On Finitistic Flat Dimension of Rings and Schemes

Assume that (X, OX) is an arbitrary scheme. The concept of the big (resp. little) finitistic at dimension FFD(X) (resp. fFD(X)) of X will be introduced. It is shown that if X is affine and any at quasi-coherent OX-module has finite projective dimension, then finitistic at dimensions are finite if and only if the finitistic projective dimensions are finite. We will find the minimum requirements for FFD(X) (resp. fFD(X)) to be finite. Furthermore, if R is a commutative n-perfect ring, we prove that fPD(R) +∞ if and only if sup m∈MaxR fPD(Rm) +∞ where fPD(R) (resp. fPD(Rm)) is the little finitistic projective dimension of R (resp. Rm).