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 authors address 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. Building on earlier work that established compact generation of (D_{qct}(X)) but did not control the number of generators, they prove that one compact object suffices.

The paper begins with a concise review of formal schemes, adic rings, and the categories of modules involved. For a formal scheme ((X,\mathcal O_X)) they define the Grothendieck category (\mathcal A_{qct}(X)) of quasi‑coherent torsion sheaves, whose derived category (D_{qct}(X)) is the main object of study. Compact objects in this triangulated category are precisely the perfect complexes, i.e. those locally quasi‑isomorphic to bounded complexes of finite‑type locally free sheaves.

The core technical development proceeds in several steps. First, in the affine case (X=\operatorname{Spf}(A)) with (A) an (I)-adic noetherian ring, they exhibit a Koszul complex built from generators of the defining ideal (I) and show that it generates the subcategory of (I)-torsion complexes (D_I(A)). Via the equivalence (D_{qct}(X)\simeq D_I(A)) (from AJL1) this yields a compact generator for the affine formal scheme.

Next, they treat the interaction between a closed formal subscheme (Z\subset X) and its open complement (U=X\setminus Z). Lemma 2.2 proves that the derived category of complexes supported on (Z) is equivalent to the derived category of ((I+\mathfrak a))-torsion modules, where (\mathfrak a) defines (Z). Proposition 2.4 establishes a localization triangle \