13. Dedekind Domains. 13. Dedekind Domains 117

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1 3. Dedekd Domas 7 3. Dedekd Domas I the last chapter we have maly studed -dmesoal regular local rgs,. e. geometrcally the local propertes of smooth pots o curves. We ow wat to patch these local results together to obta global statemets about -dmesoal rgs (resp. curves) that are locally regular. The correspodg oto s that of a Dedekd doma. Defto 3. (Dedekd domas). A tegral doma R s called Dedekd doma f t s Noethera of dmeso, ad for all maxmal deals P R the localzato R P s a regular local rg. Remark 3.2 (Equvalet codtos for Dedekd domas). As a Dedekd doma R s a tegral doma of dmeso, ts prme deals are exactly the zero deal ad all maxmal deals. So every localzato R P for a maxmal deal P s a -dmesoal local rg. As these localzatos are also Noethera by Exercse 7.23, we ca replace the requremet Defto 3. that the local rgs R P are regular by ay of the equvalet codtos Proposto 2.4. For example, a Dedekd doma s the same as a -dmesoal Noethera doma such that all localzatos at maxmal deals are dscrete valuato rgs. Ths works partcularly well for the ormalty codto as ths s a local property ad ca thus be trasferred to the whole rg: Lemma 3.3. A -dmesoal Noethera doma s a Dedekd doma f ad oly f t s ormal. Proof. By Remark 3.2 ad Proposto 2.4, a -dmesoal Noethera doma R s a Dedekd doma f ad oly f all localzatos R P at a maxmal deal P are ormal. But by Exercse 9.3 (c) ths s equvalet to R beg ormal. Example 3.4. (a) By Lemma 3.3, ay prcpal deal doma whch s ot a feld s a Dedekd doma: t s -dmesoal by Example.3 (c), clearly Noethera, ad ormal by Example 9.0 sce t s a uque factorzato doma by Example 8.3 (a). For better vsualzato, the followg dagram shows the mplcatos betwee varous propertes of rgs for the case of tegral domas that are ot felds. Rgs that are always -dmesoal ad / or local are marked as such. It s true that every regular local rg s a uque factorzato doma, but we have ot prove ths here sce ths requres more advaced methods we have oly show Proposto.40 that ay regular local rg s a tegral doma. 25 dm = DVR local 2.4 (b) 2.4 (e) dm = PID regular local 3.4 (a) dm = Dedekd 8.3 (a) UFD ormal def. doma (b) Let X be a rreducble curve over a algebracally closed feld. Assume that X s smooth,. e. that all pots of X are smooth the sese of Example.37 ad Defto.38. The the coordate rg A(X) s a Dedekd doma: t s a tegral doma by Lemma 2.3 (a) sce I(X) s a prme deal by Remark 2.7 (b). It s also -dmesoal by assumpto ad

2 8 Adreas Gathma Noethera by Remark 7.5. Moreover, by Hlbert s Nullstellesatz as Remark 0. the maxmal deals A(X) are exactly the deals of pots, ad so our smoothess assumpto s the same as sayg that all localzatos at maxmal deals are regular. I fact, rreducble smooth curves over algebracally closed felds are the ma geometrc examples for Dedekd domas. However, there s also a large class of examples umber theory, whch explas why the cocept of a Dedekd doma s equally mportat umber theory ad geometry: t turs out that the rg of tegral elemets a umber feld,. e. a fte feld exteso of Q, s always a Dedekd doma. Let us prove ths ow. Proposto 3.5 (Itegral elemets umber felds). Let Q K be a fte feld exteso, ad let R be the tegral closure of Z K. The R s a Dedekd doma. Proof. As a subrg of a feld, R s clearly a tegral doma. Moreover, by Example.3 (c) ad Lemma.8 we have dmr = dmz =. It s also easy to see that R s ormal: f a QuotR K s tegral over R t s also tegral over Z by trastvty as Lemma 9.6 (b), so t s