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

Abstract

<jats:p> We provide an equivariant extension of Carlsson’s BGG correspondence in characteristic two. As an application we classify perfect cochain complexes of <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$(\mathbb {Z}/2\times \mathbb {Z}/2)\rtimes Q$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Z</mml:mi> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> <mml:mo>×</mml:mo> <mml:mi>Z</mml:mi> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> <mml:mo>⋊</mml:mo> <mml:mi>Q</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> -representations with four-dimensional total homology for finite groups <jats:italic>Q</jats:italic> of odd order. We deduce that cochain complexes of finite, free <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$A_4$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:math> </jats:alternatives> </jats:inline-formula> -CW complexes with four-dimensional total homology are rigid: They are determined by the degrees of the nonzero homology groups. </jats:p>