论文标题
fp-prokentive周期性
Fp-projective periodicity
论文作者
论文摘要
Benson和GoodeArl发现的周期性现象与无环复合物中共伴生物的对象的行为有关。众所周知，任何平坦的$ \ Mathsf {proj} $ - 定期模块是投射的，任何fp Injextive $ \ mathsf {inj} $ - 定期模块都是注入的，任何$ \ MATHSF {COT} $ - 定期模块都是Cotorsion。众所周知，任何纯$ \ Mathsf {pproj} $ - 定期模块都是纯标记，并且任何纯$ \ Mathsf {pinj} $ - 周期性模块都是纯净的。概括了Saroch和Stovicek的结果，我们表明每个$ \ mathsf {fpproj} $ - 定期模块都是弱fp-projective。证明是相当基本的，仅使用强大的纯标记周期性和山丘引理。更笼统地，我们证明，在本地有限的Grothendieck类别中，每个$ \ mathsf {fpproj} $ - 周期性对象都是弱fp-projective。在本地相干类别中，所有弱fp-projective对象都是fp-projective。我们还提出了反例，表明在常规有限生成的通勤代数（或遗传有限的二维关联代数）上，非纯净的$ \ mathsf {pproj} $ - 定期模块，而不是纯正的。
The phenomenon of periodicity, discovered by Benson and Goodearl, is linked to the behavior of the objects of cocycles in acyclic complexes. It is known that any flat $\mathsf{Proj}$-periodic module is projective, any fp-injective $\mathsf{Inj}$-periodic module is injective, and any $\mathsf{Cot}$-periodic module is cotorsion. It is also known that any pure $\mathsf{PProj}$-periodic module is pure-projective and any pure $\mathsf{PInj}$-periodic module is pure-injective. Generalizing a result of Saroch and Stovicek, we show that every $\mathsf{FpProj}$-periodic module is weakly fp-projective. The proof is quite elementary, using only a strong form of the pure-projective periodicity and the Hill lemma. More generally, we prove that, in a locally finitely presentable Grothendieck category, every $\mathsf{FpProj}$-periodic object is weakly fp-projective. In a locally coherent category, all weakly fp-projective objects are fp-projective. We also present counterexamples showing that a non-pure $\mathsf{PProj}$-periodic module over a regular finitely generated commutative algebra (or a hereditary finite-dimensional associative algebra) over a field need not be pure-projective.