cotaed the tegral closure R of Z K. Hece by Lemma 3.3 t oly remas to show that R s Noethera whch s fact the hardest part of the proof. We wll show ths three steps. (a) We clam that R/pR < for all prme umbers p Z. Note that R/pR s a vector space over Z p = Z/pZ. It suffces to show that dm Zp R/pR dm Q K sce ths dmeso s fte by assumpto. So let a,...,a R/pR be learly depedet over Z p. We wll show that a,...,a K are also depedet over Q, so that dm Q K. Otherwse there are λ,...,λ Q ot all zero wth λ a + + λ a = 0. After multplyg these coeffcets wth a commo scalar we may assume that all of them are tegers, ad ot all of them are dvsble by p. But the λ a + + λ a = 0 s a o-trval relato R/pR wth coeffcets Z p, cotradcto to a,...,a beg depedet over Z p. (b) We wll show that R/mR < for all m Z\{0}. I fact, ths follows by ducto o the umber of prme factors m: for oe prme factor the statemet s just that of (a), ad for more prme factors t follows from the exact sequece of Abela groups 0 R/m R m 2 R/m m 2 R R/m 2 R 0, sce ths meas that R/m m 2 R = R/m R R/m 2 R <. (c) Now let I R be ay o-zero deal. We clam that m I for some m Z\{0}. Otherwse we would have dmr/i = dmz/(i Z) = dmz = by Lemma.8, sce R/I s tegral over Z/(I Z) by Lemma 9.7 (a). But dmr has to be bgger tha dmr/i, sce a cha of prme deals R/I correspods to a cha of prme deals R cotag I, whch ca always be exteded to a loger cha by the zero deal sce R s a tegral doma. Hece dmr >, a cotradcto. Puttg everythg together, we ca choose a o-zero m I Z by (c), so that mr I. Hece I/mR R/mR < by (b), so I/mR = {a,...,a } for some a,...,a I. But the the deal I = (a,...,a,m) s ftely geerated, ad hece R s Noethera. Example 3.6. Cosder aga the rg R = Z[ 5] of Example 8.3 (b). By Example 9.6, t s the tegral closure of Z Q( 5). Hece Proposto 3.5 shows that R s a Dedekd doma. We see from ths example that a Dedekd doma s geeral ot a uque factorzato doma, as e. g. by Example 8.3 (b) the elemet 2 s rreducble, but ot prme R, so that t does ot have a factorzato to prme elemets. However, we wll prove ow that a Dedekd doma always has a aalogue of the uque factorzato property for deals,. e. every o-zero deal ca be wrtte uquely as a product of o-zero prme deals (whch are the also maxmal sce Dedekd

3 3. Dedekd Domas 9 domas are -dmesoal). I fact, ths s the most mportat property of Dedekd domas practce. Proposto 3.7 (Prme factorzato of deals Dedekd domas). Let R be a Dedekd doma. Proof. (a) Let P R be a maxmal deal, ad let Q R be ay deal. The Q s P-prmary Q = P k for some k N >0. Moreover, the umber k s uque ths case. (b) Ay o-zero deal I R has a prme factorzato I = P k Pk wth k,...,k N >0 ad dstct maxmal deals P,...,P R. It s uque up to permutato of the factors, ad P,...,P are exactly the assocated prme deals of I. (a) The mplcato holds arbtrary rgs by Lemma 8.2 (b), so let us show the opposte drecto. Let Q be P-prmary, ad cosder the localzato map R R P. The Q e s a o-zero deal the localzato R P, whch s a dscrete valuato rg by Remark 3.2. So by Corollary 2.7 we have Q e = (P e ) k for some k, ad hece Q e = (P k ) e as exteso commutes wth products by Exercse.9 (c). Cotractg ths equato ow gves Q = P k by Lemma 8.33, sce Q ad P k are both P-prmary by Lemma 8.2 (b). The umber k s uque sce P k = P l for k l would mply (P e ) k = (P e ) l by exteso, cotradcto to Corollary 2.7. (b) As R s Noethera, the deal I has a mmal prmary decomposto I = Q Q by Corollary 8.2. Sce I s o-zero, the correspodg assocated prme deals P,...,P of these prmary deals are dstct ad o-zero, ad hece