Let R be a regular local ring, and let
A
=
R
/
(
x
)
A\, = \,R/(x)
, where x is any nonunit of R. We prove that every minimal free resolution of a finitely generated A-module becomes periodic of period 1 or 2 after at most
dim
A
\operatorname {dim} \, A
steps, and we examine generalizations and extensions of this for complete intersections. Our theorems follow from the properties of certain universally defined endomorphisms of complexes over such rings.