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<p>We provide an equivariant extension of Carlsson’s BGG correspondence in characteristic two. As an application we classify perfect cochain complexes of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\((\mathbb {Z}/2\times \mathbb {Z}/2)\rtimes Q\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">/</mo> <mn>2</mn> <mo>×</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">/</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>⋊</mo> <mi>Q</mi> </mrow> </math></EquationSource> </InlineEquation>-representations with four-dimensional total homology for finite groups <i>Q</i> of odd order. We deduce that cochain complexes of finite, free <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(A_4\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>A</mi> <mn>4</mn> </msub> </math></EquationSource> </InlineEquation>-CW complexes with four-dimensional total homology are rigid: They are determined by the degrees of the nonzero homology groups.</p>

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An Equivariant BGG Correspondence and Perfect Complexes for Extensions by \(\mathbb {Z}/2\times \mathbb {Z}/2\)

  • 期刊论文

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  • 发表日期：

    2026年7月20日

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  • Henrik Rüping,
  • Marc Stephan

摘要

We provide an equivariant extension of Carlsson’s BGG correspondence in characteristic two. As an application we classify perfect cochain complexes of \((\mathbb {Z}/2\times \mathbb {Z}/2)\rtimes Q\) ( Z / 2 × Z / 2 ) ⋊ Q -representations with four-dimensional total homology for finite groups Q of odd order. We deduce that cochain complexes of finite, free \(A_4\) A 4 -CW complexes with four-dimensional total homology are rigid: They are determined by the degrees of the nonzero homology groups.