Galois theory and integral models of Lambda-rings

GALOIS THEOR Y AND INTEGRAL MODELS OF Λ -RINGS JAMES BORGER, BAR T DE SMIT Abstract. W e sho w that an y Λ-ring, i n the sense of Riemann–Roch theory , which is finite ´ etale ov er the rational num bers and has an integral m odel as a Λ-ring is con tained in a product of cyclotomic fields. In fact, we sho w t hat the category of them is desc rib ed in a Galois-theoretic w a y in te rms of the monoid of pro-finite i n tegers under multiplication and the cycloto mic c haract er. W e also s tudy the maximalit y of these integral mo dels and give a more precise, int egral v ersion of the result abov e. These results r ev eal an interesting relation betw een Λ-r ings and class field theory . Introduction According to the most common definition, a Λ-ring structure on a commutativ e ring R is a sequence o f set maps λ 1 , λ 2 , . . . from R to itse lf that s a tisfy certain complex implicitly stated axioms. This no tion was intro duced by Gr othendieck [5], under the name sp ecial λ -r ing, to give a n abstra ct setting for studying the structure on Grothendiec k groups inherited from exterior pow er op er ations; and as far as w e are aw are, with just one exception [2], Λ-rings hav e b een studied in the literature for this purp os e only . How ev er, it seems that the study of abstract Λ-rings—those having no appar en t relation to K -theory—will have something to sa y ab out nu mber theo ry . One e x- ample of suc h a r elationship is the likely existence of strong arithmetic restr ic tions on the complexity of finitely gener ated r ings that admit a Λ-ring structure. The primary purp ose of this pap er is to inv estigate this issue in the zer o-dimensional case. Precisely , what finite ´ etale Λ-rings o ver Q are of the form Q ⊗ A , where A is a Λ-ring that is finite flat ov er Z ? W e will co nsider Λ-actions only on rings whose underlying ab elian gr o up is tor- sion free, a nd giving a Λ-a ction on such a ring R is the same a s giving commuting ring endomorphisms ψ p : R → R , one for ea c h prime p , lifting the F rob enius map mo dulo p —that is, such that ψ p ( x ) − x p ∈ pR for all x ∈ R . (The e q uiv a lence of this with Grothendieck’s original definitio n is prov ed in Wilk erson [8].) An example that will be imp orta n t her e is Z [ µ r ] = Z [ z ] / ( z r − 1), wher e r is a p ositive integer and ψ p sends z to z p . A morphism of tors io n-free Λ -rings is the same a s a ring map f that sa tis fies f ◦ ψ p = ψ p ◦ f for all primes p . Note that if R is a Q -algebr a, the co ngruence conditions in the definition above disapp ear. Also, Galois th eory as int erpreted b y Grothendieck gives an an ti- equiv alence betw een the categ ory of finite ´ etale Q -algebras and the c a tegory of finite discrete sets equipped with a contin uo us action of the absolute Galois group G Q with r esp e ct to a fixed algebraic closur e ¯ Q . Combining these t wo rema rks, we Date : Nov ember 17, 20 18. 19:33. Mathematics Subje ct Classific ati on: 13K05 (primary); 11R37, 19L20, 16W99 (secondary). 1 2 J. BORGER, B. DE SMIT see that the catego ry o f Λ-rings that are finite ´ etale ov er Q is nothing more than the categor y of finite discrete sets eq uipped with a contin uo us action of the monoid G Q × N ′ , where N ′ is the mono id { 1 , 2 , . . . } under m ultiplication with the dis crete top ology . This is because N ′ is freely generated as a co mm uta tiv e monoid by the prime num b ers. It is not a lw ays true, how e ver, that such a Λ - ring K has an integral Λ- mo del, by which we mean a sub-Λ-ring A , finite ov er Z , such that Q ⊗ A = K . In order to formulate