maxmal as dmr =. I partcular, there are o strct clusos amog the deals P,...,P, ad thus all of them are mmal over I. By Proposto 8.34 ths meas that the deals Q,...,Q our decomposto are uque. Now by (a) we have Q = P k for uque k N >0 for =,...,. Ths gves us a uque decomposto I = P k Pk, ad thus also a uque factorzato I = P k Pk by Exercse.8 sce the deals P k,...,pk are parwse coprme by Exercse Example 3.8. Recall from Examples 8.3 (b) ad 3.6 that the elemet 2 the Dedekd doma R = Z[ 5] does ot admt a factorzato to prme elemets. But by Proposto 3.7 (b) the deal (2) must have a decomposto as a product of maxmal deals (whch caot all be prcpal, as otherwse we would have decomposed the umber 2 to prme factors). Cocretely, we clam that ths decomposto s (2) = (2, + 5) 2. To see ths, ote frst that the deal (2, + 5) s maxmal by Lemma 2.3 (b) sce the quotet Z[ 5]/(2, + 5) = Z/(2) = Z 2 s a feld. Moreover, we have (2) (2, + 5) 2 sce ad (2, + 5) 2 (2) as 2 = ( + 5) ( + 5) (2, + 5) 2, 2 2 (2), 2( + 5) (2), ad ( + 5) 2 = (2). To uderstad the geometrc meag of the prme factorzato of deals we eed a lemma frst. Lemma 3.9 (Ideals Dedekd domas). Let R be a Dedekd doma.

4 20 Adreas Gathma (a) For all dstct maxmal deals P,...,P of R ad k,...,k,l,...,l N we have P k Pk P l Pl l k for all =,...,. (b) For ay a R\{0} we have (a) = P ν (a) P ν (a), where P,...,P are the assocated prme deals of (a), ad ν deotes the valuato of the dscrete valuato rg R P (restrcted to R). Proof. By Exercse 6.29 (a) deal cotamet s a local property,. e. we ca check t o all localzatos R P for maxmal deals P. Moreover, products commute wth localzato by Exercse.9 (c), ad the localzato of P at a maxmal deal P P s the ut deal by Example 6.25 (a). Hece: (a) We have P k Pk P l Pl (P e ) k (P e ) l R P for all l k for all. (Corollary 2.7) (b) By Proposto 3.7 (b) we kow that (a) = P k Pk for sutable k,...,k N f P,...,P are the assocated prme deals of (a). To determe the expoet k for =,...,, we localze at R P to get (a) = (P e)k the dscrete valuato rg R P, ad use Proposto 2.3 to coclude from ths that k = ν (a). Remark 3.0. If a s a o-zero elemet a Dedekd doma R ad P R a maxmal deal that s ot a assocated prme deal of (a), the same argumet as the proof of Lemma 3.9 (b) shows that the valuato of a the dscrete valuato rg R P s 0. Hece the deals P,...,P the statemet of ths lemma are exactly the maxmal deals of R so that the valuato of a the correspodg dscrete valuato rg s o-zero. Remark 3. (Geometrc terpretato of the prme factorzato of deals). Let X be a rreducble smooth curve over a algebracally closed feld, so that ts coordate rg R = A(X) s a Dedekd doma by Example 3.4 (b). Now let f R be a o-zero polyomal fucto o X, ad let a,...,a X be the zeroes of f, wth correspodg maxmal deals P,...,P R. Moreover, for =,..., let k be the order of vashg of f at a as Remark 2.2,. e. the valuato of f the rg R P of local fuctos o X at a. The Lemma 3.9 (b) (together wth Remark 3.0) states that ( f ) = P k Pk. I other words, the prme factorzato of a prcpal deal ( f ) the coordate rg of X ecodes the orders of vashg of the fucto f at all pots of X. Here s a cocrete example of ths costructo that wll also be used later o Example Example 3.2. Cosder the complex plae cubc curve X = V (y 2 x(x )(x λ)) A 2 C for some λ C\{0,}. The pcture (a) below shows approxmately the real pots of X the case λ R > : the vertcal le x = c for c R tersects X two real pots symmetrc wth respect to ths axs f 0 < c < or c > λ, exactly the pot (c,0) f c {0,,λ}, ad o real pot all other cases.

