Constructions in R[x_1, ..., x_n]. Applications to K-Theory

CONSTRUCTIONS IN R[x1, . . . , xn]. APPLICATIONS TO K-THEORY JES´ US GAGO-VARGAS Abs ac . A classical esul in K-Theo y abou polynomial ings like he Quillen-Suslin heo em admi s an algo i hmic app oach when he ing o co- e icien s has some compu a ional p ope ies, associa ed wi h G ¨obne bases. The e a e se e al algo i hms when we wo k in K[x1,...,xn], Ka ield. In his pape we compu e a ee basis o a ini ely gene a ed p ojec i e module o e R[x1,...,xn], Ra p incipal ideal domain wi h addi ional p ope ies, es he eeness o p ojec i e modules o e D[x1,...,xn], wi h Da Dedekind domain like Z[√−5] and o he one a iable case compu e a ee basis i he e exis s any. 1. In oduc ion The Quillen-Suslin heo em asse s ha i A=D[x1, . . . , xn] is a polynomial ing o e a Dedekind domain D hen e e y ini ely gene a ed p ojec i e A-module is ex ended om D([21, 22]). When Dis a p incipal ideal domain e e y ini ely gen- e a ed p ojec i e A-module is ee. This is equi alen o say ha i Ris a p incipal ideal domain and = ( 1, . . . , m) is a unimodula ow o R[x1, . . . , xn]m hen he e exis s a ma ix U∈GL(m, R[x1, . . . , xn]) such ha ·U= (1,0,...,0), o ha we can comple e o an in e ible ma ix. An algo i hm o he Quillen-Suslin heo- em p oduces such ma ix, and we call i a QS-algo i hm. The las m−1 columns o he ma ix U o m a ee basis o he module de ined by ke ( )⊂R[x1, . . . , xn]m. The e a e se e al algo i hms when Ris a ield ([16, 6, 14, 15], [20] as a co olla y). The main ool in he p ocedu e is he algo i hm o compu e G ¨obne bases, which we can ind in o he ings like Z. In Sec ion 2 we gi e some algo i hmic esul s o e he ing R[x1, . . . , xn] ha we need la e , namely, he cons uc ion o a maximal ideal ha con ains an ideal o R[x1, . . . , xn] and how o compu e in S−1R[x] and R[x1, . . . , xn]M, ings ob ained om R[x1, . . . , xn]. In Sec ion 3 we p esen wo QS-algo i hms o R[x1, . . . , xn], ha a oid he no mal- iza ion s ep used in [7]. The i s one ollows [14, 15] and he second one [19, 17]. Ou s a ing poin is a p ojec i e module Pgi en as ke nel o a unimodula ow o as a submodule o a ee module. Then we can gene alize he esul s in [14] o monoid ings R[M], because he induc ion s ep educes he p oblem o a ee monoid, whe e we ha e sol ed he p oblem. In a simila way he QS-algo i hm 2000 Ma hema ics Subjec Classi ica ion. P ima y: 13C10, 13P10, 19A49, 68W30. Seconda y: 15A33. Key wo ds and ph ases. Se e Conjec u e, Quillen-Suslin Theo em, G ¨obne bases, Dedekind domains, p ojec i e modules. Pa ially suppo ed by DGICYT PB97-0723 and Jun a de Andaluc´ıa FQM-218. This is a p elimina y e sion o his a icle. 1 2 JES´ US GAGO-VARGAS o quo ien s o polynomial ings by monomial ideals, ha is, ings o he o m R[x1, . . . , xn]/I, wi h Ia monomial ideal and Ra PID, is easily ex ended, such as appea s in [13]. In Sec ion 4 we conside D he ing o in ege s o a numbe ield, a Dedekind domain in which i is possible o compu e. Fi s we gi e a new algo i hm using