Binomial Rings and their Cohomology. Shadman Rahman Kareem

Transcription

1 Biomial Rigs ad their Cohomology. Shadma Rahma Kareem Submitted for the degree of Doctor of Philosophy School of Mathematics ad Statistics March 2018 Supervisor: Prof. Sarah Whitehouse Uiversity of Sheffield

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3 Abstract A biomial( rig ) is a Z-torsio free commutative rig R, i which all the biomial r r(r 1)(r 2) (r ( 1)) operatios = R Q, actually lie i R, for all r! i R ad 0. It is a special type of λ-rig i which the Adams operatios o it all are the idetity ad the λ-operatios are give by the biomial operatios. This thesis studies the algebraic properties of biomial rigs, cosiders examples from topology ad begis a study of their cohomology. The first two chapters give a itroductio ad some backgroud material. I Chapter 3 ad Chapter 4 we study the algebraic structure ad properties of biomial rigs, focusig o the otio of a biomial ideal i a biomial rig. We study some classes of biomial rigs. We show that the rig of itegers Z is a biomially simple rig. We give a characterisatio of biomial ideals i the rig of iteger valued-polyomials It(Z {x} ). We apply this to prove that It(Z {x} ) is a biomially pricipal rig ad rigs of polyomials that are iteger valued o a subset of the itegers are also biomially pricipal rigs. Also, we prove that the rig It(Z {x,y} ) of iteger-valued polyomials o two variables is a biomially Noetheria rig. The rig It(Z {x} ) ad its dual appear as certai rigs of operatios ad cooperatios i topological K -theory. We give some o-trivial examples of biomial rigs that come from topology such as stably iteger-valued Lauret polyomials SLIt(Z {x} ) o oe variable ad stably iteger-valued polyomials SIt(Z {x} ) o oe variable. We study geeralisatios of these rigs to a set X of variables. We show that i the oe variable case both rigs are biomially pricipal rigs ad i the case of fiitely may variables both are biomially Noetheria rigs. As a mai result we give ew descriptios of these examples. I Chapter 5 ad Chapter 6 we defie cohomology of biomial rigs as a example of a cotriple cohomology theory o the category of biomial rigs. To do so, we study biomial modules ad biomial derivatios. Our cohomology has coefficiets give by the cotravariat fuctor Der Bi (, M), of biomial derivatios to a biomial module M. We give some examples of biomial module structures ad calculate derivatios for these examples. We defie homomorphisms coectig the cohomology of biomial rigs to the cohomology of λ-rigs ad to the Adré-Quille cohomology of the uderlyig commutative rigs.

4 Ackowledgemets First of all I sicerely express my gratitude to my supervisor for four years of ecouragemet, costat help, guidace ad for havig log patiece with me. Doig research uder her directio has bee a great pleasure. I wat to admit this project would ot have bee completed without her valuable suggestios ad helpful otes. Further I would like to express my gratitude to my family for providig a great support, especially my lovely wife for all her love ad great patiece which I highly appreciated ad my so ad daughter whom I have ot spet a great amout of time with over the years. I am deeply thakful for the scholarship I have received from my Kurdista Regioal Govermet (KRG). Fially, may thaks go to all of the staff ad studets i Pure Mathematics at the Uiversity of Sheffield for their support, especially my colleagues i room J25 who made my PhD time ejoyable.

5 Cotets Abstract Ackowledgemets iii iv 1 Itroductio 1 2 Biomial rigs Itroductio Itroductio to biomial rigs Biomial rigs Iteger-valued polyomials λ-rigs Adams operatios Biomial rigs as λ-rig structure Category of biomial rigs Localizatio ad completio of biomial rigs Localizatio of biomial rigs Completio of biomial rigs Biomial ideals of biomial rigs Itroductio Biomial ideals Quotiet biomial rigs Pricipal biomial ideals Biomially simple rigs Biomially pricipal rigs v

6 3.7 Biomially Noetheria rigs Biomially filtered rigs Biomial rigs arisig i topology Itroductio K-Theory Hopf algebras Cohomology operatios ad homology cooperatios Special kids of rigs of polyomials Rigs of stably iteger-valued Lauret polyomials Rigs of stably iteger-valued polyomials Some topologically derived biomial rigs Cotriple Cohomology Itroductio Cotriple cohomology Simplicial objects Triples ad Cotriples Cotriple cohomology Adré-Quille cohomology Cohomology of λ-rigs Cohomology of biomial rigs Free biomial rigs Biomial modules Biomial derivatios Cohomology of biomial rigs

7 Chapter 1 Itroductio The otio of a biomial rig was origially itroduced by Hall [29] i coectio with his work i the theory of ilpotet groups. A biomial rig is a commutative rig R with uit whose additive group is Z-torsio free such that all the biomial operatios ( ) r r(r 1)(r 2) (r ( 1)) = R Q! actually i R, for all r i R ad 0. There is aother importat type of rig called a λ-rig which is a commutative rig R with uit equipped with a sequece of fuctios λ : R R, for all 0, called λ-operatios, which satisfy certai relatios that are satisfied by the biomial operatios. These rigs were origially itroduced i algebraic geometry by Grothedieck [28] i his work i Riema-Roch theory. The λ-operatios are ot group homomorphisms. Their actio o sums is give by λ (x + y) = i+j= λ i (x)λ j (y), for all x, y i a λ-rig. Adams [2] itroduced other operatios o commutative rigs to study vector fields o spheres, ψ : R R, for 1, which are called Adams operatios. The Adams operatios also exist o a λ-rig R. Biomial rigs have several applicatios. For example Hall [29] uses a member of a biomial rig to determie a type of geeralised expoetiatio of a elemet of ay ilpotet group. Wilkerso [55] shows that a biomial rig is a special type of λ-rig i which all Adams operatios are equal to the idetity. The λ-operatios are the give by the biomial operatios ( ) r λ (r) =, for r i the rig ad 1. Yau [57] cosider filtered λ-rig, which is a λ-rig together with a decreasig sequece of λ-ideals. He shows that for a biomial rig R, the set 1

