Lambda actions of rings of integers

LAMBD A A CTIONS OF RINGS OF INTE GERS JAMES BORGER, BAR T DE SMIT Abstract. Let O be the ring o f integers of a num ber field K . F or an O - algebra R which is torsion fr e e as a n O -mo dule we define what w e mean b y a Λ O -ring structure o n R . W e ca n determine whether a finite ´ etale K -algebra E with Λ O -ring structure has an int egral mo del in terms of a Deligne-Ribet monoid of K . This a commutativ e monoid whose inv ertible elements form a r ay class group. 1. Intr oduction Let O b e a D edekind domain with quotien t field K . Denote the set of maximal ideals of O by M . W e assume that k p p q  O { p is a finite field for eac h p P M . Let E be a torsion-free comm utativ e O -alg ebra. Then for eac h p P M the algebra E { p E  E b O k p p q o v er k p p q has a natural k p p q -algebra endomorphism F p : x ÞÑ x # k p p q , whic h is c alled the F rob e nius endomor- phism. By a F rob enius lift of E at p w e mean an O -algebra endomor- phism ψ p suc h that ψ p b k p p q  F p . W e define a Λ O -structure on E to b e a map M Ñ End O - alg p E q , denoted p ÞÑ ψ p , suc h tha t (1) ψ p is a F rob enius lift at p for eac h p P M . (2) ψ p ψ q  ψ q ψ p for a ll p , q P M . By a Λ O -ring w e mean a torsion-free O -a lgebra with Λ O -structure. If O is lo cal then the comm utation condition (2) is v acuous. F or all p P M for which E b k p p q  0 the lift ing condition (1) is v a cuous. In particular, if E is an alg ebra o v er K , then any comm uting collection of K -automor phisms of E indexed by the maximal ideals of O is a Λ O -structure o n E . A Λ Z -structure on a ring without Z -tor sion is the same as a λ -ring structure [2]. F or instance, for a n y ab e lian group A we hav e a natural Λ Z -stucture o n the group ring Z r A s giv en by ψ p p a q  a p for a P A and p a prime n um b er. If O is the ring of integers o f a n um b er field K , and E is the ring of integers of a subfield L of the strict Hilb ert class field of K , then E has a unique Λ O -structure: ψ p is the Artin sym b ol of p in the field extension K  L . This preprint is a prelimina ry version da ting from 2006. W e are making it av ailable in this for m b ecause some p eo ple would like to cite it now. The fina l version should be av a ilable before long. 1 2 J. BORGER, B. DE SMIT In an earlier pap er [1], w e sho we d that a Λ Z -ring that is reduced and finite flat ov er Z is a Λ Z -subring of Z r C s n for some finite cyclic group C and p o sitiv e in teger n . The pro of uses the explicit description of ray class fields o ver Q a s cyclotomic fields. Ov er a n um b e r field class field theory is less explicit, and the generalizations we presen t in t he presen t pap er are b y consequence less explicit. Ho wev er, w e can still giv e a v ery similar criterio n for a Λ O -structure on a finite ´ etale K -algebra E to come from an Λ O -subring whic h is finite flat as an O - mo dule see Theorem 1.2 b elow . Suc h a Λ O -subring is called an integral Λ O -mo del of the Λ O -ring E . Let I p O q b e the monoid of non- zero ideals of O , with ideal multipli- cation as the monoid op eration. It is the fr ee comm utativ e monoid on M . Let K sep b e a separable closure of K , and let G K b e the Ga lois group of K sep o ver K . It is a profite group. By a G K -set X w e mean a finite discrete se t with a contin uous G K -action. By G rothendiec k’s form ulation of G alois theory , a finite ´ etale K -alg ebra E is determined b y the G K -set S consisting of all K -algebra homomorphisms E Ñ K sep . Giving a Λ O -structure on E then translates to giving a monoid map I p O q Ñ Map G K p S, S q . By giving I p O q the discrete t o p ology , w e see that the category of Λ O -rings whose underlying O -algebra is a finite ´ etale K - algebra, is anti-equiv alen t to the catego ry of finite discrete sets with a con tin uous actio n of the monoid I p O q  G K . Let us first supp ose that O is complete discrete v a luation ring with maximal ideal p . Then I p O q is isomorphic as a monoid to the monoid of non-negative in tegers with addition. Let I K  G K b e the inertia subgroup. Then I K is no