On Finitistic Flat Dimension of Rings and Schemes

Assume that (X, O-X) is an arbitrary scheme. The concept of the big (resp. little) finitistic flat dimension FFD (X) (resp. fFD(X)) of X will be introduced. It is shown that if X is affine and any flat quasi-coherent O-X-module has finite projective dimension, then finitistic flat 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) < + infinity if and only if sup(m is an element of MaxR)fPD (R-m) < + infinity where fPD (R) (resp. fPD (R-m)) is the little finitistic projective dimension of R (resp. R-m).

Automated summary

This article introduces the big (or small) finitistic flat dimension of schemes and their properties, proving the necessary and sufficient conditions for the flat dimensions of affine schemes to be finite are finite projective dimensions. It also identifies the minimum requirements for finite flat dimensions, and demonstrates that under certain conditions, the little finitistic projective dimension is finite.