交换环的w-弱finitistic维数的注记

Note on the w-weak finitistic dimension of a commutative ring

设R是交换环,M是R-模.引入了模M的w-投射维数w-pd_R(M)和环R的w-弱finitistic维数w-f PD(R).给出w-f PD(R)=0的充分必要条件.证明了若R是w-凝聚环,M是有限表现R-模,则M有w-投射分解…→P_n→P_(n-1)→…→P_1→P_0→M→0,其中P_i是有限型的w-投射模,这里i=0,1,….最后,证明了若R是w-半遗传环,w-f PD(R)#1.

Let R be a commutative ring and M be an R -module. This paper introduces and studies the w -projective dimension w -pdR （M） of an R -module M and the w -weak finitistic dimension w - fPD （R） of R. The sufficient and necessary condition of w - fPD （R） = 0 is given. As an application, it is shown that if R is a w -coherent ring, M is of finitely presented type, then M has a w -projective resolution ... →Pn →Pn-1→P1→P0→ M→0 , where Pi is w -projective of finite type for i = 0,1 ,.... Finally, if R is w -semi-hereditary, then w - fPD （R）≤1.