Study On Homology、Relative Homology And Hom-algebras

The paper mainly studies some relative homological dimensions and relative problems of Hom-algebras.Firstly, a ring is left Noetherian if and only if the left FP-projective dimen-sion of a ring is zero. So the left FP-projective dimension of a ring measures how far away a ring is from being Noetherian. In this paper, for H is finitely dimen-sional Hopf algebra and A is a algebra over a field k, it is proved that the left FP-projective dimension is invariant under cleft extensions when H*is semisim-ple and A is left coherent. And we study the relations between FP-projective dimensions in A#H-Mod and that in AH-Mod. Moreover, we characterize the FP-projective preenvelope (resp.. precover)under the H-extension and cleft ex-tensions, respectively.Secondly, the global cotorsion dimension of a ring measures how far away a ring is from being perfect. We discuss that the relationship between the global cotorsion dimensions of the smash product A#H and A. And we give some sufficient conditions such that the left global cotorsion dimension of A#H is equal to that of A. As applications, we study the invariants of IF properties and Gorenstein global dimensions.Finally, we investigate the structure of Hom-type generalization of associa-tive algebras. We define Hom dimodule and Hom D-equation. Moreover, we give some solutions of the Hom D-equation from Hom-dimodules over Hom-bialgebras. And we construct the FRT type theorem in the category of Hom-dimodules.Monoidal Hom-algebras is a kind of Hom-structure which is obtained from the point of view of monoidal categories. We define the smash product of Hom type as the generalization of smash products in Hopf algebras. We check that it is the monoidal Hom-algebra, and prove that H#H*is isomorphic to Endk(H) as monoidal Hom-algebras under some conditions.