Research On The Homological Dimensions Of A Class Of Algebras

Homological dimension is one of powerful tools for studying algebra.The Skew group algebra,which is a natural generalization of the group algebra,is an important kind of Artin algebras,and it is one of research hotspots in the representation theory of algebras.In this thesis,we introduce the tilting dimensions of algebras and modules.We discuss the basic properties on tilting dimension,Gorenstein dimension and n-presented dimension.Moerover,we study the relation between Artin R-algebra ? and skew group algebra ?G under the three different circumstances.The research contents of this thesis are mainly divided into three parts:In the first part,we firstly introduce the concept of tilting projective module,we give the criterion to judge one module to be a tilting projective module as well as e-quivalent conditions on tilting projective modules.We prove that the tensor product of?G and X over ? is a tilting module over ?G when X is a tilting module over ?.And we also prove that the tensor product of ?G and X over ? is a tilting projective module relative to the tensor product of ?G and T over ? when X is a tilting projec-tive module relative to T,where T is a tilting module over ?.Secondly,we introduce the definition of tilting projective dimension,and discuss its properties and the relation among the tilting projective dimensions of ?-module A,B and C over the exact se-quence 0?A?B ?C? 0.At last,we give the relation of tilting global dimension between Artin R-algebra ? and skew group algebra ?G.In the second part,we firstly study the relation of Gorenstein projective dimension between the ?-module X and the tensor product of ?G and X over ?,where the tensor product is a ?G-module.Based on this discussion,we give the relation of Gorenstein global dimension between Artin R-algebra ? and skew group algebra ?G.At last,we study the relation of Gorenstein flat dimension between ?-module X and the tensor product of ?G and X over ?.In the third part,we firstly discuss the n-presented dimension of skew group al-gebra.Let ? be an Artin algebra,we obtain the relation of n-presented dimensions between ?-module X and the tensor product of ?G and X over ?.Finally,we prove that the n-presented dimension between Artin R-algebra ? and skew group algebra ?G is equal.