Splitting rational incomplete Mackey functors

Abstract

<p> Inspired by equivariant homotopy theory, equivariant algebra studies generalisations 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> -Mackey functors that do not have all transfer maps (also known as induction maps), for <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> a finite group. These incomplete Mackey functors have interesting and subtle properties that are more complicated than classical algebra. The levels of incompleteness that occur are indexed by simple combinatorial data known as transfer systems for <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> , which are refinements of the subgroup relation satisfying certain axioms. The aim of this paper is to generalise the Greenlees–May and Thévenaz–Webb splitting result of rational <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> -Mackey functors to the incomplete case. </p> <p> By calculating idempotents of the rational incomplete Burnside ring 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> , we find the maximal splitting of the category of rational incomplete <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> -Mackey functors. These splittings are determined by maps of the form <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H right-arrow upper G"> <mml:semantics> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mi>G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">H \to G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the transfer system. We give an intrinsic definition of the split pieces beyond the idempotent description in order to understand what is the minimal information needed to determine an arbitrary rational incomplete <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> -Mackey functor. We end the paper with a series of examples of possible splittings and illustrate how simpler transfer systems have fewer terms in the splitting but the split pieces are more complicated. </p>