tambara functors on profinite groups and generalized burnside functors
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Tambara Functors on Profinite Groups and GeneralizedBurnside FunctorsHiroyuki Nakaoka aa Graduate School of Mathematical Sciences , The University of Tokyo, Komaba , Meguro,Tokyo, JapanPublished online: 22 Sep 2009.
To cite this article: Hiroyuki Nakaoka (2009) Tambara Functors on Profinite Groups and Generalized Burnside Functors,Communications in Algebra, 37:9, 3095-3151, DOI: 10.1080/00927870902747605
To link to this article: http://dx.doi.org/10.1080/00927870902747605
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Communications in Algebra, 37: 30953151, 2009Copyright Taylor & Francis Group, LLCISSN: 0092-7872 print/1532-4125 onlineDOI: 10.1080/00927870902747605
TAMBARA FUNCTORS ON PROFINITE GROUPSAND GENERALIZED BURNSIDE FUNCTORS
Hiroyuki NakaokaGraduate School of Mathematical Sciences, The University of Tokyo,Komaba, Meguro, Tokyo, Japan
The Tambara functor was defined by Tambara in the name of TNR-functor, to treatcertain ring-valued Mackey functors on a finite group. Recently Brun revealed theimportance of Tambara functors in the WittBurnside construction. In this article,we define the Tambara functor on the Mackey system of Bley and Boltje. Yoshidasgeneralized Burnside ring functor is the first example. Consequently, we can considera Tambara functor on any profinite group. In relation with the WittBurnsideconstruction, we can give a Tambara-functor structure on Elliotts functor VM , whichgeneralizes the completed Burnside ring functor of Dress and Siebeneicher.
Key Words: Mackey functor; Tambara functor; WittBurnside ring.
2000 Mathematics Subject Classification: Primary 19A22; Secondary 18B40, 20J05.
The Tambara functor was defined by Tambara in  for any finite group G, inthe name of TNR-functors. Roughly speaking, a Tambara functor on G is a ring-valued Mackey functor with multiplicative transfers, satisfying certain compatibilityconditions for exponential diagrams. Recently, Brun revealed that Tambara functorsplay an important role in the WittBurnside construction .
As Mackey functors admit a Lindner-type description (see ), the categoryof Tambara functors is equivalent to the category of product-preserving functorsU Set0 from a certain category U to the category of sets . This enables us amore functorial treatment of fixed point functors, cohomology ring functors, andBurnside ring functors, as examples of Tambara functors.
On the other hand, to consider Mackey functors on a possibly infinite groupG, Bley and Boltje defined in  general Mackey systems for arbitrary groups, onwhich Mackey functors are defined. This general class of functors include ordinaryMackey functors on finite groups, Mackey functors on profinite groups (so-called
Received October 15, 2007; Revised June 17, 2008. Communicated by D. K. Nakano.Address correspondence to Hiroyuki Nakaoka, Graduate School of Mathematical Sciences, The
University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8914, Japan; Fax: +81-3-3422-4371. E-mail:email@example.com
G-modulations), and has several applications in number theory as shown in .A Mackey system for an arbitrary group G is a pair of family of subgroupsin G with certain conditions, each of them is closed under conjugation and finiteintersections.
Independently, for any finite group G and any conjugation-closed family ofsubgroups in G, Yoshida has defined in  the generalized Burnside ring G,which has several properties similar to the ordinary Burnside ring G. It is shownin  that if is, moreover, closed under (necessarily finite) intersections, Gis equal to the Grothendieck ring of a category associated to G, and becomesa subring of G. We only consider the case where = is also closed under(finite) intersections, and in this article we generalize G to a Mackey functor on any Mackey system for an arbitrary group G.
In this article, we consider a generalization of Tambara functors, namely,we define a Tambara functor on any Mackey system with certain conditions.As a consequence, we can consider a Tambara functor on a profinite group.Our main theorem (Theorem 5.16) enables us to construct Tambara functors, forexample, we make the above Burnside functor into a Tambara functor. In relation with the WittBurnside construction, on any profinite groupG, we give a Tambara-functor structure to Elliotts functor VM , where M is anarbitrary multiplicative monoid. This functor is closely related to the WittBurnsideconstruction as shown in , which generalizes the completed Burnside ring functorconsidered by Dress and Siebeneicher in .
In Section 2, after fixing our notation, we introduce some known resultsand preparative properties concerning Mackey functors on Mackey systems. InSection 3, we show any Mackey functor on a Mackey system admit a Lindner-typedefinition. In this context, the above Burnside functor can be easily regardedas a Mackey functor. In Section 4, we define Tambara systems and (semi-)Tambarafunctors on them, generalizing the case of finite groups. In Section 5, we show howa semi-Tambara functor gives rise to a Tambara functor in Theorem 5.16. Thetheorem is as follows, and proven in a similar way as the finite-group case.
Theorem 5.16. Let S be a semi-Tambara functor on . With SX =K0SX and +, , appropriately defined, S becomes a Tambara functor.
By virtue of this theorem, we can show that the Burnside functor becomes a Tambara functor. As a further example concerning the WittBurnsideconstruction, we make VM into a Tambara functor on a profinite group G.
First we fix a notation. For any group G, H G means that H is asubgroup of G. For any subgroup H G and any g G, define gH = gHg1 andHg = g1Hg. GSet denotes the category of G-sets and equivariant maps, and Gsetdenotes the category of finite G-sets, which is a full subcategory of GSet. If X is aG-set and x X, let Gx denote the stabilizer group of x in X. In this article, monoidsare assumed to be commutative and have an additive unit 0. A homomorphismof monoids preserves 0. Semi-rings are assumed to be commutative both for theaddition and the multiplication, and have an additive unit 0 and a multiplicative
TAMBARA FUNCTORS ON PROFINITE GROUPS 3097
unit 1. A homomorphism of semi-rings preserves 0 and 1. For any category andany objects X, Y Ob, the set of morphisms from X to Y in is denoted byX Y.
The following definitions are based on . When we consider a Mackeyfunctor, we will only treat the case of a -Mackey functor and call it simply aMackey functor.
Defintion 2.1 (Definition 2.1 in ). Let G be an arbitrary group. A Mackeysystem for G is a pair with the following property:
a) is a set of subgroups of G, closed under conjugation and finite intersections,b) = HH is a family of subsets H H = U U H,which satisfies:
(i) H U < ;(ii) U H;(iii) gHg1 = gHg1;(iv) U V V,for all H , U H, V H, and g G.
Example 2.2. Let be a set of subgroups of G, closed under conjugation andfinite intersections.
(1) If we define by
H = U H H U < H
then becomes a Mackey system for G.(2) If we define d by
dH = H H
then d becomes a Mackey system for G.
Remark 2.3. Both and d satisfy H H and H dH for any H .In the following, we often impose the condition
H H H (2.1)
to a Mackey system . If we fix , then (resp., d) is thelargest (resp., smallest) Mackey system, among all the Mackey systems satisfying (2.1).