exactly when this ha ppens , w e write ˆ Z ◦ for the s et o f pro finite int egers viewed as a topo logical monoid under m ultiplica tion, a nd we consider the contin uous monoid map G Q × N ′ − → ˆ Z ◦ given b y the cy clotomic c haracter G Q → ˆ Z ∗ ⊂ ˆ Z ◦ on the fir st factor and the na tural inclusion on the second. Note that this map has a dense imag e . Now let K b e a finite ´ etale algebra ov er Q , a nd let S be the set of ring maps from K to ¯ Q . Suppose K is endow ed with a Λ-ring structure, so that w e ha ve an induced monoid ma p G Q × N ′ − → Map( S, S ) , where Map( S, S ) is the mono id of set maps from S to itself. With the discr ete top ology on Map( S, S ) this map is contin uous . 0.1. Theorem. The Λ -ring K has an inte gr al Λ -mo del if and only if the action of G Q × N ′ on S factors (ne c essarily uniquely) thr ough ˆ Z ◦ ; in mor e pr e cise t erm s , if and only if ther e is a c ontinuous monoid map ˆ Z ◦ → Map( S, S ) so that the dia gr am G Q × N ′ / / & & M M M M M M M M M M ˆ Z ◦   Map( S, S ) c ommutes. It follows that the category of s uch Λ- rings is an ti-eq uiv a lent to the categor y of finite disc r ete sets with a contin uo us action of ˆ Z ◦ and that ev ery s uch Λ-ring is contained in a pro duct of cyclotomic fields. It also sugg e sts there is an interest- ing theory of a Λ-algebra ic fundamen tal monoid, analog ous to that of the usual algebraic fundament al gr oup, but we will leav e this for a la ter date. Another co nsequence is that the elements − 1 , 0 ∈ ˆ Z ◦ give a n in volution ψ − 1 (complex conjuga tion) and a ide mp otent endomor phism ψ 0 on any K with an int egral Λ-mo del. In K -theor y , the dual and rank also give such o per ators, but here they come automatically from the Λ -ring structure. Also observe that any element of the subset 0 S ⊆ S is Galois inv ariant and hence co rresp onds to a dir e ct factor Q of the a lgebra K . Therefor e K cannot b e a field unles s K = Q . In the first sectio n, we give some ba sic facts and also show the s ufficiency of the condition in the theorem ab ov e. In the second section, w e show the necessity . The pr o of com bines a simple applica tion o f the Kro neck er– W eb er theorem and the Chebo tarev dens ity theor em with some elementary but s lig ht ly intricate work on actions of the mo no id ˆ Z ◦ . The third s ection gives a proo f o f the following theorem: 0.2. Theorem. The Λ -ring Z [ µ r ] is the maximal inte gr al Λ -mo del of Q ⊗ Z [ µ r ] . GALOIS THE OR Y AND INTEGRAL MODELS OF Λ-RINGS 3 Of co urse, the non-Λ version of this statement is false—the usual maximal order of Q ⊗ Z [ µ r ] is a pro duct of rings of integers in cyclotomic fields and str ictly co n tains Z [ µ r ], if r > 1. A direct c onsequence of these theore ms is: 0.3. Corollary . Every Λ -ring t hat has finite r ank as an ab elian gr oup and has no non-zer o nilp otent elements is a sub- Λ -ring of a Λ -ring of the form Z [ µ r ] n . W e do not need to requir e that the ring b e tors ion free b ecause a ny torsion element in a Λ-r ing is nilp otent, by an easy lemma a ttributed to G. Sega l [3, p. 295]. W e emphasize that while the definition of Λ-r ing that we gav e ab ove do es not literally require the r ing to b e torsion free , it is not the correct definition in the absence of this assumption. In particular , Segal’s lemma and the corolla ry above are false if the naive definition is used; for example, take a finite field. F or the definition of Λ-r ing in the genera l case, see [5], [8], o r [1]. Finally , many of the questions answered in this pape r hav e analogues over general nu mber fields. Ther e o ne would use F rob enius