5 3. Dedekd Domas 2 x = c l P 2 P 3 0 λ x P P 2 l X P (a) (b) It s easy to check as Example.39 (b) that all pots of X are smooth, so that the coordate rg R = C[x,y]/(y 2 x(x )(x λ)) of X s a Dedekd doma by Example 3.4 (b). Now let l R be a geeral lear fucto as pcture (b) above. O the curve X t vashes at three pots (to order ) sce the cubc equato y 2 = x(x )(x λ) together wth a geeral lear equato x ad y wll have three solutos. If P, P 2, ad P 3 are the maxmal deals R correspodg to these pots, Remark 3. shows that (l) = P P 2 P 3 R. Note that for specal lear fuctos t mght happe that some of these pots cocde, as the case of l above whch vashes to order 2 at the pot P. Cosequetly, ths case we get (l ) = P 2 P 2. For computatoal purposes, the uque factorzato property for deals allows us to perform calculatos wth deals Dedekd domas very much the same way as prcpal deal domas. For example, the followg Proposto 3.3 s etrely aalogous (both ts statemet ad ts proof) to Example.4. Proposto 3.3 (Operatos o deals Dedekd domas). Let I ad J be two o-zero deals a Dedekd doma, wth prme factorzatos I = P k Pk ad J = P l Pl as Proposto 3.7, where P,...,P are dstct maxmal deals ad k,...,k,l,...,l N. The I + J = P m P m wth m = m(k,l ), I J = P m P m wth m = max(k,l ), I J = P m P m wth m = k + l for =,...,. I partcular, I J = (I + J) (I J). Proof. By Proposto 3.7 (b) we ca wrte all three deals as P m P m for sutable m,...,m (f we possbly elarge the set of maxmal deals occurrg the factorzatos). So t oly remas to determe the umbers m,...,m for the three cases. The deal I + J s the smallest deal cotag both I ad J. By Lemma 3.9 (a) ths meas that m s the bggest umber less tha or equal to both k ad l,. e. m(k,l ). Aalogously, the tersecto I J s the bggest deal cotaed both I ad J, so ths case m s the smallest umber greater tha or equal to both k ad l,. e. max(k,l ). The expoets m = k + l for the product are obvous. As a Dedekd doma s geeral ot a uque factorzato doma, t clearly follows from Example 8.3 (a) that t s usually ot a prcpal deal doma ether. However, a surprsg result followg from the computatoal rules Proposto 3.3 s that every deal a Dedekd doma ca be geerated by two elemets. I fact, there s a eve stroger statemet: Proposto 3.4. Let R be a Dedekd doma, ad let a be a o-zero elemet a deal I R. The there s a elemet b R such that I = (a,b). I partcular, every deal R ca be geerated by two elemets.

6 22 Adreas Gathma 26 Proof. By assumpto (a) I are o-zero deals, so we kow by Proposto 3.7 (b) ad Lemma 3.9 (a) that (a) = P k Pk ad I = P l Pl for sutable dstct maxmal deals P,...,P R ad atural umbers l k for all =,...,. By the uqueess part of Proposto 3.7 (b) we ca pck elemets b P l + P l P l + P l + P l + P l + for all. The b P l j+ j for all j, but b / P l +, sce otherwse by Proposto 3.3 b P l + ( P l + P l P l + cotradcto to our choce of b. Hece ) = P l + P l + b := b + + b / P l + for all, but certaly b I. We ow clam that I = (a,b). To see ths, ote frst that by Proposto 3.3 the prme factorzato of (a,b) = (a) + (b) ca cota at most the maxmal deals occurrg (a), so we ca wrte (a,b) = P m P m for sutable m,...,m. But by Lemma 3.9 (a) we see that: l m for all sce (a,b) I; m l for all sce b / P l +, ad hece (a,b) P l +. Therefore we get m = l for all, whch meas that I = (a,b). We have ow studed prme factorzatos of deals Dedekd domas some detal. However, recall that the uderlyg valuatos o the local rgs are defed orgally ot oly o these dscrete valuato rgs, but also o ther quotet feld. Geometrcally, ths meas that we ca equally well cosder orders of ratoal fuctos,. e. quotets of polyomals, at a smooth pot of a curve. These orders