G ¨obne bases o ge he ac o iza ion o an ideal o Das p oduc o p ime ideals, and we apply i o ind a ee basis o a p ojec i e module o e D, i he e exis s one. The nex s ep is o s udy he eeness o a p ojec i e module Po e D[x1, . . . , xn]. We can do i by educing he p oblem o a module o e D, and o one a iable, we gi e an algo i hm o compu e a ee basis when he e exis s one. 2. P elimina y algo i hmse Le Rbe a ing. We ecall ha linea equa ions a e sol able in Ri we ha e an algo i hm o decide he membe ship p oblem o a elemen wi h espec o an ideal and we can compu e a se o gene a o s o he module Syz(a1, . . . , am), wi h a1, . . . , am∈R. Wi h hese condi ions we can build G ¨obne bases in he ing R[x1, . . . , xn] ([1, chap e 4]). We need o add ano he one. De ini ion 1. Le Rbe a ing. We say ha Ris an MC- ing i we can sol e linea equa ions in Rand, gi en I⊂Ra p ope ideal, i is possible o compu e a se o gene a o s o a maximal ideal ha con ains I. Fo example, Z,Z[√−5] a e MC- ings. Addi ionally, we need he ac o iza ion o polynomials in (R/hpi)[x], p∈Ra p ime elemen , and Q(R)[x], Q(R) he ield o ac ions o R. De ini ion 2. Le Rbe a ing. We say ha Rhas e ec i e cose ep esen a i es i gi en Jan ideal o Ri is possible o ind a comple e se Co cose ep esen a i es o R/J, and he e is a p ocedu e o ind, o all a∈R, an elemen c∈ C such ha a≡c(mod J). This de ini ion appea s in [1, p. 226], and we need his p ope y in R o compu e he no mal o m o a polynomial wi h espec o an ideal. We include he e he algo i hm desc ibed in [7] o compu e a se o gene a o s o a maximal ideal o R[x1, . . . , xn], Ran MC-PID, ha con ains an ideal. Algo i hm 1. Inpu : F={ 1, . . . , }se o gene a o s o an ideal Io R[x1, . . . , xn]. Ou pu : H={g1, . . . , gm}se o gene a o s o a maximal ideal M ⊂ R[x1, . . . , xn] ha con ains I. (1) Compu e hsi=hFi∩R. (2) I s6= 0, le p∈Rbe a p ime elemen such ha pdi ides s. (a) Compu e ¯g1,...,¯gk∈(R/hpi)[x1, . . . , xn]gene a o s o a maximal ideal M ha con ains ¯ Iin (R/hpi)[x1, . . . , xn]. (b) Li o g1, . . . , gk∈R[x1, . . . , xn]and le H={p, g1, . . . , gk}.STOP. (3) I s= 0, compu e d∈R, d 6= 0 such ha I= (I, d)∩Iec, whe e Iec = IQ(R)[x1, . . . , xn]∩R[x1, . . . , xn]([9]). (4) I (I, d)6=R, se F←F∪ {d}, and go o s ep 1. O he wise, compu e ˜g1,...,˜gk∈Q(R)[x1, . . . , xn]gene a o s o a maximal ideal M ha con- ains ˜ Iin Q(R)[x1, . . . , xn]. CONSTRUCTIONS IN R[x1,...,xn]. APPLICATIONS TO K-THEORY 3 (5) Le Jbe ideal o R[x1, . . . , xn]such ha Je= M. Compu e he leas common mul iple o he coe icien s o a G ¨obne basis o J. Le p∈Rbe a p ime elemen ha does no di ide . Se F←F∪{p}, and go o s ep 1. Example 1.Le I=hxy +1ibe ideal o Z[x, y]. Then I∩Z= 0, so he e exis s d∈ R, d 6= 0 such ha I= (I, d)∩Iec. In his case, d= 1. The ideal M=hx−1, y +1i is a maximal ideal in Q[x, y] ha con ains Ie. Se J1=hx−1, y + 1i ⊂ Z[x, y], and s= 1. Take p= 2 and se I0= (J1,2) ⊃I. Applying he algo i hm o I0, we ge M=h2, x −1, y −1imaximal ideal o Z[x, y] ha con ains I. Le Sbe he se o monic