r mal in G K and G K { I K is the absolute Galois group of k p p q , whic h con tains the F rob enius elemen t F P G K { I K giv en b y x ÞÑ x # k p p q . Thus, F acts on an y G K -set on whic h I K acts trivially . Theorem 1.1. Supp ose O is c omplete discr ete v aluation ring with maximal id e al p . L et E b e a finite ´ etale K -algebr a with Λ O -structur e, and let S b e the se t of K - algebr a maps fr om E to K sep . Then K has an inte gr al Λ O -mo del if and only if the action of I p O q  G K on S satisfies the two c ondi tion s (1) the gr oup I K acts trivia l ly on S unr   a P I p O q a S ; (2) p P I p O q and F P G K { I K act in the same way on S unr . See Section 2 for the pro of. Next, let us assume that O is the ring of in tegers in a n umber field. In order to phra se our global result w e first recall the definition of the Deligne-Rib et monoid. A cycle of K is a forma l pro duct f  ± p p n p , where the pro duct rang es o ve r all primes of K , b oth finite and infinite, all n p are non- negativ e integers, only finitely man y of whic h a r e non- zero, and w e hav e n p P t 0 , 1 u for r eal primes p , and n p  0 fo r complex LAMBDA ACTIONS OF RI NGS OF INTEGERS 3 primes p . The finite part of f is f fin  ± p 8 p n p , which can b e view ed as an elemen t of I p O q . W e write ord p p f q  n p . F or a cycle f w e sa y tha t t w o non-zero O -ideals a and b are f - equiv alen t if x a  b for some x P K  with x ¡ 0 at all real pla ces p with ord p p f q ¡ 0, and ord p p x  1 q  ord p p a q ¥ ord p p f q at all finite places p . One can c hec k that this is an equiv alence relation, and that the m ultiplicatio n of ideals is we ll-defined on the quotien t set. Th us, the quotien t set is a monoid, the D eligne-Rib et-monoid, and w e denote it by DR p f q . It is not hard to see that the ra y class group Cl p f q is the group of in vertible elemen ts of DR p f q . Also, DR p 1 q is a group: it is the class group of O . More generally , fo r eac h ideal d dividing f fin w e can consider the map i d : C l p f { d q Ñ DR p f q that sends the class of an ideal a to t he class of a  d . These maps give rise to a bijection i  ² i d : º d | f fin Cl p f { d q  Ý Ñ DR p f q . Theorem 1.2. Supp ose O is the ring o f i n te ge rs of a numb er field K . L e t E b e a fin i te ´ etale K - a lgebr a with Λ O -structur e, an d let S b e the set of K -algebr a maps fr o m E to K sep . Then K ha s a n inte gr al Λ O -mo del if and o n ly if ther e is a cycle f of K so that the action of G K sep  I p O q on S factors (ne c essa rily uniq uely) thr ough the map G K  I p O q Ý Ñ DR p f q , which is the pr o duct of the Artin symb o l G K Ñ Cl p f q  DR p f q on the first c o or din a te, and the quotient map I p O q Ñ DR p f q on the se c o n d. It follo ws that the category of suc h Λ-ring s is an ti- equiv alent to the category of finite discrete sets with a con tin uous action b y the profinite monoid lim  Ý DR p f q , where the limit is tak en ov er all cycles f with resp ect to the canonical maps D R p f q Ñ DR p f 1 q when f 1 | f . When K  Q this limit is the multiplic ativ e monoid of profinite in tegers. 2. The local case Supp ose that O is a complete discrete v aluation ring with maximal ideal p . W e write k  k p p q . L et A b e a reduced finite flat O -algebra. Let us supp ose first tha t A is unramified ov er O , i.e., that k b O A is ´ etale ov er k . Then k b O A is a pro duct of finite fields. Since A is complete in its p -adic top olog y , idemp otents of A { p A lift to A , so that A is a finite pro duct of rings of integers in finite unramified extensions of K . W rite S  Hom O - alg p A, K sep q  Hom K - alg p A b O K, K sep q . Then the inertia group I K  G K acts trivially on S . Ev ery finite unramified field extension L of K is Galois with an ab elian Galois group, and it s rings of integers has a unique F rob enius lift, whic h is 4 J. BORGER, B. DE SMIT othen called the F ro b enius elemen t of the G alois group of L o v er K . It follo ws that when A is unramified o ve r O , it has a unique Λ O -structure. This is summarized in the next Prop osition. Prop osition 2.1. Supp os e that O is a c om plete discr ete valuation ring, and that A is an unr amifie d finite flat r e duc e d O -a lgebr a. Then A has a unique Λ O -structur e, and the induc e d action of I p O q  G K on S  Hom K - alg p A b O K, K sep q has the pr op erty that the intertia gr oup I K acts trivial ly and that p P I K p O q acts in the s a me way on S as F P G K { I K . Pr o o f o f The or em 1.1. Put S 0  S unr and for i  1 , 2 , . . . 