8 2 λ(r[x]/(x 2 )) of isomorphism classes of filtered λ-rig structures o the rig (R[x]/(x 2 )) is ucoutable. The rig of iteger-valued polyomials o a set X of variables, is the set of polyomials with coefficiets i Q that are iteger-valued over itegers. This is deoted by It(Z X ) = {f Q[X] : f(z X ) Z}, where Z X = Hom(X, Z) see [15]. It is a example of a biomial rig. This plays a importat role i our thesis. We show that the rig of iteger-valued polyomials over a subset K Z, which is deoted by It(K X, Z) = {f Q[X] : f(k X ) Z}, where K X = Hom(X, K) is also a biomial rig. Elliott [25] shows that It(Z X ) is the free biomial rig o the set X. He also defies a right adjoit to the iclusio fuctor from the category of biomial rigs to the category of λ-rigs. Usig Adams operatios o λ-rigs we defie a left adjoit Q λ to this iclusio fuctor. Ideed, Adams operatios give aother type of rig closely related to λ-rigs. This is a commutative rig R with uit equipped with a sequece of rig homomorphisms ψ : R R, for all 1. They are required to satisfy ψ 1 (r) = r ad ψ i (ψ j (r)) = ψ ij (r). Such a rig is called a ψ -rig. Wilkerso i [55] shows that there exists a λ-rig structure o a Z-torsio free ψ -rig R satisfyig the coditio ψ p (r) r p (mod pr), for r i R ad prime p, whose Adams operatios are give by the ψ -rig structure o R. Our results by applyig Wilkerso s theorem ad Theorem we show that biomial rigs are preserved uder localizatio ad completio. Theorem Let S be a multiplicative closed subset of the biomial rig R. The the localizatio S 1 R is a biomial rig. Theorem Let d be a metric o a biomial rig R. The the rig ˆR d is a biomial rig. We study the algebraic structure of biomial rigs. We start with the otio of a biomial ideal of a biomial rig. A ideal I of a biomial rig R is called a biomial ideal if it is closed uder the biomial operatios; that is ( ) a I, for a i I ad 1. There is ot much work o biomial ideals. Xatcha [56] gives a short survey o biomial ideals i his work o biomial rigs: axiomatisatio, trasfer ad classificatio. This ecouraged us to ivestigate classes of biomial rigs by properties of their biomial ideals. At the begiig, we show that the quotiet rig of a biomial rig by a biomial ideal is also a biomial rig. This will be a very useful tool i our work. A well-kow example of a o-noetheria commutative rig

9 CHAPTER 1. INTRODUCTION 3 is It(Z X ). We itroduce the otio of a biomially pricipal rig ad a biomially Noetheria rig. We show the followigs. Theorem The biomial rig It(Z {x} ) is a biomially pricipal rig. Theorem The rig It(Z {x,y} ) o two variables x ad y is a biomially Noetheria rig. The complex topological K -theory built out from vector budle of a space X by apply the Grothedieck costructio to the semi-rig Vect(X) with additio operatio the direct sum ad multiplicatio the teser product o the equivalece classes of vector budles over X. The origi of K -theory goes back to Grothedieck i algebraic geometry i his first work o the Riema-Roch theorem [28]. Atiyah ad Hirzebruch i [7] published the first work o K -theory i algebraic topology which is called topological K -theory. This is a extraordiary cohomology theory. For a good space X, (for example para-compact Hausdorff space) K 0 (X) is a λ-rig with λ-operatios give by exterior powers o vector budles E over X, λ (E) = Λ E. Kutso i [38] shows that for a biomial rig R which comes with a particular type of geeratig subset, there is a isomorphism R = Z. He applies this result to K -theory. It shows that if K 0 (X) for a good space X is a biomial rig, the K 0 (X) = Z. However o-trivial examples of biomial rigs do arise i relatio to topological K - theory. The rig It(Z {x} ) ad its dual appear as various types of operatios ad cooperatios i topological K -theory. Some ice works i this directio ca be foud i [4, 17, 18, 20]. Bases of this kid of rig of cooperatios i K -theory are give i [19]. Bases of the dual of this rig, related to operatios i K -theory, ca be foud i [50]. We use K 0 (X) as a dual to K 0 (X) for a good space X ad the properties of the rig It (Z {x} ), to give some o-trivial examples of biomial rigs arisig from topology. Our mai results give ew descriptios of these examples. The most importat oe is Theorem Let It(Z {x,y} ) be the rig of iteger-valued polyomials over two x, y variables ad let It((Z {x} )[x 1 ] be the localizatio of the rig It(Z {x} ) with respect to the multiplicatively closed set {x : N}. The we have a isomorphism of biomials rigs, It(Z {x,y} ) ((xy 1)) = It(Z {x} )[x 1 ]. Simplicial methods were itroduced by Dold ad Ka aroud They played a importat part i the developmet of homological algebra ad led to o-abelia derived fuctors. The simplicial method provides a way to defie cohomology i a categorical settig. The cocept of a triple o a category traces back to Godemet [26] ad cotriple to Huber [35] as a dual of triples. It is well kow that a cotriple C = (C, ε, δ) i a

10 4 category yields a simplicial object i this category, built out of iteratig C, with face ad degeeracy maps determied by ε ad δ. Cohomology theories have bee defied i differet areas of abstract algebra. For example for associative algebras a cohomology theory is defied via the theory of Hochschild [33], for groups via the theories of Eileberg ad Mac Lae [23] ad for Lie algebras via the theories of Chevalley ad Eileberg [16]. Barr ad Beck [22] use a cotriple that comes from a adjoit pair of fuctors, to itroduce a cohomology theory which is called the cotriple cohomology theory. Adré [6] ad Quille [46] separately itroduced a cohomology theory o the category of commutative algebras usig a cotriple o the category of commutative algebras that comes from the composite of a free fuctor ad a forgetful fuctor. This is ow called Adré-Quille cohomology theory. Robiso i [48] itroduced the cohomology of λ-rigs with coefficiets i the cotravariat fuctor Der λ (, M), which is the set of all λ-derivatios with values i a λ-module M over the λ-rig. This is a example of a cotriple cohomology theory o the category of λ-rigs. We apply Robiso s otios of λ-module ad λ-derivatio to biomial rigs to itroduce the cohomology of biomial rigs as aother example of a cotriple cohomology theory o the category of biomial rigs, with values i the cotravariat fuctor Der Bi (, M), which is the set of all λ-derivatios with values i a λ-module M over the λ-rig. our mai result Theorem Let R be a biomial rig ad let M be a biomial module over R with module structure give by ϕ M Id M. The = ( 1) 1 Der(R, M) = Der Bi (R, M). Theorem Let R be a biomial rig ad let M be a biomial module over R. The there exists a R-module homomorphism, for each 0 ϱ : H Bi(R, M) H λ (I BiR, I Bi M). This thesis is orgaized as follows. I chapter 2 we provide a overview about special classes of commutative rigs which are called biomial rigs ad λ-rigs. I geeral we show that for K Z, It(K X, Z) is a biomial rig. The we itroduce the otio of Adams operatio o a λ-rig. It is show that a biomial rig is a special type of λ-rig i which all Adams operatios are the idetity ad the λ-operatios are give by the biomial operatios. I 2.8, we itroduce the fuctor Q λ from the category of λ-rigs to the category of biomial rigs. We show that it is left adjoit to the iclusio fuctor I Bi from the category of biomial rigs to the category of λ-rigs. At the ed of this chapter we show that biomial rigs are preserved by localizatio ad completio. I chapter 3 we focus o the otio of a biomial ideal of a biomial rig. We start with the defiitio alogside some examples ad properties. The proof that the quotiet rig of a biomial rig by a biomial ideal is a biomial rig is give i 3.3. The we itroduce the otio of a pricipal biomial ideal. We use this to give some examples

