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

中文翻译：

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关于弱负子类别，权重结构和（弱）近似三角分类

摘要

在本说明中，我们证明了某些三角分类类别在A. Neeman的意义上是（弱）近似的。我们证明，如果\（\ underline {C}（G，G [i]）= \ {}，则由单个对象\（G \）紧凑生成的三角分类\（\ underline {C} \）几乎是近似的。 0 \} \）为\（I> 1 \） （我们说\（G \）是如果这种假设被满足弱负;的情况下平等\（\下划线{C}（G，G [1] ）= \ {0 \} \）也被Neeman自己提到）。此外，如果\（G \ cong \ bigoplus_ {0 \ leq i \ leq n} G_ {i} \）和\（\下划线{C}（G_ {i}，G_ {j} [1]）= \ { 0 \} \）每当\（I \当量Ĵ\）然后\（\ underline {C} \）也是可近似的。后一种结果可能是有用的，因为（在更多其他假设下）它可以将\（\ underline {C} \）的某些显式子类别表征为有限从子类别同调仿函数\（\下划线{C} ^ {C} \）的紧凑对象\（\下划线{C} \）到\（R \） -模块（为一个诺特交换环\（R \ ），以使\（\下划线{C} \）为\（R \）-线性）。可以将此语句应用于某些伴随函子和\（t \）-结构的构造。我们的（弱）逼近度的证明在上述假设下的\（\下划线{C} \）与某些（弱）权重结构（的权重分解）密切相关，我们将详细讨论这种关系