Brown--Adams representability for triangulated categories with locally coherent cohomology

TL;DR

This paper extends Brown and Brown-Adams representability theorems to triangulated categories with locally coherent cohomology, showing that certain cohomological functors are representable by objects with coherent cohomology.

Contribution

It introduces new variants of Brown and Brown-Adams representability for triangulated categories with coherent cohomology, providing explicit conditions for representability.

Findings

Every cohomological functor satisfying certain conditions is representable.

Representing objects have coherent cohomology.

Results apply to dg-algebras over noetherian rings with coherent cohomology.

Abstract

In this paper, we deal with two types of representability. The first is a variant of the Brown representability theorem in the spirit of Rouquier and Neeman. The second is a variant of the Brown-Adams representability. If $A$ is a dg-algebra over a commutative noetherian ring $R$, such that $A$ has coherent cohomology, it is shown that every cohomological (contravariant) functor $M:\mathbf{D}_{perf}(A)\to\mathrm{Mod}\textrm{-}R$, also satisfying $M(A[-n])\in\mathrm{mod}\textrm{-}R$, for all $n\in\mathbb{Z}$ is isomorphic to $\mathbf{D}(A)(-,X)|_{\mathbf{D}_{perf}(A)}$, where $X\in\mathbf{D}(A)$ is such that $H^n(X)$ is coherent for all $n\in\mathbb{Z}$.