1 let S i b e the set of all s P S with s R S i  1 and p s P S i  1 . Supp ose that S n  ∅ , and S n  1  ∅ . Let E i  Map G K p S i , K sep q b e the corresp onding finite ´ etale K -algebra fo r each i . Then multiplic ation b y p giv es rise to K -algebra homomorphisms f i : E i  1 Ñ E i for i  1 , 2 , . . . , and f 0 : E 0 Ñ E 0 . f 0 ü E 0 f 1 Ý Ñ E 1 f 2 Ý Ñ    f n  1 Ý Ñ E n Since S  S 0 ² S 1 ²    ² S n is a decomp osition o f G as a G K -set, and w e ha v e a corresp onding pro duct decomp osition of the finite ´ etale K -algebras E  E 0       E n . In terms of this decomp osition ψ p is giv en b y ψ p p e 0 , e 1 , . . . , e n q  p f 0 p e 0 q , f 1 p e 0 q , . . . , f n  1 p e n  1 qq . Since S 0 is closed under m ultiplicatio n by p , t he quotien t ring E 0 of E is a quotien t Λ O -ring of E , with F rob enius lift f 0 at p . W e will sho w that the Λ O -ring surjection E Ñ E 0 splits. Note that now p k S  S 0 for sufficien tly lar g e k , so p act as a bijection on S 0 . Thus, f 0 is a n automor phism of E 0 . F or s P S i w e hav e p i s P S 0 and p acts inv ertibly o n S 0 , so w e can define a map S Ý Ñ S 0 b y sending s P S i to p  i p p i s q . This map comm utes with the I p O q  G K -action, and it splits the inclusion S 0 Ñ S . Th us, E 0 is not only a quotien t Λ O -ring of E , but a lso a sub-Λ O -ring: i : E 0 Ý Ñ E e 0 ÞÝ Ñ p e 0 , f 1 f 1 0 e 0 , f 2 f 1 f  2 0 e 0 , . . . , f n  1    f 1 f  n  1 0 e 0 q . No w supp ose that the Λ O -ring E has an integral mo del, i.e., that E has a n O - sub algebra A whic h statisfies (1) A is finite flat ov er O ; (2) ψ p p A q  A ; (3) ψ p b O k is the F rob enius x ÞÑ x # k on A b k . The image A 0 of A in t he quotient ring E 0 of E is a sub-Λ O -ring of E 0 whic h is reduced and finitely g enerated as an O -mo dule and O -t o rsion free. Thus , E 0 has an integral Λ O -mo del. Since f 0 is an auto morphism of E 0 the rings A 0 and its subring f p A 0 q ha ve the same discriminan t. LAMBDA ACTIONS OF RI NGS OF INTEGERS 5 Th us, f 0 p A 0 q  A 0 and f 0 is an automorphism of A 0 . This implies that the map x ÞÑ x # k on A 0 b O k is an automorphism, so that A 0 is unramified o v er O . Conditions (1 ) a nd (2) of Theorem 1.1 now follow b y Prop o sition 2 .1 . F or the conv erse, supp ose t ha t conditions (1) and (2) hold. W e will pro duce an integral Λ O -mo del of E  E 0      E n . Let R i b e the inegral closure of O in E i . Since I K acts trivially on S 0 the ring R 0 has a unique Λ O -structure by Prop osition 2.1. No w suypp ose that A  i p R 0 q ` p 0  a 1      a n q with a i an ideal in R i . Then the condition ψ p p a q  a # k p p q P p A f o r all a P A is equiv alen t to a # k p p q i  pa i and f i p a i  1 q  pa i . This holds, fo r instance if a i  p i R i , in whic h case A is an in tegr a l Λ O -mo del of E .  The in tegra l mo del that is supplied by the pro of is not alw ays opti- mal. F or instance, for the Λ Z -ring Z r C 4 s w e get a strict subring. Ho w- ev er for the Λ Z -ring Z r V 4 s the pro of pro vides a Λ Z -subring of Q r V 4 s whic h is strictly larger than Z r V 4 s . 3. G lobal argum ents No w assume that K is a global field with ring of in tegers O . Let E b e a finite ´ etale K -algebra with a Λ O -sturcture. W riting S  Hom K p E , K sep q w e thus get an action of I p O q  G K on S . F or eac h maximal ideal p of O w e consider the completion O p , and its quotien t field K p . Then w e obtain an Λ O p -structure on the finite ´ etale K p -algebra E p  E b K K p . I f A is an integral Λ O -mo del of E , then A b O O p is a n in tegral Λ O p -mo del of E p . Fixing an embedding K sep Ñ K sep p for eac h p we can view G p as a subgroup of G K . The finite ´ etale K p -algebra E p then