Tambara Functors on Profinite Groups and Generalized Burnside Functors

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<ul><li><p>This article was downloaded by: [Columbia University]On: 14 November 2014, At: 10:49Publisher: Taylor &amp; FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK</p><p>Communications in AlgebraPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/lagb20</p><p>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.</p><p>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</p><p>To link to this article: http://dx.doi.org/10.1080/00927870902747605</p><p>PLEASE SCROLL DOWN FOR ARTICLE</p><p>Taylor &amp; Francis makes every effort to ensure the accuracy of all the information (the Content) containedin the publications on our platform. 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Terms &amp; Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions</p><p>http://www.tandfonline.com/loi/lagb20http://www.tandfonline.com/action/showCitFormats?doi=10.1080/00927870902747605http://dx.doi.org/10.1080/00927870902747605http://www.tandfonline.com/page/terms-and-conditionshttp://www.tandfonline.com/page/terms-and-conditions</p></li><li><p>Communications in Algebra, 37: 30953151, 2009Copyright Taylor &amp; Francis Group, LLCISSN: 0092-7872 print/1532-4125 onlineDOI: 10.1080/00927870902747605</p><p>TAMBARA FUNCTORS ON PROFINITE GROUPSAND GENERALIZED BURNSIDE FUNCTORS</p><p>Hiroyuki NakaokaGraduate School of Mathematical Sciences, The University of Tokyo,Komaba, Meguro, Tokyo, Japan</p><p>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.</p><p>Key Words: Mackey functor; Tambara functor; WittBurnside ring.</p><p>2000 Mathematics Subject Classification: Primary 19A22; Secondary 18B40, 20J05.</p><p>1. INTRODUCTION</p><p>The Tambara functor was defined by Tambara in [7] 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 [2].</p><p>As Mackey functors admit a Lindner-type description (see [5]), the categoryof Tambara functors is equivalent to the category of product-preserving functorsU Set0 from a certain category U to the category of sets [7]. This enables us amore functorial treatment of fixed point functors, cohomology ring functors, andBurnside ring functors, as examples of Tambara functors.</p><p>On the other hand, to consider Mackey functors on a possibly infinite groupG, Bley and Boltje defined in [1] 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</p><p>Received October 15, 2007; Revised June 17, 2008. Communicated by D. K. Nakano.Address correspondence to Hiroyuki Nakaoka, Graduate School of Mathematical Sciences, The</p><p>University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8914, Japan; Fax: +81-3-3422-4371. E-mail:deutsche@ms.u-tokyo.ac.jp</p><p>3095</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Col</p><p>umbi</p><p>a U</p><p>nive</p><p>rsity</p><p>] at</p><p> 10:</p><p>49 1</p><p>4 N</p><p>ovem</p><p>ber </p><p>2014</p></li><li><p>3096 NAKAOKA</p><p>G-modulations), and has several applications in number theory as shown in [1].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.</p><p>Independently, for any finite group G and any conjugation-closed family ofsubgroups in G, Yoshida has defined in [8] the generalized Burnside ring G,which has several properties similar to the ordinary Burnside ring G. It is shownin [8] 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.</p><p>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 [4], which generalizes the completed Burnside ring functorconsidered by Dress and Siebeneicher in [3].</p><p>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.</p><p>Theorem 5.16. Let S be a semi-Tambara functor on . With SX =K0SX and +, , appropriately defined, S becomes a Tambara functor.</p><p>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.