Derived Igusa-Todorov Functions And Finitistic Dimension Conjecture

This paper mainly studies the generalization of Igusa-Todorov functions in the bounded derived category and some results of the finitistic dimension under the extension of algebras.In the first part,we define the notion of the syzygy dimension and the pro-jective dimension of a bounded complex(see Definition 3.5),and prove thatΩDdim(X·)= pdD(X·),for any X·∈Db(A),here A is an artin algebra.Follow-ing this result,the Igusa-Todorov function in mod(A)is generalized into derived category.Thus,we get the D-IT functions.Then we extend the related conclu-sions in mod(A)to the derived category and get the generally result(see The-orem4.10):Let Db(A)satisfy the Fitting’s property.Then for any distinguished triangle X·→Y·→Z·→X·[1]in Db(A)such that pd(Z·)<∞,we have that pd(Z·)≤ΨD(X·(?)Y·)+ 1.Recently,the finitistic dimension conjecture is related to studying finitistic dimensions for extensions of algebras.In the second part,we consider the rela-tion between the finitistic dimension and the extension of algebra,and get the following results under different conditions of algebraic extension:Let B(?)A be an extension of artin algebras and I be an ideal of B.(1)If pdBA<∞,Irad(B)is a left ideal in A,the full subcategory of B/I-modules is B-syzygy-finite and A is A-syzygy-finite,then findim(B)<∞;(2)If I is a left ideal of A,the full sub-category of B/I-modules and A-modules are torsionless-finite respectively,then findim(B)<∞.