lift s modulo prime ide a ls of the ring of in teger s, a nd then general class fie ld theo ry and the clas s gr oup co me in, as w ell as the theory of co mplex multiplication in pa rticular cases. Be c ause these ana logues of Λ- rings a re no t ob jects o f prio r interest, we hav e not included anything ab out them her e. But it is clear that finite-ra nk Λ-r ings, in this g e neralized s ense or the original, a re fundamen ta lly ob jects of class field theory and that they offer a slightly different p ersp ective on the sub ject. It would b e in teresting to explore this further. 1. Basics The category of Λ-r ings has all limits and colimits, and they agree, as rings, with those taken in the category of rings. (E .g . [1]) W e will only need to take tensor pro ducts, intersections, and images of mor phis ms, and it is quite ea sy to show their existence on the sub categ ory of torsion- fr ee Λ-rings using the equiv alent definition given in the intro duction. F or any ring R , let R [ µ r ] denote R [ z ] / ( z r − 1). Because R [ µ r ] = R ⊗ Z [ z ] / ( z r − 1), the r ing R [ µ r ] is natura lly a Λ-ring if R is. Given a Λ-ring R , it will be conv enient to ca ll a Λ-ring K equipp ed with a map R → K of Λ-r ings an R Λ -ring. (Compare [1 , 1.13 ].) When we say K is flat, or ´ etale, or so on, we mean as R -algebra s in the usual sense. W e call a sub-Λ-ring of a Q Λ-ring a Λ-or der if it is finite over Z . W e do not requir e that it hav e full ra nk. 1.1. Prop osition. L et K b e a fin ite ´ etale Q Λ -ring. Then K has a Λ -or der that c ontains al l others. W e call this Λ -order the max imal Λ-order of K . Pr o of. Beca use any Λ-or der A is co nt ained in the us ua l maximal order of K , which is finite o ver Z , it is eno ugh to show that any tw o Λ-orders A and B a re cont ained in a third. But A ⊗ B is a Λ- ring tha t is finite ov er Z . Since A ⊗ B is the co pr o duct in the ca teg ory of Λ-r ings, the map A ⊗ B → K coming fr om the universal prop erty of co pro ducts (i.e., a ⊗ b 7→ ab ) is a Λ-ring map. Therefore its image is a Λ-ring that is finite over Z , is contained in K , and c ontains A a nd B .  1.2. Prop osition. L et K ⊆ L b e an inclusion of fi nite ´ etale Q Λ -rings. L et A ⊆ K and B ⊆ L b e their maximal Λ -or ders. Then A = K ∩ B . 4 J. BORGER, B. DE SMIT Pr o of. The in ter section K ∩ B is on the o ne hand a s ub- Λ-ring of K and, o n the other, finite ov er Z . It is maximal among suc h r ings beca use of the maximality of B .  W e can now prove the sufficiency of the conditions of theo rem 0.1. F or any ring R , let R ◦ denote R itself but viewed o nly as a monoid under m ultiplication. So the group R ∗ of units is just the group of inv er tible element s of the mo noid R ◦ . 1.3. Prop osition. L et r b e a p ositive inte ger, let S b e a finite ( Z /r Z ) ◦ -set, and let K b e the c orr esp onding finite ´ etale Q Λ -ring. Then K has an inte gr al Λ -mo del. Pr o of. T ake a set T (such as S ) admitting a surjectio n ` T ( Z /r Z ) ◦ → S of ( Z /r Z ) ◦ - sets, the left side denoting the free ( Z /r Z ) ◦ -set generated by T . Let L b e the corres p onding finite ´ etale Q Λ-ring. Then K is na turally a s ub-Λ-ring of L . On the other hand L is Q [ µ r ] T and s o has a Λ-mo del Z [ µ r ] T . The intersection o f this with K is then b oth a Λ-ring and an orde r of full rank in K .  