11 CHAPTER 1. INTRODUCTION 5 of quotiets of It(Z {x} ), (Theorem ad Theorem ). The relatio with usual pricipal ideals is give. As a mai aim of this chapter we itroduce some classes of biomial rigs by properties of their biomial ideals. First, we itroduce the otio of biomially simple rig. We show that the rig of itegers Z is a biomially simple rig. The we itroduce the otio of biomially pricipal rig. As the first step we give the characterisatio of biomial ideals i It(Z {x} ). We use it ad the fact that Q[x] is a pricipal itegral domai to show that It(Z {x} ) is a biomially pricipal rig. Fially, we defie the otio of biomially Noetheria rig. We use a characterisatio of biomial ideals i It(Z X ) o a set X of variables ad the fact that Q[x, y] is a Noetheria rig to show that It(Z {x,y} ) is a biomially Noetheria rig. I the last sectio of this chapter we defie the otio of biomially filtered rigs. We show that the power series rig ( ) ( ) ( ) x x x Z,,, is a biomial rig. Bhargava [12], for S Z, gives a regular basis of the rig It(S {x}, Z). We use this i the case where S has a p-orderig simultaeously for all primes p to give a descriptio of a particular completio of this rig. Chapter 4 is devoted to givig some o-trivial examples of biomial rigs arisig from topology. We start with the costructio of K -theory geometrically i terms of classes of vector budles over the space X, Vect(X) ad some basic results o it. The we itroduce the spectrum K associated with the spaces BU Z ad U, which defies a cohomology theory called complex K -theory. The various types of cohomology operatios ad related cooperatios are give. We also give all the ecessary backgroud o Hopf algebras. I 4.5, we start with discussio of stably iteger-valued Lauret polyomials. We show that the rig SLIt(Z {x} ) = {f(x) Q[x, x 1 ] : z m f(z) Z for all z Z ad some m 0} is a biomial rig. Also, we itroduce the rig of stably iteger-valued polyomials SIt(Z {x} ) = {f(x) Q[x] : z m f(z) Z for all z Z ad some m 0}. We show that it is a biomial rig. At the ed of this chapter, we explai how these examples of biomial rigs come from topology. The mai ew results i this chapter are Theorem ad Theorem 4.6.9, givig ew descriptios of these examples. I chapter 5 we provide backgroud material o cotriple cohomology theory. Also we give a overview of Adré-Quille cohomology theory for commutative algebras as a example of cotriple cohomology. At the ed of this chapter, we give a summary of cohomology of λ-rigs. There is o origial work i this chapter. Chapter 6 is devoted to itroducig the cohomology of biomial rigs as aother example of a cotriple cohomology theory, o the category of biomial rigs. We itroduce the otio of a biomial module over a biomial rig by applyig the otio of a λ- module to the special case of a biomial rig. We give examples of differet biomial module structures. I the same way we apply the otio of a λ-derivatio of a λ-rig

12 6 with values i a λ-module to itroduce the otio of a biomial derivatio of a biomial rig with values i a biomial module M. We look at derivatios of the biomial polyomials. We use this to uderstad derivatios of the iteger-valued polyomial rig. We ivestigate biomial derivatios o a biomial rig with differet biomial module structures. At the ed of this chapter we defie the cohomology of biomial rigs usig cotriple cohomology with values i the cotravariat fuctor Der Bi (, M). We show that for a biomial rig R ad biomial modules M over R, with a particular biomial module structure, give by ϕ M = ( 1) 1 Id M, we have Der Bi (R, M) = 0. As a cosequece for this kid of module structure, we get H Bi(R, M) = 0, for all 0. However, other biomial module structures give o-zero cohomology at lest i degree zero see Propositio We defie homomorphisms coectig the cohomology of biomial rigs to the cohomology of λ-rigs ad to the Adré-Quille cohomology of the uderlyig commutative rigs.

13 Chapter 2 Biomial rigs 2.1 Itroductio The mai purposes of this chapter are as follows. 1. To give the defiitios ad review some basic properties ad well kow results about special classes of rigs which are called biomial rigs ad λ-rigs. 2. To ivestigate the relatioship betwee them usig Adams operatios o λ-rigs. Our mai result is to costruct the fuctor deoted by Q λ from the category of λ- rigs to the category of biomial rigs. We show that this fuctor is left adjoit to the iclusio fuctor from the category of biomial rigs to the category of λ-rigs (Theorem ). We exted well kow results about λ-rigs to biomial rigs. It is show that biomial rigs are closed uder localizatio (Theorem 2.9.5) ad completio (Theorem ). I 2.2 we give a short itroductio to biomial rigs. The defiitio ad some basic properties of biomial rigs used throughout the whole thesis alogside some examples are give i 2.3. I 2.4 we discuss the rig of iteger-valued polyomials o a set X of variables, It(Z X ). We show that the rig It(Z X ) is a biomial rig. We use the rig It(Z X ) to give aother descriptio of biomial rigs: a Z-torsio free rig which is the homomorphic image of the rig It(Z X ) is a biomial rig ad all biomial rigs are of this form. I 2.5 the defiitio of λ-rigs is preseted alog with some examples that will be referred to later i this thesis. The otio of Adams operatios o λ-rigs give i 2.6. The proof that the biomial rig is special type of λ-rigs i which all Adams operatios are the idetity maps as biomial rigs (Theorem 2.7.1) give i 2.7. I 2.8 we itroduce the category of biomial rigs whose objects are biomial rigs ad morphisms are rig homomorphisms. We costruct the left adjoit fuctor to the iclusio fuctor from the category of biomial rigs to the category of λ-rigs. We use Theorem to show that biomial rigs are closed uder localisatio ad completio. 7