</p><p>2. PRELIMINARIES</p><p>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</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Col</p><p>umbi</p><p>a U</p><p>nive</p><p>rsity</p><p>] at</p><p> 10:</p><p>49 1</p><p>4 N</p><p>ovem</p><p>ber </p><p>2014</p></li><li><p>TAMBARA FUNCTORS ON PROFINITE GROUPS 3097</p><p>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.</p><p>The following definitions are based on [1]. When we consider a Mackeyfunctor, we will only treat the case of a -Mackey functor and call it simply aMackey functor.</p><p>Defintion 2.1 (Definition 2.1 in [1]). Let G be an arbitrary group. A Mackeysystem for G is a pair with the following property:</p><p>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:</p><p>(i) H U &lt; ;(ii) U H;(iii) gHg1 = gHg1;(iv) U V V,for all H , U H, V H, and g G.</p><p>Example 2.2. Let be a set of subgroups of G, closed under conjugation andfinite intersections.</p><p>(1) If we define by</p><p>H = U H H U &lt; H </p><p>then becomes a Mackey system for G.(2) If we define d by</p><p>dH = H H </p><p>then d becomes a Mackey system for G.</p><p>Remark 2.3. Both and d satisfy H H and H dH for any H .In the following, we often impose the condition</p><p>H H H (2.1)</p><p>to a Mackey system . If we fix , then (resp., d) is thelargest (resp., smallest) Mackey system, among all the Mackey systems satisfying (2.1).</p><p>Definition 2.4. In (1) in Example 2.2, if in particular G is a topological groupand is the set of all closed (resp., open) subgroups of G, we call thenatural (resp., open-natural) Mackey system for G. If G is a profinite group, thenthis definition of the (resp., open-) natural Mackey system agrees with the definitionof the (resp., finite) natural Mackey system in [1].</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Col</p><p>umbi</p><p>a U</p><p>nive</p><p>rsity</p><p>] at</p><p> 10:</p><p>49 1</p><p>4 N</p><p>ovem</p><p>ber </p><p>2014</p></li><li><p>3098 NAKAOKA</p><p>Definition 2.5 (cf. Definition 2.3 in [1]). Let be a Mackey system for anarbitrary group G. A semi-Mackey functor M on is a function which assigns:</p><p>a) a monoid MH to each H ;b) a homomorphism of monoids cgH MH MgH to each H and each</p><p>g G;c) a homomorphism of monoids rHI MH MI to each pair I H in ;d) a homomorphism of monoids tHI MI MH to each H and each</p><p>I H,in a compatible way as in [1]. If all the MH are abelian groups, then M is calleda Mackey functor. The maps cgH , r</p><p>HI , t</p><p>HI are called conjugations, restrictions, and</p><p>transfers, respectively.A morphism of (semi-)Mackey functors f M N is a set of monoid</p><p>homomorphisms f = fH MH NHH, which are compatible with theconjugations, restrictions, and transfers in the obvious sense. We write the categoryof semi-Mackey functors (resp., Mackey functors) as Mack (resp., Mack).Note that Mack is a full subcategory of Mack.</p><p>Remark 2.6. For a finite group G, if we regard G as a discrete topological group,both the natural and open-natural Mackey systems are</p><p> = subgroup of GH = H = subgroup of H H G</p><p>A (resp., semi-) Mackey functor on this Mackey system is nothing other than a(resp., semi-) Mackey functor on G. Thus the Mackey functor theory on finitegroups is contained in that on Mackey systems.</p><p>Definition 2.7 (Definition 2.6 in [1]). Let be a Mackey system for G.</p><p>(1) GSet is defined to be a full subcategory of GSet, whose objects are thoseX ObGSet which satisfy</p><p>Gx for any x X</p><p>(2) GSet is defined to be a category with the same objects as GSet, whosemorphisms from X to Y are those f GSetX Y satisfying the followingproperties:</p><p>(i) f has finite fibers (i.e., f1y is a finite set for any y Y );(ii) Gx Gfx for any x X</p><p>Remark 2.8. Let be a set of subgroups of G closed under conjugationand finite intersections, and consider the Mackey system . Then for anyf GSetX Y, we have</p><p>f GSet X Y f has finite fibers.