  In this article we study nonzero modules of finite length and finite
projective dimension over a local ring. We show the Loewy length of such a
module is larger than the regularity of the ring whenever the ring is strict
Cohen-Macaulay, extending work of Avramov-Buchweitz-Iyengar-Miller beyond the
Gorenstein setting. Applications include establishing a conjecture of
Corso-Huneke-Polini-Ulrich and verifying a Lech-like conjecture, comparing
generalized Loewy length along flat local extensions, for strict Cohen-Macaulay
rings. We also give significant improvements on known lower bounds for Loewy
lengths of modules of finite projective dimension without any assumption on the
associated graded ring. The strongest general bounds are achieved over complete
intersection rings.
