Flat complexes, pure periodicity and pure acyclic complexes
Flat complexes, pure periodicity and pure acyclic complexes
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DOI：
10.1016/j.jalgebra.2017.02.013
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发表时间：
2017-06
影响因子：
0.9
通讯作者：
D. Simson
中科院分区：
文献类型：
--
作者：
D. Simson
该论文可以被视为 Emmanouil (2016)[5] 最近论文的补充和延伸。除此之外，还提出了 Emmanouil 结果的另一种函子类别方法。通过应用 Neeman (2008)[16] 的思想和其中获得的主要结果，我们证明局部有限呈现的 Grothendieck 范畴 A 中的链复形 F 是纯无环的，当且仅当从 A 到 F 中的纯射影对象的复形 P 的任何链映射 f: P→ F 是零同伦的。因此我们证明 A 中的任何纯周期对象都是纯射影的。此外，我们证明 A 是纯半单的当且仅当 A 具有纯 QF 性质，即 A 中的每个纯内射对象都是纯射影。
The paper can be viewed as an addition and extension of the recent paper of Emmanouil (2016)[5]. Among others, an alternative functor category approach to Emmanouil's results is presented. By applying an idea of Neeman (2008)[16] and the main results obtained there, we prove that a chain complex F in a locally finitely presented Grothendieck category A is pure acyclic if and only if any chain map f: P→ F from a complex P of pure-projective objects in A to F is null-homotopic. As a consequence we prove that any pure periodic object in A is pure-projective. Moreover, we show that A is pure semisimple if and only if A has the pure QF-property, that is, every pure-injective object in A is pure-projective.