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<p>In his work on the Bass series of syzygy modules of modules over a commutative noetherian local ring <i>R</i>, Lescot introduces a numerical invariant, denoted <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\sigma (R)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, and asks whether it is finite for any <i>R</i>. He proves that this is so when <i>R</i> is Gorenstein or Golod. In the present work many new classes of rings <i>R</i> for which <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\sigma (R)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is finite are identified. The new insight is that <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\sigma (R)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is related to the natural map from the usual cohomology of the module to its stable cohomology, which permits the use of multiplicative structures to study the question of finiteness of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\sigma (R)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.</p>

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Unstable elements in cohomology and a question of Lescot

  • 期刊论文

  • 发表日期：

    2026年3月3日

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  • Srikanth B. Iyengar,
  • Sarasij Maitra,
  • Tim Tribone

摘要

In his work on the Bass series of syzygy modules of modules over a commutative noetherian local ring R, Lescot introduces a numerical invariant, denoted \(\sigma (R)\) σ ( R ) , and asks whether it is finite for any R. He proves that this is so when R is Gorenstein or Golod. In the present work many new classes of rings R for which \(\sigma (R)\) σ ( R ) is finite are identified. The new insight is that \(\sigma (R)\) σ ( R ) is related to the natural map from the usual cohomology of the module to its stable cohomology, which permits the use of multiplicative structures to study the question of finiteness of \(\sigma (R)\) σ ( R ) .