LAUSR

Assume that $$(X,{\mathcal {O}}_X)$$ ( 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 $${\mathcal {O}}_X$$ 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 $$\mathrm {fPD}(R)<+\infty $$ fPD ( R ) < + ∞ if and only if $$\mathrm {sup}_{{\mathfrak {m}}\in \mathrm {Max}R}\mathrm {fPD}(R_{\mathfrak {m}})<+\infty $$ sup m ∈ Max R fPD ( R m ) < + ∞ where fPD( R ) (resp. fPD $$(R_{\mathfrak {m}})$$ ( R m ) ) is the little finitistic projective dimension of R (resp. $$R_{\mathfrak {m}}$$ R m ).