An ideal criterion for torsion freeness

Abstract

Auslander and Bridger have shown that, under conditions somewhat weaker than finite projective dimension, the “torsion freeness” properties of a module <italic>M</italic> (e.g. being reflexive, being the <italic>k</italic>th syzygy of another module) are determined by certain arithmetic conditions on the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Ext Superscript i Baseline left-parenthesis upper M comma upper R right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mtext>Ext</mml:mtext> <mml:mi>i</mml:mi> </mml:msup> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>,</mml:mo> <mml:mi>R</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{\text {Ext}^i}(M,R)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In this paper it is shown that a single ideal, the intersection of the annihilators of these modules, gives this same information. This ideal is then related to the Fitting invariants and invariant factors of <italic>M</italic>, and a computation is made of certain syzygies of a quotient of <italic>M</italic> (by a regular <italic>M</italic>-sequence).

Locations

• Proceedings of the American Mathematical Society - View - PDF