A Relative Telescope-Type Conjecture: Support and Generation in Half-Bounded Derived Categories

Abstract

Let $R$ be a commutative noetherian ring. We ask whether complexes with finitely generated cohomology and a fixed support can be generated by relatively compact objects, using coproducts which exist in the half-bounded derived category. We prove that $\mathbf D^+_{\mathrm{fg},W}(R)=\operatorname{Loc}_{\mathrm{ex}}(\mathbf D^b_{\mathrm{fg},W}(R))$ whenever $W\subseteq\operatorname{Spec} R$ is specialization closed and has finite dimension. The same equality holds for every specialization-closed subset of $\operatorname{Spec} B$ when $B$ is a finite-type algebra over a noetherian domain $A$ of finite character and either $A$ or $B$ is regular. These equalities classify the thick Hom-submodules closed under existing coproducts. We also prove the corresponding bounded-above statement from the support results of Matsui--Takahashi. For an arbitrary noetherian ring, the bounded-below generation conjecture reduces to torsion complexes over quotient domains, with their support unchanged. Finally, over a regular noetherian domain of infinite dimension, we give a bounded-below object which is generated by bounded objects with the same support but has no bounded-below representative by finitely generated modules. The comparison functor from the bounded-below derived category of finite modules is therefore neither full nor essentially surjective.