corresponds to the G p -set that one gets b y restricting the action of G K on S to G p . Let us assume that an in tegral Λ O -mo del A of E exists. Let ¯ G b e the image of G K in Map p S, S q . Cheb ot arev’s theorem no w implies the follo wing: for each g P ¯ G there is a maximal ideal p  p g of O so that (1) the image of I p is tr ivial in ¯ G ; (2) the image of F p P G p { I p in ¯ G is g ; (3) A is unramified at p . By Prop osition 2.1, the action of g on S is the same a s the action of p g on S . Since the p g comm ute with eac ho t her, it follo ws that ¯ G is ab elian. It remains top sho w that the I p O q  G K -action on S factors through the D eligne-Rib et monoid of some cycle f . By class field theory , any con tinuous action of G K on a finite discrete set T , whose image is ab elian, fa ctors, b y the Artin map, thro ugh the ra y class group Cl p c p T qq for a minimal cycle c p T q of K , whic h w e call the conductor of T . 6 J. BORGER, B. DE SMIT Define r P I p O q b y setting ord p p r q  inf t i ¥ 0 : p i  1 S  p i S u for all maximal ideals p of O . This is w ell defined b ecause p S  S whenev er p is unramified in A b y Prop osition unramified. W e no w define the cycle f b y f  lcm d | r d  c p d S q . Note first that c p S q | f , so the G K -action on S factors through the Artin map G K Ñ Cl p f q . Next, w e claim that fo r a P I p O q coprime to f the action of a on S is equal that of its class r a s in Cl p f q . It suffices to pr ov e this for a  p prime. Then one notes that r | f so p ∤ r so p acts as a bijection on S . By our lo cal result, A is unramified at p and p acts as F p P G p { I p on S . By the defnition of the Artin sym b o l, the action of r p s P Cl p f q is the same. This show s the claim. No w suppose that d P I p O q with d | f . Let us write I d for the submonoid of I p O q consisting of all a P I p O q that are coprime to f { d . W e now claim that I d acts by bijections on d S , b y the definition of f is quotient of Cl p f { d q . a nd that the action factors through Cl p f { d q . T o see this, let a  gcd p d , r q and write d  ab . By definition of r all prime divisors of b a ct bijectiv ely on d S , so c p a S q  c p d S q is coprime t o b . By definition o f f w e hav e c p a S q | f { g a , a nd it fo llo ws t hat c p a S q | f { d . Th us, the G K -action on d S factors through Cl p f { d q and the claim holds. Since m ultiplication b y an y divisor d P I p O q of f giv es a bijection Cl p f { d q Ñ r d s DR p f q  this sho ws that the action of t a P I p O q : gcd p a , f q  d u on S factors through r d s DR p f q  . T aking the union ov er all d we see that the I p O q -actions factors thro ugh DR p f q . F or the con v erse, assume t hat the G K  I p O q -action on S factors through D R p f q for some cycle f . W e first sho w the existence, for eac h p P M of an in tegral Λ O p -mo del for the Λ O p -ring E b K K p . F or p ∤ f this follo ws from the definition of the Artin map and Prop osition 2.1. So a ssume p | f , and write f  p n f 1 with p ∤ f 1 . Then r p k s P r p n s Cl p f 1 q  DR p f q for all k ¥ n . This implies that the action of p on  i p i S  p n S is give n by the Artin sym b ol of r p s P Cl p f 1 q , which b y Theorem 1.1 guarantees existence of an integral Λ O p -mo del. No w let R be the in tegral closure of O in E . Then R is finite flat o ver O and E  R b O K . F or all p ∤ f w e are in the unra mified case, and o ur in tegral Λ O p -mo dule is equal to R b O O p . It follow s that the in tersection A ov er all p of o ur in tegral Λ O p -mo dule giv es a sub- O - algebra of R , whic h is of finite index, and whic h is closed under all ψ p . Also, eac h ψ p are F rob enius lifts, since A b O O p is a Λ O p -ring. This pro v es Theorem 1.2 LAMBDA ACTIONS OF RI NGS OF INTEGERS 7 Reference s [1] James Bor ger and Bart de Smit. Galois theory and int egra l mo dels of Λ-rings . Bul l. L ond. Math. So c. , 40(3):43 9 –446 , 2008 . [2] Clarence Wilk erson. Lam b da- rings, binomial domains, and v ecto r bundles ov er C P p8q . Comm. Algebr a , 10(3):3 11–32 8, 1982. E-mail addr ess : ja mes.bo rger@a nu.edu.au, des mit@ma th.lei denuniv.nl