Essential support of Green biset functors via morphisms
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Archiv der Mathematik
Essential support of Green biset functors via morphisms Benjam´ın Garc´ıa
Abstract. In this note, we introduce a criterion to detect vanishing of essential algebras of a Green biset functor by means of morphisms. We introduce the inflation morphism and restriction morphism to prove that the essential supports of a Green biset functor and its shifted functors are the same. We use the kernel of the restriction to give a characterization of the seeds of the shifted functors. Mathematics Subject Classification. 20C99. Keywords. Green biset functor, Essential algebra, Yoneda-Dress construction.
1. Introduction. A natural problem when studying a category of linear functors is to give a classification of its simple objects by means of parameters which help to better understand the effect of these functors on the domain category. For biset functors over an admissible subcategory D, Bouc provided a bijection between the isomorphism classes of simple biset functors and equivalence classes of pairs (H, V ) called seeds [3, Ch. 4]. Bouc extended these ideas to the case of modules over a general Green biset functor A, for which a seed is then a pair (H, V ) consisting of a group H such that the essential algebra A(H) is non-zero and a simple A(H)-module V , with an appropriate notion of isomorphism of seeds. Then for any seed (H, V ), there is a simple module SH,V for which H is a minimal group and SH,V (H) ∼ isomor= V as A(H)-modules, phic seeds give rise to isomorphic simple modules, and any simple module is isomorphic to one of the form SH,V . Bouc conjectured that the assignment (H, V ) → SH,V would induce a oneto-one correspondence between the set of isomorphism classes of seeds and the set of isomorphism classes of simple A-modules. Romero disproved this conjecture in [7], where she provides a simple kB C4 -module having non-isomorphic This work was supported by UC MEXUS and CONACYT.
B. Garc´ıa
Arch. Math.
minimal groups. However, Romero herself had proved in [8] that if A satisfies that minimal groups for simple modules are unique up to isomorphism, then [(H, V )] → [SH,V ] is indeed a bijection. Some well-known Green biset functors satisfy such condition, e.g., kB [3], kRF for fields k and F of characteristic zero and its shifted functors ([1,6]), and kB Cp for a prime number p [7]. If simple A-modules admit a parametrization by seeds, the next step is to determine the class of groups H for which A(H) is non-zero, the problem to which this note is devoted. In Section 3, we call this class the essential sup port of A and denote it by Supp(A). We observe that a morphism of Green biset functors f : A −→ C induces morphisms between the essential alge bras, implying Supp(C) ⊂ Supp(A). A further consequence is presented in Section 4, where we prove that Supp(A G ) = Supp(A) for any G by means of the morphisms Res and Inf . Finally, we use the kernel κG of Res to give a characterization of the seeds of simple AG -modules that cannot be obtained as pullback of A-modules. 2. Generalitie
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