ca the be postve (f the fucto has a zero), egatve (f t has a pole), or zero (f the fucto has a o-zero value at the gve pot). Let us ow trasfer ths exteso to the quotet feld to the global case of a Dedekd doma R. Istead of deals we the have to cosder correspodg structures (. e. R-submodules) that do ot le R tself, but ts quotet feld QuotR. Defto 3.5 (Fractoal deals). Let R be a tegral doma wth quotet feld K = QuotR. (a) A fractoal deal of R s a R-submodule I of K such that ai R for some a R\{0}. (b) For a,...,a K we set as expected I (a,...,a ) := Ra + + Ra ad call ths the fractoal deal geerated by a,...,a (ote that ths s fact a fractoal deal sce we ca take for a (a) the product of the deomators of a,...,a ). Example 3.6. (a) A subset I of a tegral doma R s a fractoal deal of R f ad oly f t s a deal R (the codto Defto 3.5 (a) that ai R for some a s vacuous ths case sce we ca always take a = ). (b) ( 2 ) = 2 Z Q s a fractoal deal of Z. I cotrast, the localzato Z (2) Q of Example 6.5 (d) s ot a fractoal deal of Z: t s a Z-submodule of Q, but there s o o-zero teger a such that az (2) Z. Remark 3.7. Let R be a tegral doma wth quotet feld K. (a) Let I be a fractoal deal of R. The codto ai R of Defto 3.5 (a) esures that I s ftely geerated f R s Noethera: as ai s a R-submodule R t s actually a deal R, ad hece of the form ai = (a,...,a ) for some a,...,a R. But the also I = ( a a,..., a ) a s ftely geerated.

7 3. Dedekd Domas 23 (b) The stadard operatos o deals of Costructo. ca easly be exteded to fractoal deals, or more geerally to R-submodules of K. I the followg, we wll maly eed products ad quotets: for two R-submodules I ad J of K we set IJ := { a b : N, a,...,a I, b,...,b J = I : K J := {a K : aj I}. Note that IJ s just the smallest R-submodule of K cotag all products ab for a I ad b J, as expected. The dex K the otato of the quotet I : K J dstgushes ths costructo from the ordary deal quotet I : J = {a R : aj I} ote that both quotets are defed but dfferet geeral f both I ad J are ordary deals R. Exercse 3.8. Let K be the quotet feld of a Noethera tegral doma R. Prove that for ay two fractoal deals I ad J of R ad ay multplcatvely closed subset S R we have: (a) S (IJ) = S I S J, (b) S (I : K J) = S I : K S J. Our goal the followg wll be to check whether the multplcato as Remark 3.7 (b) defes a group structure o the set of all o-zero fractoal deals of a tegral doma R. As assocatvty ad the exstece of the eutral elemet R are obvous, the oly remag questo s the exstece of verse elemets,. e. whether for a gve o-zero fractoal deal I there s always aother fractoal deal J wth IJ = R. We wll see ow that ths s deed the case for Dedekd domas, but ot geeral tegral domas. Defto 3.9 (Ivertble ad prcpal deals). Let R be a tegral doma, ad let I be a R-submodule of K = Quot R. (a) I s called a vertble deal or (Carter) dvsor f there s a R-submodule of J of K such that IJ = R. (b) I s called prcpal f I = (a) for some a K. Lemma As above, let K be the quotet feld of a tegral doma R, ad let I be a R- submodule of K. Proof. (a) If I s a vertble deal wth IJ = R, the J = R: K I. (b) We have the mplcatos }, I o-zero prcpal I vertble I fractoal. (a) By defto of the quotet, IJ = R mples R = IJ I (R : K I) R, so we have equalty IJ = I (R: K I). Multplcato by J ow gves the desred result J = R: K I. (b) If I = (a) s prcpal wth a K the (a) ( a ) = R, hece I s vertble. Now let I be a vertble deal,. e. I (R: K I) = R by (a). Ths meas that = a b = for some N ad a,...,a I ad b,...,b R: K I. The we have for all b I b = = a b b. }{{} R So f we let a R be the product of the deomators of a,...,a, we get ab R. Therefore ai R,. e. I s fractoal.