14 8 I particular the facts that p-local iteger rig Z (p) ad p-adic itegers Ẑp both are biomial rigs is the subject of Itroductio to biomial rigs The cocept of biomial rig was origially itroduced by Hall [29] i coectio with his groudbreakig work i the theory of ilpotet groups. His origial defiitio is as follows. Let R be a commutative rig with uity. It is a biomial rig if it is Z-torsio-free ad closed uder the biomial operatios ( ) r r(r 1)(r 2) (r ( 1)) = R! for every r R ad 1. Alogside the origial referece today there are three other basic refereces for biomial rigs. The most recet oe is the book [57] by Doald Yau. I Chapter 5 of this book he gives a elemetary itroductio to biomial rigs with a few examples, basic properties ad theorems. He explais that the uiversal λ-rig o a biomial rig R is isomorphic to the ecklace rig Nr(R) of R, where Nr(R) of R is the rig with uderlyig set Nr(R) = =1 R. Fially he itroduces the cocept of a filtered λ-rig. He shows that for a biomial rig R the set λ(r[x]/(x 2 )) of isomorphism classes of filtered λ-rig structures o the rig (R[x]/(x 2 )) with the x-adic filtratio is ucoutable. The secod basic referece is the paper [25] by Elliott. The mai theme of this paper is to elucidate the coectio betwee biomial rigs ad λ-rigs. He defies the free biomial rig o the set X via the iteger-valued polyomial rig It(Z X ) o the set X. He applies this to give aother characterisatio of biomial rigs, that they are homomorphic images of the rigs of iteger-valued polyomials that are Z-torsio free rigs. More geerally he itroduce the otio of quasi biomial as a homomorphic image of a biomial rig to describe aother characterisatio of a biomial rig. Furthermore, he costructs both left ad right adjoit fuctors to the iclusio fuctor from biomial rigs to rigs ad from the poit of view of Adams operatios, he describes a right adjoit for the iclusio fuctor from biomial rigs to λ-rigs. The third basic referece is the paper [55] by Wilkerso. I this paper from the poit of view of Adams operatios he shows that a biomial rig is equivalet to a λ-rig i which all Adams operatios are the idetity. There is also a ice paper [38] by Kutso, which applies the Adams operatios to show that the triviality of Adams operatios i group represetatio rigs ad topological K -theory of spaces lead to triviality of the whole rig. It shows that if the rig R(G) of a fiite group is a biomial rig the ecessarily G = {e}. O the other had, later i other chapters we will see some examples of o-trivial biomial rigs arisig from topology.

15 CHAPTER 2. BINOMIAL RINGS Biomial rigs Sice this thesis deals with biomial rigs a lot, we begi with a sectio o them icludig the defiitio, some basic properties ad examples. Let us begi with the Z-torsio free property. Oe of the coditios for ay rig to be a biomial rig is that it should be Z-torsio free as a Z-module. Thus first we give the defiitio ad some examples of Z-torsio free rigs. Defiitio A elemet r i a rig R is called a Z-torsio elemet, if r = 0 for some Z +. Example The rig Z 3 = {0, 1, 2} of itegers modulo 3 has three Z-torsio elemets. Defiitio A rig R is called a Z-torsio free rig, if 0 is the oly Z-torsio elemet i R. Some examples of Z-torsio free rigs iclude biomial rigs, polyomial rigs over Z ad ay subrig of the ratioals Q. Let R be a rig. Cosider the rig homomorphism give by R R Q r r 1. The property of R be Z-torsio free meas exactly that this rig homomorphism is ijective. Defiitio A biomial rig is a commutative rig R with uit whose additive group is Z-torsio free ad that cotais all the biomial operatios ( ) r r(r 1)(r 2) (r ( 1)) = R Q! ( ) r actually i R for every r R ad 0, where = 1. 0 I other words, a Z-torsio free commutative rig R is a biomial rig if ad oly if it is closed uder takig biomial operatios. Note that the biomial rig structure o a Z-torsio free commutative rig R is uique (if it exists). But later we will see whe we defie the otio of biomial module over biomial rig the structure of biomial module is ot uique. Example Some examples of biomial rigs are the followig. 1. The simplest biomial rig is the rig of itegers Z. It is clear Z is a Z-torsio free rig. Sice the biomial operatios are itegers this implies that the rig Z is preserved by biomial operatios.

16 10 2. Ay field R of characteristic 0. Sice R has o zero-divisors ad its characteristic is 0 this implies R is a Z-torsio free rig. Sice R is a field, every o-zero elemet i R is ivertible. This implies that the biomial operatios lie i R for every elemet i R. 3. Ay Q-algebra is a biomial rig. Other examples of biomial rigs will appear i the followig sectios whe we itroduce the rig of iteger-valued polyomials ad cosider Adams operatios o λ-rigs ad i followig chapters whe we defie the cocept of biomial ideal i a biomial rig. Here is a preview of some of them. Example Every rig of iteger-valued polyomials It(Z X ) o a set X of variables is a biomial rig (Theorem 2.4.7). Example Every rig of stable iteger-valued Lauret polyomials SLIt(Z X ) o a set X of variables is a biomial rig (Theorem 4.5.6). Example I geeral every rig of stable iteger-valued polyomials SIt(Z X ) o a set X of variables is a biomial rig (Theorem ). Example The rig Z (p) of p-local itegers (Corollary 2.9.6) ad the rig Ẑp of p-adic itegers (Corollary ) both are biomial rigs. Example I geeral every rig of the iteger-valued polyomial rigs over a subset K Z, It(K X, Z) o a set X of variables is a biomial rig (Theorem ) ad the geeralizatio of a iteger-valued polyomial rig o a biomial domai R with quotiet field F, It(R) is a biomial rig (Propositio ). Example Ay λ-rig whose Adams operatios all are the idetity is a biomial rig (Theorem 2.7.1). Example The quotiet rig R/I of a biomial rig R by a biomial ideal I is a biomial rig (Theorem 3.3.1). Example The power series rig Z ( x 1), ( x 2), ( x 3), is a biomial rig (Propositio 3.8.9). We ow state without proof some basic properties of biomial operatios, which follow from stadard facts about biomial coefficiets. Theorem [56] Let R be a biomial rig. For all a, b R ad m,, k N the followig hold ( ) a + b = ( ) ab = m=0 =p+q ( ) a m ( )( ) a b. p q q 1 +q 2 + +q m= ( ) ( ) b b. q 1 q m

17 CHAPTER 2. BINOMIAL RINGS ( )( ) a a = m k=0 ( a m + k ( ) 1 = 0 whe 2. ( ) a = a. 1 )( m + k )( ). k Next here are some good properties of biomial rigs. Propositio Let R ad K be biomial rigs. 1. The direct product rig R K is a biomial rig with biomial operatios give by ( ) (( ) ( )) (r, k) r k =,, for r R, k K ad 0. So, if R 1,..., R m are biomial rigs, the product rig m R i is a biomial rig. i=1 2. The tesor product rig R K is a biomial rig with biomial operatios determied by ( ) ( ) r 1 r = 1, ( ) ( ) 1 k k = 1, for r R, k Kad The itersectio R K is a biomial rig. More geerally, if {R i } i I is a family of biomial rigs the the rig i R i is a biomial rig. Proof. Property 1 is clear. We are goig to prove property 2. We ca write ( ) ( ) r k (r 1)(1 k) =. Sice R ad K both are biomial rigs, by Theorem (2) ad above formula this implies that ( ) r k R K. For Property 3 see [25]. [ ] Defiitio The Stirlig umber of the first kid, deoted by for 0 i ad i N, is defied as the umber of ways to permute elemets ito exactly i cycles.