2. N ecessar y conditions Let K b e a finite ´ etale Q Λ-ring admitting an integral Λ-mode l A , a nd let S = Hom( K, ¯ Q ) be the corre s po nding G Q × N ′ -set. The purp ose of this section is to show tha t there is an integer r > 0 s uch that this a ction factors throug h the map G Q × N ′ → ( Z /r Z ) ◦ given by the cyclotomic character o n the first fac tor and reduction mo dulo r o n the second. As us ua l, we say a prime n umber p is unramified in A if A/pA has no non-zero nilpo ten t elements. A prime is r a mified in A if and only if it divides the discr iminant of A . Therefore the set of primes that ramify in A is finite and contains the set of primes that ra mify in the usual ma x imal order of K . 2.1. Prop osition. The endomorphism ψ p of A is an automorph ism if and only if p is u nr amifie d in A . In this c ase, ψ p is the unique lift of the F r ob enius endomorphism of A/ pA . Pr o of. If ψ p is an a utomorphism, then the F rob enius endomo rphism x 7→ x p of A/pA is an auto morphism, and so p is unramified. Suppo se instead that p is unra mified. Then A/pA is a finite pro duct of finite fields, a nd so the F rob enius endomorphism of A/pA is an automor phism of finite order. The ca tegory o f finite ´ etale Z p -algebra s is equiv alent to the categ ory of finite ´ etale F p -algebra s, b y wa y of the functor F p ⊗ Z p − . (See [4 , IV (18.3 .3)], say). Thu s the endomorphism 1 ⊗ ψ p of Z p ⊗ A is the unique F rob enius lift and it is an automorphism of finite o rder. It follows that ψ p : A → A is the unique F ro benius lift to A , and is also an automor phis m.  2.2. Prop osition. The r e is a p ositive inte ger c , divisible only by primes that r amify in A , s u ch that the action of G Q on S factors thr ough the cyclotomic char acter G Q → ( Z /c Z ) ∗ . If p is unr amifie d in A , then p ∈ N ′ and ( p mo d c ) ∈ ( Z /c Z ) ∗ act in t he same way on S . Pr o of. Define the n um be r field N to b e the in v ariant field of the k ernel of the map G Q → Ma p( S, S ). W rite ¯ G = Ga l( N/ Q ) and let O N be the ring of integers of N . T ake an y element g ∈ ¯ G . By Cheb otarev’s theor em [7, V.6] there is an unr amified prime p of N lying over a prime num b er p such that g ( x ) ≡ x p mo d p for all x ∈ O N , i.e., g is the F ro b enius element o f p in the extension Q ⊂ N . Since Chebo tarev’s GALOIS THE OR Y AND INTEGRAL MODELS OF Λ-RINGS 5 theorem provides infinitely many s uch p , we may also assume that A is unr amified at p . W e now claim that for all s ∈ S = Hom( A, O N ) the maps s ◦ ψ p and g ◦ s from A to O N are equal. Since A is unr a mified at p , the map Hom( A, O N ) → Hom( A, O N / p ) is injective, so it suffices to show that their comp ositions with the map O N → O N / p are e qual. But this follows from 2.1 and our c hoic e of p . Thus, g ∈ ¯ G a nd p ∈ N ′ act in the s a me wa y on S . It follows that the image of G Q in Map( S, S ) is contained in the image of N ′ , so ¯ G is ab elian. By the Kr oneck er- W eber theo rem [7, II I.3.8 ], N is con tained in a cyclo tomic field Q ( µ c ), wher e c is divisible only by pr imes that ramify in N . Since N is the common Galois closur e of the comp onents of A ⊗ Q , such pr imes are ra mified in A as well. The last statemen t follows from the fact that for any prime num b er p ∤ c the element ( p mo d c ) ∈ ( Z /c Z ) ∗ corres p onds to the F r ob enius element o f an y prime ov er p in the extension Q ⊂ Q ( µ c ).  It follows that our map of top olo gical mono ids G Q × N ′ → Map( S, S ) fa c to rs through ˆ Z ∗ × N ′ . W e will show that it factor s further thro ugh ˆ Z ◦ with the following criterion. 