Brown--Adams representability for triangulated categories with locally 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 $n\in\mathbb{Z}$.