18 12 Propositio The Stirlig umber of the first kid satisfy the liear recurrece, with iitial coditios [ ] = ( 1) i [ 1 [ ] 0 = 1 ad for i < 0, 0 i ] + [ ] i for 0 ad i N, [ ] 1, (2.1) i 1 [ ] = 0. i Proof. For proof see [27, p. 247]. We let x = x(x 1)(x 2)....(x ( 1)), (2.2) for 0, be the -th fallig power of x ad be the th risig powers of x. Actually, the Stirlig umber of the first kid differet ways. They appear as the coefficiets i x. x = x(x + 1)(x + 2)....(x + ( 1)), (2.3) Propositio [27, p. 249] For 0 we have [ ] ca be expressed i may equivalet i x = [ ] ( 1) i x i. (2.4) i i=0 Proof. We will prove this by iductio o. Suppose that = 0. The both sides are equal 1. Assume that the result holds for 1. The we have 1 [ ] 1 x 1 = ( 1) ( 1) i x i. (2.5) i i=0 We will prove it for. We have The by our assumptio we obtai x = x 1 (x ( 1)) = xx 1 ( 1)x 1.

19 CHAPTER 2. BINOMIAL RINGS 13 1 [ ] 1 1 [ ] 1 x = x ( 1) ( 1) i x i ( 1) ( 1) ( 1) i x i, by (2.5) i i i=1 i=1 1 [ ] 1 1 [ ] 1 = ( 1) ( 1) i x i+1 ( 1) ( 1) ( 1) i x i, i i i=1 i=1 [ ] 1 1 [ ] 1 = ( 1) ( i) x i ( 1) ( 1) ( 1) i x i, i 1 i i=2 i=1 [ ] = ( 1) ( i) x i by (2.1). i i=0 Propositio For 0 we have x = i=0 [ ] x i. i Proof. The proof similar to the proof of Propositio 2.5. We use the expressio of Stirlig umbers of the first kid This implies that [ ] i (2.4) to expad i ( ) x. ( ) x = 1 ( [ ) ( 1) ]x i i. (2.6)! i i=0 I combiatorics see [40, p. 56], the Stirlig umber of the first kid ca be expressed as the sum over (c 1,..., c ) of the umber of permutatios of type [c 1, c 2,, c ] [ ] = ( 1) +i! i 1 c 1 2 c c c1!c 2!... c!. (2.7) i=c 1 +c 2 + +c =c 1 +2c c Lemma For a prime p, the Stirlig umber of the first kid by p for 1 < i < p. Proof. From (2.7) we have, [ ] p = ( 1) +i i i=c 1 +c 2 + +c p p=c 1 +2c 2 + +pc p p! 1 c 1 2 c 2... p c p c 1!c 2!... c p!. [ ] p i is divisible The umerator is divisible by p. Sice 1 < i < p ad we have c p = 0 each factor i the deomiator [ ] is less tha p ad so the prime p is ot caceled i the umerator. p Therefore is divisible by p. i

20 14 Propositio I a commutative rig R we have, for r R ad a prime p. r p r r(r 1)(r 2)... (r (p 1)) (mod pr), (2.8) Proof. From (2.4) we have Ad by Lemma , r(r 1)(r 2).(r (p 1)) = [ ] p i p [ ] p ( 1) i r i. i i=1 is divisible by p for 1 < i < p. We obtai [ ] p 0 (mod pr). i Also by Wilso s theorem see ([30, p. 85]) we have ad [ ] p = 1. p [ ] p = (p 1)! 1 (mod p), 1 Propositio [57, Lemma 5.5]I a biomial rig R the cogruece coditio holds for all r R ad p prime. r p r (mod pr) (2.9) Proof. By (2.8) we have r p r r(r 1)(r 2).(r (p 1)) (mod pr) ( ) r = p! by (2.6) p 0 (mod pr) Therefore r p r (mod pr).

21 CHAPTER 2. BINOMIAL RINGS Iteger-valued polyomials Most of the examples i this thesis are related to rigs of iteger-valued polyomials o a set X of variables. So we will begi with a sectio o rigs of iteger-valued polyomials. This meas rigs of polyomials with ratioal coefficiets that are itegervalued o itegers. We prove that the rig It(Z X ) o a set X of variables is a biomial rig (Theorem 2.4.7). Precisely later i 2.8 we will show that the rig It(Z X ) o a set X of variables is the free biomial rig o the set X. The result is give i [25]. Later we itroduce the otio of iteger-valued polyomials It(K {x}, Z) over a subset K Z. As a result we show that It(K X, Z), is a biomial rig (Theorem ). For a more thorough descriptio of iteger-valued polyomials we refer to [15]. We begi with the defiitio of the rig of iteger-valued polyomials o a set X of variables. Defiitio Let Q[X] be the rig of polyomials o a set X of variables with ratioal coefficiets. We defie the set of iteger-valued polyomials o X by It(Z X ) = {f Q[X] : f(z X ) Z}. This is a subrig of Q[X] ad it is called the rig of iteger-valued polyomials o X, where Z X = Hom(X, Z), which is the set of fuctios (x). We computig f at ay by replacig each x X with iteger (x) i f. The the coditio f(z X ) Z meas that f() Z. I particular we have It(Z {x} ) = {f Q[x] : f(z) Z}, (2.10) the rig of iteger-valued polyomials i oe variable x. Defiitio The biomial polyomial i oe variable x is defied by ( ) x x(x 1)(x 2) (x ( 1)) = Q[x].! for all 0, where ( x 0) = 1. Notatio For some o-empty set X of variables, we defie a multi-idex to be J = (j x ) x X x X Z 0. (2.11) With this multi-idex J we defie the geeralized biomial polyomial to be ( ) X = ( ) x. (2.12) J j x x X Note that for aother multi-idex J = (j x ) the biomial operatio defie by, ( ) J = I x X ( jx i x ) Z. (2.13)

22 16 Notatio Let I = (j 1,..., i ) k=1 Z 0. ad let J = (j 1,..., j ) be aother multi-idex. The we mea by I > J if ad oly if i 1 = j 1,..., i k = j k ad i k+1 > j k+1 for some k with 0 k 1. I particular, for ay multi-idexes J 1 < J 2 < J 3 < < J, we have ( ) { Jt 1 if t = k, = 0 if t < k. Lemma For 0 the biomial polyomial is a iteger-valued polyomial i oe variable x. J k ( ) x (2.14) I geeral the product of biomial polyomials each i oe variable ( )( ) ( ) x1 x2 xi, (2.15) 1 2 i i the polyomial rig Q[x 1, x 2,..., x i ] i i variables for 1, 2,..., i iteger-valued polyomial i i variables. 0 is a Theorem [57]The geeralized biomial polyomials i a set of variables X { ( ) X : J = (j x ) Z 0 }, (2.16) J x X ( ) x is a Z-basis of the rig It(Z X ). I particular, the polyomials, for 0, form a Z-module basis of the rig It(Z {x} ) i oe variable, ad the set { ( x 1 1 ) ( xi i ) } : 1,..., i 0 is a Z-module basis of the rig It(Z {x 1,,x i } ) i i variables. (2.17) Proof. First the set {( ) X : J = (j x ) Z 0 }, J x X is a Q-vector space basis of the polyomial rig Q[X] by [57, Propositio 5.31]. The for f It(Z X ), with f 0, we ca write f = ( ) X a t J t t=1

