Homological algebra on a complete intersection, with an application to group representations

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 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.