PGF-complexes And Strongly PGF-modules(Complexes)

In chapter three,the concept of projectively coresolved Gorenstein flat complexes is introduced.If there exists an exact sequence of complexes(?)such that(?)remains exact for any injective complex I and G=Ker u0,where Pi is a projective complex for all(?),then G is called a projectively coresolved Gorenstein flat complex.Over arbitrary associate ring R,it is proved that G is a projectively coresolved Gorenstein flat complex if and only if module Gm of G for each degree is a projective coresolved Gorenstein flat module.Meanwhile,some basic properties of projectively coresolved Gorenstein flat complexes are studied and the relationship of projectively coresolved Gorenstein flat dimension of complexes and the projectively coresolved Gorenstein flat dimension of modules for each degree is discussed.In chapter four,the concepts of strongly projective coresolved Gorenstein flat modules and strongly projective coresolved Gorenstein flat complexes are introduced.If there exists an exact sequence of modules(?)such that(?)remains exact for any injective module E and M=Ker f,where P is a projective module,then M is called a strongly projective coresolved Gorenstein flat module.If there exists an exact sequence of complexes(?)such that(?)remains exact for any injective complex I and G=Ker f,where P is a projective complex,then G is called a strongly projective coresolved Gorenstein flat complex.It is proved that every projective module(complex)is a strongly projective coresolved Gorenstein flat module(complex),and M(G)is a projectively coresolved Gorenstein flat module(complex)if and only if it is a direct summand of a strongly projective coresolved Gorenstein flat module(complex)over arbitrary associate ring R.Meanwhile,some basic properties of strongly projective Gorenstein coresolved flat modules(complexes)are studied.