Homological Dimensions Of Derived Categories And Abelian Categories

Let A be an artin algebra.We give an upper bound for the dimension of the bounded derived category of the category mod A of finitely generated right ?-modules in terms of the projective and injective dimensions of certain class of simple right A-modules as well as the radical layer length of ?�In addition,we give an upper bound for the dimension of the singularity category of mod A in terms of the radical layer length of A.Let,A be an abelian category having enough projective objects and enough injec-tive objects.We prove that if A admits an additive generating object,then the extension dimension and the weak resolution dimension of,A are identical,and they are at most the representation dimension of A minus two.By using it,for a right Morita ring?,we establish the relation between the extension dimension of the category mod ?of finitely generated right ?-modules and the representation dimension as well as the right global dimension of ?.In particular,we give an upper bound for the extension di-mension of mod ? in terms of the projective dimension of certain class of simple right A-modules and the radical layer length of A.In addition,we investigate the behavior of the extension dimension under some ring extensions and recollements.Let ? be an artin algebra and O=IO(?)I1(?)I2(?)�(?)In a chain of ideals of ? such that(Ii+1//Ii)rad(?/Ii,)=0 for any 0?i?n-1 and A/In is semisimple.If the cardinality of the set[i]the projective dimension of Ii-is infinite for 1?i?n }is at most two,then the finitistic dimension of ? is finite.As a consequence,we have that if the cardinality of the set {i]the projective dimension of radi(A)is infinite for 1?i?n-1 is at most two(where n is the Loewy length of ?),then the finitistic dimension of A is finite.Some known results are obtained as corollaries.