2.3. Prop osition. A c ontinu ous action of ˆ Z ∗ × N ′ on a finite discr ete set T factors thr ough a c ontinuous action of ˆ Z ◦ if and only if (i) al l but finitely many primes p ∈ N ′ act as aut omorphisms on T , and (ii) for al l d ∈ N ′ ther e exist s an inte ger c d such that the action of ˆ Z ∗ on dT factors thr ough ( Z /c d Z ) ∗ and for e ach n ∈ N ′ with ndT = dT we have — n is r elatively prime t o c d , and — the elements ( n mo d c d ) ∈ ( Z /c d Z ) ∗ and n ∈ N ′ act on dT in t he same way. Pr o of. T o s how the necess ity of (i) and (ii), assume that the action of ˆ Z ∗ × N ′ factors thr o ugh ( Z /r Z ) ◦ for some integer r > 0 . T he n a ll primes not dividing r , when viewed as elements of N ′ , ac t as a uto morphisms on T . This es tablishes (i). T o show (ii), take any d ∈ N ′ and let c d be the smallest p os itive integer c for which the ˆ Z ∗ -action o n dT fac to rs thro ugh ( Z /c Z ) ∗ . Note that c d divides all c with this prop erty . Now supp ose that p is a prime with pdT = dT . W rite r = p n e , with p ∤ e . Then for any x, y ∈ ( Z /r Z ) ◦ with x ≡ y mod e and a ny s ∈ dT (in fact any s ∈ T ), we have p n  xs  = ( p n x ) s = ( p n y ) s = p n ( y s ) . Since p a cts bijectively o n dT , this implies tha t x a nd y act in the same wa y on dT , so the action of ˆ Z ◦ on dT factors through ( Z /e Z ) ◦ . In particular, c d | e , so p ∤ c d , and the elements p ∈ N ′ and ( p mo d e ) ∈ ( Z /e Z ) ∗ and ( p mo d c d ) ∈ ( Z /c d Z ) ∗ all act in the s a me wa y on dT . Since N ′ is gener ated b y the primes, part (ii) follows. F or the conv erse, suppo se (i) and (ii) ho ld. F or every pr ime n umber p , let a p be the smallest integer a ≥ 0 such that p a T = p a +1 T . By (i) w e hav e a p = 0 for all but finitely many p , so r 0 = Q p p a p is an int eger. Note that fo r an y n ∈ N ′ we hav e nT = gcd( n, r 0 ) T . Now le t r be a n y in teger divisible by dc d for every d | r 0 . W e will show that the action of ˆ Z ∗ × N ′ on T factors through ( Z /r Z ) ◦ . T o do this, we will show directly that an y t wo elements ( a 1 , d 1 ) , ( a 2 , d 2 ) ∈ ˆ Z ∗ × N ′ satisfying a 1 d 1 ≡ a 2 d 2 mo d r act in the same wa y on T . 6 J. BORGER, B. DE SMIT Since r 0 | r the co ngruence implies that d 1 and d 2 hav e the same gr eatest common divisor d with r 0 , so we hav e d 1 T = dT = d 2 T . F or i = 1 , 2 we define d ′ i ∈ N ′ by d i = dd ′ i and deduce that d ′ i ( dT ) = dT . Using (ii) one sees that d ′ i is coprime to c d , and that the a c tion of d ′ i on dT is that o f ( d ′ i mo d c d ) ∈ ( Z /c d Z ) ∗ . By the defining prop erty of r we have dc d | r , so a 1 d ′ 1 d ≡ a 2 d ′ 2 d mo d dc d , which implies a 1 d ′ 1 ≡ a 2 d ′ 2 mo d c d . It follows that ( a 1 , d ′ 1 ) and ( a 2 , d ′ 2 ) are mapp e d to the sa me element of ( Z /c d Z ) ◦ , which in fact lies in ( Z /c d Z ) ∗ . Thus, ( a 1 , d ′ 1 ) and ( a 2 , d ′ 2 ) act ident ically o n dT , and comp osing with (1 , d ) we see that ( a 1 , d 1 ) and ( a 2 , d 2 ) ac t in the same way on T .  In or der to finish the pro o f of theorem 0.1 one chec k s the conditions of 2.3 fo r T = S . Condition (i) follows fro m 2.1 a nd the fact that A is ra mified a t only finitely many primes. F or condition (ii), suppo se d ∈ N ′ is given and consider the s ub-Λ- ring ψ d ( A ) o f A which corre s po nds to the G Q × N ′ -set dS of S . Prop osition 2.2 applied to ψ d ( A ) now pro vides an in teger c d so that the G Q -action on dS facto r s through ( Z /c d Z ) ∗ . Any n ∈ N ′ with ndS = dS is a pro duct of primes that ar e unramified in ψ d ( A ) by 2.1, and so 2 .2 tells us tha t ( n mo d c d ) ∈ ( Z /c d Z ) ∗ and that this element acts on dS in the s ame wa y as n . This gives condition (ii). 