Finite Flat and Projective Dimension

known as the left big finitistic projective dimension of A, is finite. Here pdM denotes the projective dimension of M . Unfortunately, this number is not known to be finite even if A is a finite dimensional algebra over a field, where, indeed, its finiteness is a celebrated conjecture. On the other hand, for such an algebra, finite flat certainly implies finite projective dimension, simply because each flat module is projective. So it seems that there might be results based on other conditions than finiteness of FPD(A). The germ of such a result is in [2, cor. 3.4] which shows that finite flat implies finite projective dimension for a ring which is a homomorphic image of a noetherian commutative Gorenstein ring with finite Krull dimension. Now, for such a ring, FPD(A) is in fact finite, and so formally, [2, cor. 3.4] follows from [3, prop. 6]. However, the method of proof of [2, cor. 3.4] lends itself to generalization, and in this note I will use it to show that finite flat implies finite projective dimension for any right-noetherian algebra which admits a dualizing complex. This includes finite dimensional algebras, but also more interesting cases such as noetherian complete semi-local PI algebras, and filtered algebras whose associated graded algebras are connected and noetherian, and either PI, graded FBN, or with enough normal elements.