch(𝒢) where the cofibrant objects are precisely the dg‑deconstructible complexes (complexes whose each degree lies in 𝔇(𝒮) and whose differentials respect the filtration). This yields both projective‑type and injective‑type model structures, as well as compatible t‑structures on the derived category D(𝒢).

The authors illustrate the power of their framework by revisiting several recent results. Gillespie’s flat‑cotorsion model structure, Murfet–Neeman’s derived category of flats, and the work of Estrada, Guil Asensio, Trlifaj on deconstructible classes of quasi‑coherent sheaves all appear as special cases of the general theory. Moreover, the paper clarifies how the Hill Lemma underlies many of these constructions, providing a unifying perspective.

Finally, the paper outlines future directions: (i) developing invariants that measure the “size’’ or complexity of a deconstructible class, (ii) extending the theory beyond Grothendieck categories to algebraic stacks or ∞‑categories, and (iii) exploring deeper interactions between t‑structures and cotorsion pairs in derived settings. In sum, the work establishes deconstructibility and the generalized Hill Lemma as fundamental tools for structural and homotopical investigations in Grothendieck categories.