3. Explicit maximal Λ -orders Given a prime num b er p , there is a notion of Λ p -action on a ring R , and as befo re, this ha s a simple description if R ha s no p -tor sion: a r ing endomor phism ψ p of R that lifts the F robenius endomorphism, that is, s uc h that ψ p ( x ) − x p ∈ pR for all x ∈ R . Also as b efor e, a sub-Λ p -ring A of a Q p Λ p -ring K is called a Λ p -order if it is finite ov er Z p . It is said to b e maximal if it contains e very other Λ p -order in K . W e have t wo natural ways o f making Λ p -orders . First, for any ab elian gr oup V the gr oup ring Z p [ V ] is a Λ p -ring whe n w e s et ψ p ( r ) = r for r ∈ Z p and ψ p ( v ) = v p for v ∈ V . Secondly , if A is the ring of integers of a finite unramified extension K of Q p , then A has a unique Λ p -ring structure, wher e ψ p is the F rob enius map (cf. 2.1). By extending ψ p to K w e see that A is the maximal Λ p -order of K . T aking tensor pro ducts of these tw o building blo cks we see that for any in teger q the ring A [ µ q ] = A [ z ] / ( z q − 1) is a Λ p -order in K [ µ q ]. 3.1. Lemma. If q is a p ower of p , then A [ µ q ] is the maximal Λ p -or der of K [ µ q ] . Pr o of. By inductio n, it is eno ugh to assume A [ µ q ] is maximal and then prove A [ µ pq ] is. Let k denote the r esidue field of A , and let ζ denote a primitive pq - th r o ot o f unit y in so me extension o f K . Then w e have K [ z ] / ( z pq − 1) = K ( ζ ) × K [ y ] / ( y q − 1), the element z corres po nding to ( ζ , y ). In these terms, the Λ p -action is given by ψ p ( b, f ( y )) = ( f ∗ ( ζ p ) , f ∗ ( y p )), where f ∗ ( y ) denotes p o lynomial obtained by applying the F ro benius map ψ p co efficient-wise to f ( y ). Now c o nsider the following diagram of ring s: A [ z ] / ( z pq − 1) z 7→ y / / / / z 7→ ζ     A [ y ] / ( y q − 1) y 7→ y     A [ ζ ] ζ 7→ y / / / / k [ y ] / ( y q − 1) , GALOIS THE OR Y AND INTEGRAL MODELS OF Λ-RINGS 7 where A [ ζ ] denotes the r ing of integers in K ( ζ ). As noted for example in Kerv aire– Murthy [6], this is a pull-back diagra m. This is just an instance of the easy fa c t that for ideals I a nd J in any ring R , we hav e R/ ( I ∩ J ) = R /I × R/ ( I + J ) R/J. In our case, take R = A [ z ], I = ( z q − 1), and J = (1 + z q + · · · + z q ( p − 1) ). Now supp ose R is a Λ p -order in K [ µ pq ]. Then the ima g e of R in K [ y ] / ( y q − 1) is contained, by induction, in A [ y ] / ( y q − 1). Therefore R is c o nt ained in A [ ζ ] × A [ y ] / ( y q − 1); we will view e le men ts of R as elements o f this pro duct without further comment. Because the diag ram above is a pull-back dia g ram, we need only show that the tw o maps R ⇒ k [ y ] / ( y q − 1 ) g iven by mapping to the tw o factors a nd then pro jecting to k [ y ] / ( y q − 1) agree. Let v denote the v alua tion on A [ ζ ] nor malized so that v ( p ) = 1. Let a 6 1 / ( p − 1) be the larges t n umber such that (3.1.1) for all ( b, f ( y )) ∈ R w e have v ( b − f ( ζ )) > a. Note that the expression f ( ζ ) makes sense only mo dulo ζ p − 1, and so the condition ab ov e is meaningless if a > v ( ζ p − 1) = 1 / ( p − 1). Let ( b, f ( y )) b e an element of R . Then b ecaus e R is a Λ p -ring, there is an elemen t ( c, g ( y )) ∈ pR such that ( c, g ( y )) = ( b, f ( y )) p − ψ p ( b, f ( y )) = ( b p − f ∗ ( ζ p ) , f ( y ) p − f ∗ ( y p )) . On the other ha nd, b ecause of our a ssumption on a , we hav e 1 + a 6 v ( c − g ( ζ )) = v (( b p − f ∗ ( ζ p )) − ( f ( ζ ) p − f ∗ ( ζ p )) = v ( b p − f ( ζ ) p ) . But b ecause the in teg ral p olynomial p ( X − Y ) divides ( X − Y ) p − ( X p − Y p ), we hav e v (( b − f ( ζ )) p − ( b p − f ( ζ ) p )) > 1 + a and hence v ( b − f ( ζ )) > (1 + a ) /p. That is, (1 + a ) /p satisfies (3.1 .1). But then a = 1 / ( p − 1 ) b eca us e otherwise this would violate the maximality of a . In other words, for any element ( b, f ( y )) ∈ R , the element ζ p − 1 divides b − f ( ζ ). This is just another way of saying b a nd f ( y ) hav e the same ima ge in A [ ζ ] / ( ζ p − 1) = k [ y ] / ( y q − 1), and hence the element ( b, f ( y )) lie s in the fib er pro duct, which w e showed ab ov e is A [ µ pq ].  