Brown--Adams representability for triangulated categories with locally coherent cohomology
George Ciprian Modoi
TL;DR
The paper develops a general framework for representability of contravariant functors on $R$-linear categories with weak kernels, linking representability to finite presentation and weak exactness. It then extends Brown–Adams representability to triangulated categories via a finite-step construction from a generating subcategory and shows how this extends to big categories with coproducts. By introducing $G$-locally coherent cohomology and pure dimension, the authors characterize the restricted Yoneda image and establish fullness for relevant subcategories, including $s$-bounded variants. A central application to derived categories of dg-algebras shows that for a proper, regular $A$ with coherent cohomology, a cohomological functor $M$ on $\mathbf{D}_{perf}(A)$ is representable precisely when $M^{*}(A)$ is finitely generated, with analogous results for $\mathbf{D}_{coh}(A)$ and scheme-theoretic contexts. The results unify and generalize Brown–Adams type representability beyond approximability, with implications for both algebraic and geometric settings.
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 $n\in\mathbb{Z}$.