Freyd’s generating hypothesis with almost split sequences

Abstract

Freyd’s generating hypothesis for the stable module category of a non-trivial finite group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the statement that a map between finitely generated <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k upper G"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mi>G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">kG</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -modules that belongs to the thick subcategory generated by the field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> factors through a projective module if the induced map on Tate cohomology is trivial. In this paper we show that Freyd’s generating hypothesis fails for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k upper G"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mi>G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">kG</mml:annotation> </mml:semantics> </mml:math> </inline-formula> when the Sylow <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -subgroup of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has order at least <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="4"> <mml:semantics> <mml:mn>4</mml:mn> <mml:annotation encoding="application/x-tex">4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> using almost split sequences. By combining this with our earlier work, we obtain a complete answer to Freyd’s generating hypothesis for the stable module category of a finite group. We also derive some consequences of the generating hypothesis.