论文标题
Gorenstein扁平模块和复合物相对于二元性对的同源和同位方面
Homological and homotopical aspects of Gorenstein flat modules and complexes relative to duality pairs
论文作者
论文摘要
我们研究了Gorenstein Flat模块的同源和同位方面，相对于双重性对$（\ Mathcal {l，a}）$。这些模块被定义为$ \ MATHCAL {l} $组件的精确链复合物的循环，这些组件在$ \ Mathcal {a} \ cap {}^\ perp \ perp \ mathcal {a} = a} = \ mathcal {a} = \ mathcal {a} \ cap \ cap \ big big big big big big big biccap_ In中被对象张开后保持精确\ Mathbb {Z} _ {> 0}}} {\ rm ker}（{\ rm selt}^i_ {r^{\ rm o}}}（ - ，\ mathcal {a}}）\ big）\ big）$。 In the case where $(\mathcal{L,A})$ is product closed and bicomplete (meaning in addition that $\mathcal{L}$ is closed under extensions, (co)products, $R \in \mathcal{L}$, $(\mathcal{A,L})$ is also a duality pair, and $\mathcal{A}$ is the right half of a遗传性完整的合并对）我们证明，这些相对的Gorenstein平坦模块在扩展下关闭，并且相应的Gorenstein平面维度的表现很好，因为它恢复了其（绝对）Gorenstein Flat Part的许多属性和特征（例如，可以用Torsion functors）进行描述）。后者反过来是我们在这些相对Gorenstein Flat模块与某些Gorenstein Injextive模块相对于$ \ Mathcal {a} $之间显示的Pontryagin二元性关系的结果。我们还发现，相对于$（\ Mathcal {l，a}）$，从这些Gorenstein Flat模块和复合物中发现了几个遗传性和协同生成的Abelian模型结构。在链络合物的层面上，我们发现这些模型结构的同型类别之间的三个回忆，以及连接这些回忆的几个派生辅助。
We study homological and homotopical aspects of Gorenstein flat modules over a ring with respect to a duality pair $(\mathcal{L,A})$. These modules are defined as cycles of exact chain complexes with components in $\mathcal{L}$ which remain exact after tensoring by objects in $\mathcal{A} \cap {}^\perp\mathcal{A} = \mathcal{A} \cap \Big( \bigcap_{i \in \mathbb{Z}_{> 0}} {\rm Ker}({\rm Ext}^i_{R^{\rm o}}(-,\mathcal{A})) \Big)$. In the case where $(\mathcal{L,A})$ is product closed and bicomplete (meaning in addition that $\mathcal{L}$ is closed under extensions, (co)products, $R \in \mathcal{L}$, $(\mathcal{A,L})$ is also a duality pair, and $\mathcal{A}$ is the right half of a hereditary complete cotorsion pair) we prove that these relative Gorenstein flat modules are closed under extensions, and that the corresponding Gorenstein flat dimension is well behaved in the sense that it recovers many of the properties and characterizations of its (absolute) Gorenstein flat counterpart (for instance, it can be described in terms of torsion functors). The latter in turn is a consequence of a Pontryagin duality relation that we show between these relative Gorenstein flat modules and certain Gorenstein injective modules relative to $\mathcal{A}$. We also find several hereditary and cofibrantly generated abelian model structures from these Gorenstein flat modules and complexes relative to $(\mathcal{L,A})$. At the level of chain complexes, we find three recollements between the homotopy categories of these model structures, along with several derived adjunctions connecting these recollements.
