The converse to a theorem of Sharp on Gorenstein modules

Abstract

Let <italic>A</italic> be a commutative local Noetherian ring with identity of Krull dimension <italic>n, m</italic> its maximal ideal. Sharp has proved that if <italic>A</italic> is Cohen-Macauley and a homomorphic image of a Gorenstein local ring, then <italic>A</italic> has a Gorenstein module <italic>M</italic> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="dimension Subscript upper A slash m Baseline upper E x t Superscript n Baseline left-parenthesis upper A slash m comma upper M right-parenthesis equals 1"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>dim</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>A</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>m</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:msup> <mml:mi>Ext</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>⁡</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>A</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>m</mml:mi> <mml:mo>,</mml:mo> <mml:mi>M</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">{\dim _{A/m}}\operatorname {Ext}^n(A/m,M) = 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The aim of this note is to prove the converse to this theorem.

Locations

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