论文标题
Frobenius函子，稳定等价和$ K $ - Gorenstein Projective模块的理论
Frobenius functors, stable equivalences and $K$-theory of Gorenstein projective modules
论文作者
论文摘要
由于$ k $理论的差异，杜格（Dugger）和希普利（Shipley）的一个例子意味着，稳定类别的戈伦斯坦（Gorenstein）投射模块等效性不应是quillen等价性。我们给出了足够和必要的条件，使弗罗贝尼乌斯对两个阿贝尔类别之间的忠实函子是quillen等价性，这也等同于frobenius functors诱导Gorenstein投射物体稳定类别之间的相互反向等价。 我们表明，Gorenstein Projective对象的类别是Waldhausen类别，然后引入并表征Gorenstein $ k $ -groups。作为应用程序，我们表明，莫里塔型的稳定等效性保留Gorenstein $ k $ -groups，CM填充性和CM繁琐。提出了两个特定的路径代数示例，以说明结果，为此计算了Gorenstein $ k_0 $和$ k_1 $ groups。
Owing to the difference in $K$-theory, an example by Dugger and Shipley implies that the equivalence of stable categories of Gorenstein projective modules should not be a Quillen equivalence. We give a sufficient and necessary condition for the Frobenius pair of faithful functors between two abelian categories to be a Quillen equivalence, which is also equivalent to that the Frobenius functors induce mutually inverse equivalences between stable categories of Gorenstein projective objects. We show that the category of Gorenstein projective objects is a Waldhausen category, then Gorenstein $K$-groups are introduced and characterized. As applications, we show that stable equivalences of Morita type preserve Gorenstein $K$-groups, CM-finiteness and CM-freeness. Two specific examples of path algebras are presented to illustrate the results, for which the Gorenstein $K_0$ and $K_1$-groups are calculated.