23 CHAPTER 2. BINOMIAL RINGS 17 for some 1 with a t Q ad some multi-idexes {J t } t T. We obtai f(j) = ( J a t ). J t t=1 I the same way i (2.14) whe we compute f for orderig the idexes J 1 < J 2 < < J k at X = J 1, we obtai f(j 1 ) = a 1 Z. So by iductio o k assume that a 1, a 2, a k Z. Now we are goig to show that a k+1 Z, to see that we calculate f at X = J k+1, also by (2.14) we have f(j k+1 ) = k+1 ( ) Jk+1 a t t=1 = a 1 ( Jk+1 J 1 J t ) + + a k ( Jk+1 J k ) + a k+1. We kow that f(j k+1 ) is a iteger. So by the iductio we coclude that a k+1 is also a iteger. This shows that the geeralized biomial polyomials i a set X of variables spas It(Z X ) over Z. Also we kow from [57, Propositio 5.31] that the geeralized biomial polyomials i the set of variables X are a Q-vector space basis of the polyomial rig Q[X]. So they are liearly idepedet over Q. Hece they are also liearly idepedet over Z. Here is the mai purpose of this sectio, which shows that the rig It(Z X ) o a set X of variables is a biomial rig. Theorem [57] The rig It(Z X ) o a set X of variables is a biomial rig. Proof. First we eed to show that It(Z X ) is Z-torsio free, which is clear sice It(Z X ) is a subrig of Q[X]. To see the other coditio of a biomial rig, cosider f It(Z X ). We have ( ) f Q[X]. The for a m Z X, we have f(m) Z. Notice ( ) ( ) f f(m) (m) =. The by Lemma 2.4.5, So ( ) f(m) Z. ( ) f It(Z X ). Now we tur attetio to a rig of iteger-valued polyomials over a subset.

24 18 Defiitio For a subset K Z, we say that a polyomial f Q[X] o a set X of variables which satisfies that f(k X ) Z is a iteger-valued polyomial over subset K, where K X = Hom(X, K), which is the set of fuctios as i Defiitio We computig f at ay by replacig each x X with k K, (x) i f. The the coditio f(k X ) Z meas that f() Z. It(K X, Z) = {f Q[X] : f(k X ) Z}. (2.18) This is a subrig of Q[X] ad it is called the rig of iteger-valued polyomials over K o set X. I particular we have It(K {x}, Z) = {f(x) Q[x] : f(k) Z}, (2.19) called the rig of iteger-valued polyomials over subset K i oe variable x. Note that the iteger-valued polyomial rig It(Z {x} ) is iteger-valued over Z, that is It(Z {x} ) = It(Z {x}, Z). So we have iclusio Z[x] It(Z {x} ) It(K {x}, Z) Q[x]. (2.20) Example I particular from the rig of iteger-valued polyomials over {0} o oe variable x we have It({0} {x}, Z) = {f(x) Q[x] : f(0) Z}. I other words, the rig It({0} {x}, Z) is the set of all polyomials i Q[x] with costat term is a iteger. I example for each o-zero subset K Z we have Z[x] It(Z {x} ) It(K {x}, Z) It({0} {x} ) Q[x]. (2.21) Here is our mai result of this sectio. Theorem For K Z, the rig It(K {x}, Z) is a biomial rig. Proof. We kow from (2.20) that It(K {x}, Z) is a subrig of Q[x], so it is clearly a Z-torsio free rig. To see the other coditio of a biomial rig, pick a elemet g(x) It(K {x}, Z). We have for 0. The ( ) g(x) = g(x)(g(x) 1)(g(x) 2) (g(x) ( 1))! Q[x],

25 CHAPTER 2. BINOMIAL RINGS 19 ( ) g(x) g(k)(g(k) 1)(g(k) 2) (g(k) ( 1)) (k) =! ( ) g(k) = Z by Lemma 2.4.5, ( ) g(x) for k K. Therefore It(K {x}, Z) as desired. Theorem For a subset K Z the rig of iteger-valued polyomials over subset K, It(K X, Z) o a set X of variables is a biomial rig. Proof. The proof is aalogues to the proof of Theorem λ-rigs I this sectio we discuss λ-rigs, we give some well kow results o them ad some of their properties. A λ-rig is a commutative rig R with idetity equipped with a sequece of fuctios λ i : R R for i 0 which are called λ-operatios, satisfyig certai relatios that are satisfied by the biomial operatios. The λ-rigs were first itroduced i algebraic geometry by Grothedieck [28] uder the ame special λ-rig. They have bee show to play importat roles i various field of mathematics. For example i group theory, the paper [7] used λ-rigs to study group represetatios. I algebraic topology, the K -theory of a good space is a λ-rig. I both cases the λ-operatios are iduced by exterior powers of vector spaces. Kutso i [37] used λ-rigs to study represetatios of the symmetric group. I pure algebra, Doald Yau published a book uder the ame λ-rigs [57]. The mai aim of this is to study λ-rigs purely algebraically. For example if R is a commutative rig with uit, The the rig W (R) of big witt vectors o R has caoical λ-rig structure. Also the otio of λ-rig R uses the classical essetial theorem of symmetric fuctios to describe the actio of λ-operatios o a product λ (r 1 r 2 ) ad the compositio of λ-operatios λ λ m (r 1 ) for r 1, r 2 R. We begi with the defiitio of λ-rigs. Defiitio A λ-rig is a commutative with uit rig R together with a sequece of fuctios λ : R R (called λ-operatios) for each 0 such that the followig axioms are satisfied. 1. λ 0 (x) = 1, 2. λ 1 (x) = x, 3. λ (1) = 0 for 2, 4. λ (x + y) = λ i (x)λ j (y), i+j=