</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Col</p><p>umbi</p><p>a U</p><p>nive</p><p>rsity</p><p>] at</p><p> 10:</p><p>49 1</p><p>4 N</p><p>ovem</p><p>ber </p><p>2014</p></li><li><p>TAMBARA FUNCTORS ON PROFINITE GROUPS 3099</p><p>Proof. The necessity is trivial. Conversely, assume f has finite fibers. For anyx X, let Xx denote the orbit through x in X. Since Gfx/Gx Xx f1fx andf1fx is finite by the assumption, we have Gfx Gx = Xx &lt; , i.e., Gx Gfx. </p><p>Remark 2.9. If the Mackey system satisfies 21, then for any two objectsX Y in GSet, any injective map from X to Y in GSet</p><p> X Y</p><p>belongs to GSet .In particular, isomorphisms, inclusions are morphisms in GSet .Moreover, the folding maps</p><p> X X X</p><p>are morphisms in GSet .</p><p>Remark 2.10. Let be an arbitrary Mackey system. For any pullbackdiagram in GSet</p><p>if g belongs to GSet , then g also belongs to GSet .</p><p>Definition 2.11 (cf. Definition 2.6 in [1]). Category Bif (resp., Bif) isdefined as follows.</p><p>An object M in Bif (resp., Bif) is a function which assigns:</p><p>a) a monoid (resp., abelian group) MX to each X ObGSet;b) a monoid morphism f MY MX to each f GSetX Y;c) a monoid morphism g MX MY to each g GSetX Y,in such a way that the following conditions are satisfied:</p><p>(i) We have</p><p>g g = g g f f = f f </p><p>for all composable pairs in GSet and GSet respectively, and</p><p>id = id = id</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Col</p><p>umbi</p><p>a U</p><p>nive</p><p>rsity</p><p>] at</p><p> 10:</p><p>49 1</p><p>4 N</p><p>ovem</p><p>ber </p><p>2014</p></li><li><p>3100 NAKAOKA</p><p>(ii) (Mackey axiom) If</p><p>is a pullback diagram in GSet where g GSetX Y, then</p><p>f g = g f </p><p>(iii) For any direct sum decomposition X = X in GSet, the natural mapi MX </p><p>MX</p><p>is an isomorphism of sets, where i X X are the inclusions.M consists of a single element.For any MN ObBif (or Bif), a morphism from M to N is a</p><p>collection of monoid homomorphisms X MX NX X GSet, compatiblewith all g and f . Remark that Bif is a full subcategory of Bif.</p><p>Remark 2.12. An object in Bif (resp., Bif) is nothing other than apair of functors M = MM which satisfies MX = MX = MX X ObGSet, where:</p><p>a) M GSet Mon (resp., GSet Ab) is a contravariant functor withMf = f ;</p><p>b) M GSet Mon (resp., GSet Ab) is a covariant functor withMg = g,</p><p>which satisfies the above condition (ii) and (iii).In this view, a collection of monoid homomorphisms X MX </p><p>NXXObGSet is a morphism in Bif if and only if it is a naturaltransformation with respect to each of the covariant and the contravariant part.</p><p>Remark 2.13. Let be a Mackey system satisfying 21, and M be an objectin Bif. Let X =</p><p>1in Xi be a finite direct sum of objects in GSet, and let</p><p>i Xi X be the inclusion (1 i n). The inverse of the isomorphism</p><p> = i 1in MX </p><p>1inMXi =</p><p>1in</p><p>MXi</p><p>is 1in</p><p>i </p><p>1inMXi MX</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Col</p><p>umbi</p><p>a U</p><p>nive</p><p>rsity</p><p>] at</p><p> 10:</p><p>49 1</p><p>4 N</p><p>ovem</p><p>ber </p><p>2014</p></li><li><p>TAMBARA FUNCTORS ON PROFINITE GROUPS 3101</p><p>Proof. Since</p><p>are pullback diagrams, we have</p><p>i i = idi j = 0 i = j</p><p>So we have</p><p> 1in</p><p>i = id</p><p>Since is an isomorphism, this means 1 = 1in i. Corollary 2.14. Let be a Mackey system satisfying (2.1), and M be an objectin Bif. If the pullback of f GSetX Z and g GSetY Z is written as</p><p>where ki GSetWi X and hi GSetWi Y, then we have</p><p>f g =</p><p>1inki hi </p><p>Proof. Put</p><p>W = 1in</p><p>Wi</p><p>k = 1in</p><p>ki</p><p>h = 1in</p><p>hi</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Col</p><p>umbi</p><p>a U</p><p>nive</p><p>rsity</p><p>] at</p><p> 10:</p><p>49 1</p><p>4 N</p><p>ovem</p><p>ber </p><p>2014</p></li><li><p>3102 NAKAOKA</p><p>a...</p></li></ul>