论文标题
无环复合物和戈伦斯坦环
Acyclic complexes and Gorenstein rings
论文作者
论文摘要
对于给定类别的模块$ \ MATHCAL {a} $，我们用$ \ widetilde {\ Mathcal {\ Mathcal {a}} $具有$ \ Mathcal {a} $的所有循环的精确复合物$ x $的类$ \ MATHCAL {A} $。我们使用符号$ \ MATHCAL {gi} $ $（\ Mathcal {gf}，\ Mathcal {gp}）$用于Gorenstein Imentive的类（分别Gorenstein Flat，Gorenstein Poxpartive，Gorenstein partsive，分别是gorenstein partiveive） $ \ MATHCAL {PGF} $用于投影核心Gorenstein Flat模块（有关定义，请参见第2节）。我们证明，以下内容与任何环$ r $ $：（1）每个精确的注射模块的复合物都是完全无环的。 （2）Gorenstein Injective模块的每个精确复合物都位于$ \ widetilde {\ Mathcal {gi}} $中。 （3）$ dw（\ Mathcal {gi}）$中的每个复合物都是dg-gorenstein Injective。我们表明，扁平和戈伦斯坦扁平模块的复合物的模拟结果也有任意环。如果此外，对于某些整数$ n \ ge 0 $，戒指是$ n $ - 完美的，那么Flat和Gorenstein Flat Flat模块的三个等效陈述也与其对应物的投影和凝固性的Gorenstein Flat Flat模块相同。我们还证明了Gorenstein Rings的以下特征：让$ r $是一个交通连贯的环。以下陈述是等效的：（1）每一个精确的fp Injumentive模块的复合物都具有其所有循环滴入式内脏模块。 （2）每一个精确的注射剂都具有其所有循环叮叮态模块，并且每个$ r $ $ - 模块M $ m^+$是Gorenstein Flat是ding Injementive。如果ring $ r $具有有限的krull尺寸，则上述语句（1），（2）也等同于（3）$ r $是Gorenstein Ring（从Iwanaga的意义上）。
For a given class of modules $\mathcal{A}$, we denote by $\widetilde{\mathcal{A}}$ the class of exact complexes $X$ having all cycles in $\mathcal{A}$, and by $dw(\mathcal{A})$ the class of complexes $Y$ with all components $Y_j$ in $\mathcal{A}$. We use the notations $\mathcal{GI}$ $(\mathcal{GF}, \mathcal{GP})$ for the class of Gorenstein injective (Gorenstein flat, Gorenstein projective respectively) $R$-modules, $\mathcal{DI}$ for Ding injective modules, and $\mathcal{PGF}$ for projectively coresolved Gorenstein flat modules (see section 2 for definitions). We prove that the following are equivalent over any ring $R$: (1) Every exact complex of injective modules is totally acyclic. (2) Every exact complex of Gorenstein injective modules is in $\widetilde{\mathcal{GI}}$. (3) Every complex in $dw(\mathcal{GI})$ is dg-Gorenstein injective. We show that the analogue result for complexes of flat and Gorenstein flat modules also holds over arbitrary rings. if moreover, the ring is $n$-perfect for some integer $n \ge 0$, then the three equivalent statements for flat and Gorenstein flat modules are also equivalent with their counterparts for projective and projectively coresolved Gorenstein flat modules. We also prove the following characterization of Gorenstein rings: Let $R$ be a commutative coherent ring. The following statements are equivalent: (1) every exact complex of FP-injective modules has all its cycles Ding injective modules. (2) every exact complex of injectives has all its cycles Ding injective modules and every $R$-module M such that $M^+$ is Gorenstein flat is Ding injective. If moreover the ring $R$ has finite Krull dimension then statements (1), (2) above are also equivalent to (3) $R$ is a Gorenstein ring (in the sense of Iwanaga).