26 20 5. λ (xy) = P (λ 1 (x),, λ (x); λ 1 (y),, λ (y)), 6. λ (λ m (x)) = P i,j (λ 1 (x),, λ m (x)), for all x, y R ad, m 0. The polyomials P ad P,m which describe the actio of λ-operatios o products ad the compositio of λ-operatios are described below. Defiitio For a rig R, we cosider the rig R[x 1, x 2,, x ] of polyomials i idepedet variables x 1, x 2,, x. The polyomial f R[x 1, x 2,, x ] is called a symmetric fuctio if it is ualtered uder every permutatio of the variables. That is, we have f(x 1, x 2,, x ) = f(x π(1), x π(2),, x π() ), for every permutatio π o the set {1, 2,, }. We say that the polyomial g R[x 1, x 2,..., x ; y 1, y 2,..., y ] i x 1, x 2,..., x ad y 1, y 2,..., y idepedet variables is a symmetric fuctio if it is ualtered uder every permutatio of variables. That is, we have f(x 1, x 2,..., x ; y 1, y 2,..., y ) = f(x π(1), x π(2),..., x π() ; y τ(1), y τ(2),, y τ() ), for every part of permutatios π ad τ o the set {1, 2,, }. Example For each 1 k, a importat symmetric fuctio is the k th symmetric polyomial s k R[x 1, x 2,, x ] which is the sum of all products of moomial of legth k. That is we have s k = x i1 x i2 x ik. I particular, we have 1<i 1 < <i k s 1 = x 1 + x x, s 2 = x 1 x 2 + x 1 x x 1 x, s = x 1 x 2 x. Aother way to obtai the kth elemetary symmetric fuctio s k (x 1, x 2,, x ) i variables is by cosiderig the formula, where t is a extra variable, f(t) = s k t k = k=0 (1 + tx i ), (2.22) Theorem [57, p. 3]Ay symmetric fuctio f i R[x 1, x 2,, x ] ca be writte as a polyomial i the elemetary symmetric fuctios s 1, s 2,, s with coefficiets i R ad it is uique. i=1

27 CHAPTER 2. BINOMIAL RINGS 21 The polyomials P,m ad P that appeared i the defiitio of λ-rig are called uiversal polyomials. The polyomial P (s 1, s 2,, s ; α 1, α 2,, α ) is the coefficiet of t i the polyomial f(t) = (1 + x i y j t), i,j=1 where each of s i ad α i are i th elemetary symmetric fuctios i x 1,, x ad i y 1,, y respectively. The polyomialp,m (s 1, s 2,, s m ) is the coefficiet of t i the polyomial f(t) = 1 i 1 < <i m m (1 + x i1 x i2 x im t). Example P,1 (s 1, s 2,, s ) is the coefficiet of t i the polyomial so f(t) = 1 i (1 + x i t) = 1 + s 1 t + s t, P,1 (s 1, s 2,, s ) = s ad P 1,m (s 1, s 2,, s m ) is coefficiet of t i the polyomial so f(t) = 1 i 1 < <i m m (1 + x i1 x i2 x im t) = 1 + x 1 x 2 x m t P 1,m (s 1, s 2,, s m ) = s m. For more detail o symmetric fuctios see [37, chapter 1] ad for the uiversal polyomials see [57, chapter 1]. I geeral P,m P m, as λ-operatios do ot commute. For more detail ad calculatio see [34]. There the author gives several forms for P,m ad calculates P up to = 10. Here are some small values of both uiversal polyomials. 1. P 0 = P 1 (s 1, α 1 ) = s 1 α P 2 (s 1, s 2 ; α 1, α 2 ) = s 2 1 α 2 2s 2 α 2 + s 2 α P 3 (s 1, s 2, s 3 ; α 1, α 2, α 3 ) = s 3 1 α 3 3s 1 s 2 α 3 + s 1 s 2 α 1 α 2 3s 3 α 1 α 2 + s 3 α s 3α P 0,m = 1, for all m N. 6. P 1,0 = P,0 = 0, for all P 1,1 (s 1 ) = s 1.

28 22 9. P 2,2 (s 1, s 2, s 3, s 4 ) = s 1 s 3 s P 2,3 (s 1, s 2, s 3, s 4, s 5, s 6 ) = s 6 + s 2 s 4 s 1 s P 3,2 (s 1, s 2, s 3, s 4, s 5, s 6 ) = s 6 2s 2 s 4 s 1 s 5 + s 2 1 s 4 + s 2 3 s 1s P 3,3 (s 1, s 2, s 3, s 4, s 5, s 6, s 7, s 8, s 9 ) = s 9 s 1 s 8 s 4 s 5 s 2 s 7 + s 2 1 s 7 + s 2 1 s 4 + 3s 3 s 6 2s 1 s 3 s 5 s 1 s 2 s 6. Defiitio Let R 1 ad R 2 be λ-rigs. A rig homomorphism f : R 1 R 2 is called a λ-homomorphism if it commutes with the λ-operatios that is λ (f(r)) = f(λ (r)), for all r R ad 0. We write Rig λ for the category of λ-rigs, whose objects are λ-rigs ad morphisms are λ-homomorphisms. Defiitio We call R a pre λ-rig if oly the first four axioms of Defiitio are satisfied. Cosider a (pre) λ-rig R together with the homomorphism λ t from the additive group of R ito the multiplicative group of power series i t with costat term 1, defied by λ t (r) = λ (r)t R[[t]]. (2.23) =0 Now we ca use (2.23) to write additio of λ-operatios as λ t (r 1 + r 2 ) = λ t (r 1 ).λ t (r 2 ) (2.24) Example The rig Z of itegers is a pre λ-rig with λ t (r) = (1 + t) r = r =0 ( ) r t. (2.25) So It is clear by Theorem ( ) r λ (r) =. ( ) r. is satisfy all axioms. Defiitio If λ t (x) is a polyomial of degree, the we say that x has dimesio. If each r R is differece of fiite dimesio elemets, the we say that R is fiite dimesioal. Example [57, p. 9]Some examples of λ-rigs are the followig.

29 CHAPTER 2. BINOMIAL RINGS The simplest fiite dimesioal λ-rig is the rig of itegers Z with the λ- operatios defied by the biomial operatios λ (r) = ( r ). that is So λ t () = coefficietsoft i i(1 + t) = λ (r) = ( ) r. These are also the coefficiets of t i above power series. i=0 ( ) r t i. i 2. Oe ca get a λ-rig structure o the represetatio rig for a group G, i which λ is iduced from the th exterior power o represetatios of the group G, λ (V ) = Λ (V ) for V i rep(g). 3. The topological K-theory K(X) of ay good space X ( para-compact Hausdorff space). This is a λ-rig, i which λ is iduced from the th exterior power, λ (B) = Λ (B), for a vector budle B over X. I particular the K -theory of a poit is K(pt) = Z with the structure of λ-operatios give i example Adams operatios The aim of this sectio is to itroduce the otio of Adams operatios o λ-rig. The λ-operatios have complicated axioms. It ca be difficult to costruct a λ-rig structure o some kids of commutative rig ad λ-operatios are ot group homomorphisms. So i [2] Adams itroduced the ψ -operatios to study vector fields o spheres from the λ-operatios o a rig R. We will use it i the ext sectio. I fact, ψ -operatios give us aother type of rig, which is a commutative rig R, equipped with a sequece of fuctios ψ : R R, for all 1, satisfyig certai properties. These are called ψ -rigs. The ψ -rigs are much easier to deal with ad sometimes we will eed to pass to them to execute some calculatios for λ-rigs ad to costruct λ-rig structures o some particular types of rigs. For example Wilkerso i [55] explais that costructig ψ -operatios o a Z-torsio free rig R that satisfy the axiom ψ p (r) r p (mod pr) for every r R ad every prime p, is sufficiet to costruct a λ-rig structure o R (Theorem ). He also cosiders the Adams operatios to show that a λ-rig whose ψ -operatios all are the idetity for all 0 is a biomial rig (Theorem 2.7.1).