8 24 Adreas Gathma Example 3.2. (a) Let R be a prcpal deal doma. The every o-zero fractoal deal I of R s prcpal: we have ai R for some a QuotR\{0}. Ths s a deal R, so of the form (b) for some b R\{0}. It follows that I = ( b a ),. e. I s prcpal. I partcular, Lemma 3.20 (b) mples that the otos of prcpal, vertble, ad fractoal deals all agree for o-zero deals a prcpal deal doma. (b) The deal I = (x, y) the rg R = R[x, y] s ot vertble: settg K = Quot R = R(x, y) we have R: K I = { f R(x,y) : x f R[x,y] ad y f R[x,y]} = R[x,y]. But I (R: K I) = (x,y)r[x,y] R, ad hece I s ot vertble by Lemma 3.20 (a). Proposto 3.22 (Ivertble = fractoal deals Dedekd domas). I a Dedekd doma, every o-zero fractoal deal s vertble. Proof. Let I be a o-zero fractoal deal of a Dedekd doma R. Assume that I s ot vertble, whch meas by Lemma 3.20 (a) that I (R: K I) R. As the cluso I (R: K I) R s obvous, ths meas that I (R : K I) s a proper deal of R. It must therefore be cotaed a maxmal deal P by Corollary 2.7. Extedg ths cluso by the localzato map R R P the gves I e (R P : K I e ) P e by Exercse 3.8. Ths meas by Lemma 3.20 (a) that I e s ot vertble R P. But R P s a dscrete valuato rg by Remark 3.2, hece a prcpal deal doma by Proposto 2.4, ad so I e caot be a fractoal deal ether by Example 3.2 (a). Ths s clearly a cotradcto, sce I s assumed to be fractoal. Remark By costructo, the vertble deals of a tegral doma R form a Abela group uder multplcato, wth eutral elemet R. As expected, we wll wrte the verse R: K I of a vertble deal I as Lemma 3.20 (a) also as I. Proposto 3.22 tells us that for Dedekd domas ths group of vertble deals ca also be thought of as the group of o-zero fractoal deals. Moreover, t s obvous that the o-zero prcpal fractoal deals form a subgroup: every o-zero prcpal fractoal deal s vertble by Lemma 3.20 (b); the eutral elemet R = () s prcpal; for two o-zero prcpal fractoal deals (a) ad (b) ther product (ab) s prcpal; for ay o-zero prcpal fractoal deal (a) ts verse (a ) s also prcpal. So we ca defe the followg groups that are aturally attached to ay tegral doma. Defto 3.24 (Ideal class groups). Let R be a tegral doma. (a) The group of all vertble deals of R (uder multplcato) s called the deal group or group of (Carter) dvsors of R. We deote t by DvR. (b) We deote by PrR DvR the subgroup of (o-zero) prcpal deals. (c) The quotet PcR := DvR/PrR of all vertble deals modulo prcpal deals s called the deal class group, or group of (Carter) dvsor classes, or Pcard group of R. Let us restrct the study of these groups to Dedekd domas. I ths case, the structure of the deal group s easy to uderstad wth the followg proposto. Proposto 3.25 (Prme factorzato for vertble deals). Let I be a vertble deal a Dedekd doma R. The I = P k Pk for sutable dstct maxmal deals P,...,P ad k,...,k Z, ad ths represetato s uque up to permutato of the factors.