3.2. R emark. It is not true that Z p [ V ] is maximal for e very finite ab elian group V . F or ex ample, if V = Z /p Z × Z / p Z and x = 1 p P σ ∈ V σ , then ψ p ( x ) = p a nd x 2 = px . Therefor e, the subring Z p [ V ][ x ] of Q p [ V ] is a sub-Λ p -ring whic h is finite ov er Z p . The global analogue a lso holds: Z [ V ] is not the maximal integral Λ-mo del for Q [ V ]. This follo ws from the lo cal statement. 3.3. Prop osition. L et K b e a finite ´ etale Q Λ -ring, and let R b e a Λ -or der. If Z p ⊗ R is a maximal Λ p -or der in Q p ⊗ R for al l primes p , t hen R is maximal. Pr o of. Let S be the maxima l Λ-order in Q ⊗ R . Beca use Z p ⊗ R is the maximal Λ p -order in Q p ⊗ R , the inclusion Z p ⊗ R ⊆ Z p ⊗ S is an equality . Therefo re R and S have the s a me rank, and p does no t divide the index of R in S .  3.4. Theorem. L et r > 1 b e an inte ger. Then Z [ µ r ] is the maximal Λ -or der in Q [ µ r ] . 8 J. BORGER, B. DE SMIT Pr o of. By 3.3, it s uffices to show that for every prime p , the Λ p -order Z p [ µ r ] maximal. W rite r = q n , where q is the larg e st p ow er of p dividing r . Then Z p [ µ r ] = Z p [ µ n ][ µ q ] = Q A A [ µ q ], where A r uns over the irreducible factors of Z p [ µ n ], all of whic h a re unramified extensions of Z p . By 3.1, each facto r A [ µ q ] is a maximal Λ p -order, and therefor e so is their pr o duct.  3.5. R emark. Using 0.3 and 1.2, w e can als o describ e the maximal Λ- o rder in general as follows. Let S be a finite ( Z /r Z ) ◦ -set, let ζ r ∈ ¯ Q denote a pr imitiv e r -th ro ot of unity , and let K = Hom ( Z /r Z ) ∗ ( S, Q ( ζ r )) denote the co rresp onding finite ´ etale Λ-ring over Q . Consider the iso mo rphism Q [ µ r ] → Y d | r Q ( ζ d r ) given by z 7→ ( . . . , ζ d r , . . . ) d | r . Then the ma ximal Λ-or der in K is the set of elements f ∈ K s uch that for all s ∈ S , the element ( . . . , f ( ds ) , . . . ) d | r ∈ Y d | r Q ( ζ d r ) ∼ = Q [ µ r ] lies in Z [ µ r ]. References [1] James Bor ger and Ben Wieland. Pl eth ystic algebra. A dv. Math. , 194(2):246–283, 2005. [2] F. J. B. J. Clauw ens. Commuting p olynomials and λ - ring structures on Z [ x ] . J. Pur e Appl. Algebr a , 95( 3):261–269, 1994. [3] Andreas W. M. Dress. Induction and structure theorems f or orthogonal representations of finite groups. Ann. of Math. (2) , 102 (2):291–32 5, 1975. [4] A. Grothendiec k. ´ El´ ements de g´ eom´ etrie alg´ ebri que. IV. ´ Etude lo cale des sch ´ emas et des mor- phismes de sch ´ emas IV. Inst. Hautes ´ Etudes Sci. Publ. Math. , (32):361, 1967. [5] Al exander Grothendiec k. La th´ eorie des classes de Chern. Bull. So c . Math. F r anc e , 86:137–154, 1958. [6] Mi c hel A. Kerv aire and M. P av aman Murth y . On the pro jective class gr oup of cyclic groups of prime p ow er order. Comment. Math. Helv. , 52(3):415–452, 197 7. [7] J ¨ urgen Neukirch. Class field the ory , v olume 280 of Grund lehr en der Mathematischen Wis- senschaften . Springer-V erlag, Berlin, 1986. [8] Clar ence Wilke rson. Lambda-rings, binomial domains, and vec tor bundles ov er C P ( ∞ ). Comm. A lgeb r a , 10(3):311– 328, 1982. James Bor g er, Ma them a tical Sciences Institute, Build ing 27, Australian Na tional University, ACT 0 200, Australia Bar t de Smit, Ma them a tisch Instituut, Universiteit Leiden, Postbus 9512, 2300 RA Leiden, The Netherlands E-mail ad dr e ss : borger@mat hs.anu.e du.au, desmit@math. leidenuni v.nl