标题： Ischebeck的公式，阶数和准同调维数 显示英文标题

标题： Ischebeck's formula, Grade and quasi-homological dimensions

摘要： 准射影维数和准内射维数是最近引入的同调不变量，分别推广了经典的射影维数和内射维数的概念。 对于局部环$R$和有限生成的$R$-模$M$和$N$，我们提供了涉及准同调维数的条件，使得等式$\sup \lbrace i\geq 0: \operatorname{Ext}_R^i(M,N)\not=0 \rbrace =\operatorname{depth} R-\operatorname{depth} M$成立，我们将此称为 Ischebeck 公式。 在这方面的一个结果推广了 Ischebeck 关于有限内射维数模的著名结果，考虑了准内射维数。 另一方面，我们建立了一个不等式，将有限生成模的准射影维数与其阶数相关联，并引入了准完美模的概念，作为完美模的自然推广。 我们为这个新概念证明了几条类似于经典结果的结论。 此外，我们提供了在局部环上具有有限准内射维数的有限生成模的阶数公式，以及具有有限准射影维数的模的阶数不等式。 在我们的研究中，也获得了 Cohen-Macaulay 性的标准。 显示英文摘要

摘要： The quasi-projective dimension and quasi-injective dimension are recently introduced homological invariants that generalize the classical notions of projective dimension and injective dimension, respectively. For a local ring $R$ and finitely generated $R$-modules $M$ and $N$, we provide conditions involving quasi-homological dimensions where the equality $\sup \lbrace i\geq 0: \operatorname{Ext}_R^i(M,N)\not=0 \rbrace =\operatorname{depth} R-\operatorname{depth} M$, which we call Ischebeck's formula, holds. One of the results in this direction generalizes a well-known result of Ischebeck concerning modules of finite injective dimension, considering the quasi-injective dimension. On the other hand, we establish an inequality relating the quasi-projective dimension of a finitely generated module to its grade and introduce the concept of a quasi-perfect module as a natural generalization of a perfect module. We prove several results for this new concept similar to the classical results. Additionally, we provide a formula for the grade of finitely generated modules with finite quasi-injective dimension over a local ring, as well as grade inequalities for modules of finite quasi-projective dimension. In our study, Cohen-Macaulayness criteria are also obtained.