9 3. Dedekd Domas 25 Proof. By Lemma 3.20 (b) we kow that ai s a deal R for a sutable a R\{0}. Now by Proposto 3.7 (b) we have ai = P r Pr ad (a) = P s Ps for sutable dstct maxmal deals P,...,P ad r,...,r,s,...,s N, ad so we get a factorzato I = (a) ai = P r s P r s as desred. Moreover, f we have two such factorzatos P k Pk = P l Pl wth k,...,k,l,...,l Z, we ca multply ths equato wth sutable powers of P,...,P so that the expoets become o-egatve. The uqueess statemet the follows from the correspodg oe Proposto 3.7 (b). Remark Let R be a Dedekd doma. Proposto 3.25 states that the deal group DvR s fact easy to descrbe: we have a somorphsm Dv R {ϕ : mspec R Z : ϕ s o-zero oly o ftely may maxmal deals} sedg ay vertble deal P k Pk to the map ϕ : mspecr Z wth oly o-zero values ϕ(p ) = k for =,...,. Ths s usually called the free Abela group geerated by mspecr (sce the maxmal deals geerate ths group, ad there are o o-trval relatos amog these geerators). The group DvR s therefore very bg, ad also at the same tme ot very terestg sce ts structure s so smple. I cotrast, the deal class group PcR = DvR/PrR s usually much smaller, ad cotas a lot of formato o R. It s of great mportace both geometry ad umber theory. It s out of the scope of ths course to study t detal, but we wll at least gve oe terestg example each of these areas. But frst let us ote that the deal class group ca be thought of as measurg how far away R s from beg a prcpal deal doma : Proposto For a Dedekd doma R the followg statemets are equvalet: Proof. (a) R s a prcpal deal doma. (b) R s a uque factorzato doma. (c) PcR s the trval group,. e. PcR =. (a) (b) s Example 8.3 (a). (b) (c): By Exercse 8.32 (b) every maxmal deal P R (whch s also a mmal o-zero prme deal as dmr = ) s prcpal. But these maxmal deals geerate the group DvR by Proposto 3.25, ad so we have Pr R = Dv R,. e. Pc R =. (c) (a): By Defto 3.24 the assumpto PcR = meas that every vertble deal s prcpal. But every o-zero deal of R s vertble by Proposto 3.22, so the result follows. Example 3.28 (A o-trval Pcard group umber theory). Let R = Z[ 5] be the tegral closure of Z K = Q( 5) as Examples 8.3 (b) ad 3.6. We have see there already that R s a Dedekd doma but ot a prcpal deal doma: the deal I := (2, + 5) s ot prcpal. I partcular, the class of I the Pcard group PcR s o-trval. We wll ow show that ths s the oly o-trval elemet PcR,. e. that PcR = 2 ad thus ecessarly PcR = Z/2Z as a group, wth the two elemets gve by the classes of the deals I 0 := () ad I. Uwdg Defto 3.24, we therefore clam that every vertble deal of R s of the form ai 0 or ai for some a K. So let I be a vertble deal of R. The I s also a o-zero fractoal deal by Lemma 3.20 (b), ad thus there s a umber b K wth bi R. But ote that R = Z[ 5] = {m + 5 : m, Z} s just a rectagular lattce the complex plae, ad so there s a elemet c bi K of mmal o-zero absolute value. Replacg I by b c I (whch s a equvalet elemet the Pcard group) we

10 26 Adreas Gathma ca therefore assume that I s a vertble deal wth I, hece R I, ad that s a o-zero elemet of mmal absolute value I. Let us fd out whether I ca cota more elemets except the oes of R. To do ths, t suffces to cosder pots the rectagle wth corers 0,, 5, ad + 5 show the pctures below: I s a addtve subgroup of C cotag R, ad so the complete set of pots I wll just be a R-perodc repeated copy of the pots ths rectagle. To fgure out f I cotas more pots ths rectagle, we proceed three steps llustrated below. (a) By costructo, I cotas o pots of absolute value less tha except the org,. e. o pots the ope dsc U (0) wth radus ad ceter pot 0. Lkewse, because of the R-perodcty of I, the deal also does ot cota ay pots the ope ut dscs aroud the other corers of the rectagle, except these corer pots themselves. I other words, the shaded area pcture (a) below caot cota ay pots of I except the corer pots. 5 Im 5 Im 5 Im Re Re (a) (b) (c) Re (b) Now cosder the ope dsc U ( 2 2 5), whose tersecto wth our rectagle s the left dark half-crcle pcture (b) above. Aga, t caot cota ay pots of I except ts ceter: as I s a R-module, ay o-ceter pot a I ths dsc would lead to a o-ceter pot 2a I the dsc U ( 5), whch we excluded already (a). But fact the ceter pot 2 5 caot le I ether, sce the we would have = 2 I as well, cotradcto to (a). I the same way we see that the ope dsc U 2 ( + 2 5) does ot cota ay pots of I ether. Hece the complete shaded area pcture (b) s excluded ow for pots of I. (c) Fally, cosder the ope dsc U ( ), show pcture (c) above dark color. For the same reaso as (b), o pot ths dsc except the ceter ca le I. As our dscs ow cover the complete rectagle, ths meas that the oly pot our rectagle except the corers that ca be I s Ths leads to exactly two possbltes for I: ether I = R = I 0 or I = (, ) = 2 I. I fact, we kow already that ths last case 2 I s a vertble deal of R, so that ths tme ( cotrast to (b) above) t does ot lead to a cotradcto f the ceter pot of the dsc les I. Altogether, we thus coclude that PcR = 2,. e. PcR = Z/2Z, wth the class of I 0 beg eutral ad I beg the uque other elemet. We have deed also checked already that I s ts ow verse PcR, sce by Example 3.8 I I = (2)

11 3. Dedekd Domas 27 s prcpal, ad hece the eutral elemet PcR. Example 3.29 (A o-trval Pcard group geometry). Cosder aga the complex plae cubc curve X = V (y 2 x(x )(x λ)) A 2 C for some λ C\{0,} as Example 3.2. We have already see that ts coordate rg R = A(X) s a Dedekd doma. Let us ow study ts deal class group PcR. Note that there s a obvous map ϕ : X PcR, a I(a) that assgs to each pot of X the class of ts maxmal deal PcR = DvR/PrR. The surprsg fact s that ths map ϕ s jectve wth ts mage equal to PcR\{()},. e. to PcR wthout ts eutral elemet. We ca therefore make t bjectve by addg a pot at fty to X that s mapped to the mssg pot () of PcR. Ths s partcularly terestg as we the have a bjecto betwee two completely dfferet algebrac structures: X s a varety (but a pror ot a group) ad PcR s a group (but a pror ot a varety). So we ca use the bjecto ϕ to make the cubc curve X { } to a group, ad the group PcR to a varety. We caot prove ths statemet or study ts cosequeces here sce ths would requre methods that we have ot covered ths course ths s usually doe the Algebrac Geometry class. However, we ca use our defto of the Pcard group ad the map ϕ above to descrbe the group structure o the curve X { } explctly. To do ths, cosder two pots o the curve wth maxmal deals P ad P 2, as show the pcture o the rght. Draw the le through these two pots; t wll tersect X oe more pot Q sce the cubc equato y 2 = x(x )(x λ) together wth a lear equato x ad y wll have three solutos. By Example 3.2, ths meas algebracally that there s a lear polyomal l R (whose zero locus s ths le) such that (l) = P P 2 Q. Next, draw the vertcal le through Q. By the symmetry of X, t wll tersect X oe more pot P. Smlarly to the above, t follows that there s a lear polyomal l such that (l ) = Q P. But the Pcard group PcR ths meas that P P 2 Q = (l) = (l ) = Q P (ote that (l) ad (l ) both defe the eutral elemet PcR), ad thus that P P 2 = P. Hece the above geometrc costructo of P from P ad P 2 descrbes the group structure o X { } metoed above. Note that t s qute obvous wthout much theory behd t that ths geometrc two-le costructo ca be used to assocate to ay two pots o X (correspodg to P ad P 2 above) a thrd pot o X (correspodg to P). The surprsg statemet here (whch s very hard to prove wthout usg Pcard groups) s that ths gves rse to a group structure o X { }; partcular that ths operato s assocatve. The eutral elemet s the addtoal pot (correspodg to the class of prcpal deals PcR), ad the verse of a pot X s the other tersecto pot of the vertcal le through ths pot wth X (so that e. g. the above pcture we have P = Q). P P 2 X l Q P l 27