On the existence of a compact generator on the derived category of a noetherian formal scheme

In this paper, we prove that for a noetherian formal scheme X, its derived category of sheaves of modules with quasi-coherent torsion homologies D_qct(X) is generated by a single compact object. In an appendix we prove that the category of compact objects in D_qct(X) is skeletally small.

💡 Research Summary

The paper addresses a fundamental question in the homological algebra of formal schemes: whether the derived category of quasi‑coherent torsion sheaves on a Noetherian formal scheme X, denoted D_qct(X), admits a single compact generator. After recalling the basic definitions of formal schemes, quasi‑coherent sheaves, and torsion sheaves, the authors define D_qct(X) as the full triangulated subcategory of the derived category of O_X‑modules consisting of complexes whose cohomology sheaves are both quasi‑coherent and torsion. They first establish that D_qct(X) is a triangulated category with all small coproducts and that it satisfies the usual compactness criteria.

The core of the argument is the construction of a specific compact object G that generates D_qct(X). The construction proceeds locally: for each affine open U = Spf(A) in a finite affine cover of X, one chooses an ideal of definition I ⊂ A and forms the Koszul complex K(I) on a finite set of generators of I. By completing K(I) with respect to the I‑adic topology one obtains a bounded complex \widehat{K}(I) whose terms are finitely generated A‑modules and whose cohomology sheaves are quasi‑coherent torsion. The authors prove that \widehat{K}(I) is compact in D_qct(U).

Next, they glue these local complexes together using Čech techniques. The resulting global complex G is a finite direct sum of the completed Koszul complexes, shifted appropriately, and it lives in D_qct(X). The key technical point is that the gluing respects compactness: the direct sum of finitely many compact objects remains compact, and the differentials are compatible with the adic topology, ensuring that G remains in the subcategory of torsion sheaves.

Having constructed G, the authors verify that it is a strong generator. They show that for any object M in D_qct(X) there exists an integer n such that if Hom_{D_qct(X)}(G, M