Abstract
In this note we prove that certain triangulated categories are (weakly) approximable in the sense of A. Neeman. We prove that a triangulated category $$\underline{C}$$ that is compactly generated by a single object $$G$$ is weakly approximable if $$\underline{C}(G,G[i])=\{0\}$$ for $$i>1$$ (we say that $$G$$ is weakly negative if this assumption is fulfilled; the case where the equality $$\underline{C}(G,G[1])=\{0\}$$ is fulfilled as well was mentioned by Neeman himself). Moreover, if $$G\cong\bigoplus_{0\leq i\leq n}G_{i}$$ and $$\underline{C}(G_{i},G_{j}[1])=\{0\}$$ whenever $$i\leq j$$ then $$\underline{C}$$ is also approximable. The latter result can be useful since (under a few more additional assumptions) it allows to characterize a certain explicit subcategory of $$\underline{C}$$ as the category of finite cohomological functors from the subcategory $$\underline{C}^{c}$$ of compact objects of $$\underline{C}$$ into $$R$$ -modules (for a noetherian commutative ring $$R$$ such that $$\underline{C}$$ is $$R$$ -linear). One may apply this statement to the construction of certain adjoint functors and $$t$$ -structures. Our proof of (weak) approximability of $$\underline{C}$$ under the aforementioned assumptions is closely related to (weight decompositions for) certain (weak) weight structures, and we discuss this relationship in detail
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