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

Abstract

Let <italic>R</italic> be a regular local ring, and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A equals upper R slash left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mspace width="thinmathspace"/> <mml:mo>=</mml:mo> <mml:mspace width="thinmathspace"/> <mml:mi>R</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">A\, = \,R/(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where <italic>x</italic> is any nonunit of <italic>R</italic> . We prove that every minimal free resolution of a finitely generated <italic>A</italic> -module becomes periodic of period 1 or 2 after at most <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="dimension upper A"> <mml:semantics> <mml:mrow> <mml:mi>dim</mml:mi> <mml:mspace width="thinmathspace"/> <mml:mi>A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {dim} \, A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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.