Linkage module by G_K-perfect module and behavior of graded reduced

Let R be commutative Noetherian ring with identity (not necessarily local), M and N be nonzero finitely generated R- modules. Let K be a semidualizing R-module and χ_n be a subcategory of G_K- perfect R-modules of grade n . In this paper, we study the theory of linkage of module with respect to χ_n. Also, we generalize a number of statements about horizontally linked modules to linked modules with respect to χ_n. In particular, we define the reduce grade of R-module M then for linked module with respect to χ_n, over a commutative Noetherian local ring R, the connection of its reduced grade and depth of its linked module is discussed. As a consequence, over a Cohen-Macaulay local ring, we determine the attached primes of the local cohomology module〖 H〗_m^(〖cd〗_m (M)) (M), in terms of the depth of its linked module. Finally, under certain conditions, it is shown that some relation between relative reduce grade of M and relative reduce grade of its linked module.Keywords: Linkage of modules, Semidualizing module, G_K-dimension.

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