30 24 The Adams operatios o a λ-rig R, ψ : R R for 1, are defied by usig the λ-operatios o R. We costruct the group homomorphism λ t : R R[[t]]. We kow from (2.24) that the additio formula is give by λ t (r 1 + r 2 ) = λ t (r 1 ).λ t (r 2 ), for r 1, r 2 R. I order to obtai a additive homomorphism ψ t : R R a atural idea is to apply the logarithm to (2.24). Precisely, Sice λ t (r) has costat equal to 1, we apply the power series formula log(1 + x) = ( 1) i+1 x i, i=1 to log(λ t (r)) which is meas we add deomiators ito R, say by tesorig with Q. We obtai group homomorphism with coefficiets of power series of t of log(λ t (r)) which take value i R Q. Now by applyig the operator d dr to this we elimiate Q. Therefore we obtai from the above iformatio this geeratig fuctio, ψ t (r) = t d dt (log λ t(r)) = tλ t(r). (2.26) λ t (r) Defiitio Let R be a λ-rig. We defie the th Adams operatios o R ψ : R R by cosiderig the geeratig fuctio (2.26) for all 1 ad r R, where ψ t (r) = i 1 ψ i (r)t i. (2.27) O other words,ψ i (r) is the coefficiet of ( t) i i ψ t. Example We kow from the previous sectio that the rig Z of itegers is a λ-rig with λ-operatios give by λ t (a) = (1 + t) a for a Z. The the Adams operatios i the rig Z are give by This implies that So for all i 1, ψ i (a) = a. ψ t (a) = t d dt (log(1 + t)a ) = at 1 + t. ψ t (a) = at 1 t = a(t + t2 + ). Later we will show that the same thig holds i all biomial rigs. The Adams operatios satisfy the followig properties. Propositio [2]Let R be a λ-rig. followig properties hold i R. For fixed i, j 1 ad r 1, r 2 R, the

31 CHAPTER 2. BINOMIAL RINGS ψ i : R R is a rig homomorphism. 2. ψ 1 =Id. 3. ψ i ψ j = ψ ij = ψ j ψ i. 4. ψ p (r) r p (mod pr) for all prime umbers p. The Adams operatios are coected to the λ-operatios by the followig formula, which is kow as Newto s Formula ( which is quite closely related to Newto s formula for symmetric fuctios see [57, Theorem 3.9], but recursive rather that closed formula relatig Adams operatios ad λ-operatios). Theorem [57, Theorem 3.10]The followig equality holds i a λ-rig R. ψ (r) = λ 1 (r)ψ 1 (r) λ 2 (r)ψ 2 (r) + + ( 1) λ 1 (r)ψ 1 (r) + ( 1) +1 λ (r), (2.28) for r R ad 1. Proof. For a proof see [57, Theorem 3.10]. I other words, Newto s Formula gives a recursive formula for the Adams operatios i terms of λ-operatios. So we ca calculate Adams operatios recursively i terms of λ-operatios. Here are the values of Adams operatios for some small values of, i terms of λ- operatios. 1. ψ 1 (r) = λ 1 (r) = r. 2. ψ 2 (r) = r 2 2λ 2 (r). 3. ψ 3 (r) = r 3 3rλ 2 (r) + 3λ 3 (r). 4. ψ 4 (r) = r 4 4r 2 λ 2 (r) + 4rλ 3 (r) 4rλ 4 (r) + 2(λ 2 (r)) ψ 5 (r) = r 5 5r 3 λ 2 (r) + 5r 2 λ 3 (r) 5rλ 4 (r) + 5λ 5 (r) + 5(λ 2 (r)) 2 5λ 2 (r)λ 3 (r). Theorem Let R be a λ-rig, the every Adams operatio o R, ψ : R R for 1, is a λ-homomorphism. Proof. For a proof see [57, Theorem 3.6]. Next we itroduce aother type of rig closely related to λ-rigs, which is kow as a ψ -rig. Defiitio A ψ -rig is a commutative rig R with uit, together with a sequece of rig homomorphisms ψ : R R, for all 1, which are called ψ - operatios such that the followig axioms are satisfied. 1. ψ 1 =Id,

32 26 2. ψ i ψ j = ψ ij = ψ i ψ j, for all r R ad i, j 1. We say that a ψ -rig R is special if it also satisfies the axiom for each prime p. ψ p (r) r p (mod pr) (2.29) Example The rig of itegers Z is a ψ -rig with ψ -operatios give by ψ i () = for all Z ad i 0. Example I geeral every commutative rig R with uit is a ψ -rig with ψ -operatios give by ψ i (r) = r for all r R ad i 0. Defiitio Let R 1 ad R 2 be ψ -rigs, the a rig homomorphism f : R 1 R 2 is called ψ -homomorphism if it commutes with the ψ -operatios that is ψ (f(r)) = f(ψ (r)), for all r R 1 ad 0. We write the set of all ψ -homomorphisms by Hom ψ (R 1, R 2 ). We write Rig ψ for the category of ψ -rigs whose objects are ψ -rigs ad morphisms are ψ -homomorphisms. We kow from the previous sectio that the λ-operatios are either additive or multiplicative ad a λ-rig R has some complicated axioms. Thus it ca be hard to costruct a λ-rig structure o some types of rigs. However the ψ -rig axioms are easier to deal with. So Wilkerso i [55] showed that to costruct a special ψ -rig structure o a Z-torsio free rig R, it is eough to costruct a λ-rig structure o R from it is Adams operatio the oes that give the ψ -rig structure. We will use it i the comig sectio whe we show that biomial rigs are preserved by localizatio ad completio. Theorem Let R be a Z-torsio free special ψ -rig. The the Adams operatios o R which give the ψ -rig structure o R determie a λ-rig structure o R, related λ-operatio to Adams operatio by Newto s Formula as i Theorem Proof. For a proof see [57, Theorem 3.54]. 2.7 Biomial rigs as λ-rig structure The purpose of this sectio is to itroduce a special class of λ-rig structures from the poit of view of Adams operatios. The result is due to Wilkerso [55] who shows that a λ-rig R whose Adams operatios all are the idetity o R is a biomial rig (Theorem 2.7.1). Later i this sectio we use this result to give aother descriptio of biomial rigs (Propositio 2.7.4). Here is the mai aim of this subsectio. This will be a very useful tool i the ext sectio ad comig chapters i this thesis.