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 $M:{\mathbf{D}}_{perf}(A)\to \mathrm{Mod}\text{-}R$, also satisfying $M(A[-n])\in \mathrm{mod}\text{-}R$, 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 $n\in \mathbb{Z}$.

Replacement for arXiv:2405.06475. Comments are welcome!