polynomials o R[x], and w i e R0=S−1R[x]. When Ris a ield, he ing R0is he ield o a ional unc ions o e R[x]. Lemma 1. Le Rbe a p incipal ideal domain whe e we can di ide and compu e he g ea es common di iso and I=h , gi ⊂ R0be an ideal o R0. Then i is possible o compu e h, 0, g0∈R0such ha I=hhi, = 0hand g=g0h. P oo . By [11, p. 117], we know ha R0is a p incipal ideal domain. Then I= hh0i, whe e h0is he g ea es common di iso o and gin R0. We can assume , g ∈R[x] aking o denomina o s and compu e h= gcd( , g) in R[x] wi h he pseudo-di ision algo i hm ([4, algo i hms 3.2.10, 3.1.2]). E e y i educible elemen o R[x] is i educible o a uni o R0. Then I=hhi, and by di ision we ob ain 0, g0∈R[x] such ha = 0h, g =g0h. Rema k 1.In [2] i is shown ha i Ris an euclidean domain, hen S−1R[x] is an euclidean domain oo. Howe e , he di ision algo i hm passes h ough a o mal powe se ie. Co olla y 1. Le Rbe an MC-PID. Then R0is an MC-PID. P oo . Gi en I=h 1, . . . , ni ⊂ R0ideal o R0, by i e a i e applica ions o Lemma 1, we compu e ha gene a o o I. I ∈R0, we can check whe he ∈Iby educing o R[x] and making he di ision by h. I ∈I, we ob ain 0∈R0wi h = 0h. The syzygy module o a se 1, . . . , min R0is easily educed o a compu a ion o a syzygy module in R[x]. Le Ibe a p ope ideal o R0. By Lemma 1, we ind a no monic polynomial (x)∈R[x] such ha I=h (x)iR0. We ge 1(x)∈R[x] an i educible and no monic polynomial ha di ides (x) in R[x], by ac o ing in Q(R)[x] and Gauss’s Lemma. Then I⊂ h 1(x)iR0, maximal ideal in R0. P oposi ion 1. Le Rbe an MC- ing. I Mis a maximal ideal o R[x1, . . . , xn] hen R[x1, . . . , xn]Mis an MC- ing. P oo . The cons uc ion o a maximal ideal ha con ains a gi en ideal is i ial, because R[x1, . . . , xn]Mis local. We ha e o check he condi ions abou linea equa ions. No e ha h ough G ¨obne bases in R[x1, . . . , xn] we can check i a polynomial belongs o an ideal I, and i so, exp ess i as linea combina ion o gen- e a o s, and his p ocedu e is alid in R[x1, . . . , xn]M. Le IM=h 1, . . . , mibe an ideal in R[x1, . . . , xn]M, and ∈R[x1, . . . , xn]M. We can suppose , 1, . . . , m∈ R[x1, . . . , xn]. I any iis no in M, hen IM=R[x1, . . . , xn]M, and we a e done. Then assume ha I⊂ M, and ∈ M. We ha e ha ∈IMi and only i he e exis s s /∈ M such ha s· ∈I, i.e., s∈(I: ). We can compu e c1, . . . , cm∈R[x1, . . . , xn] a se o gene a o s o (I: ). I e e y ciis in M hen 4 JES´ US GAGO-VARGAS /∈IM. I , o example, c1/∈ M, hen c1· ∈I, and we can exp ess as a linea combina ion o he gene a o s o IMwi h coe icien s in R[x1, . . . ,xn]M. In a simila way o Co olla y 1, we can ge a se o gene a o s o he module Syz( 1, . . . , m) wi 1, . . . , m∈R[x1, . . . , xn]M. Rema k 2.I Rhas e ec i e cose ep esen a i es hen, o a gi en IMp ope ideal o R[x1, . . . , xn]M, we can compu e he cose s o R[x1, . . . , xn]/I and he same se is alid o R[x1, . . . , xn]M/IM. 3. QS-algo i hms in R[x1, . . . , xn] Le Rbe an MC-PID, = ( 1, . . . , m) a unimodula ow in R[x1, . . . , xn]m and P= ke ( ). Then Pis a p ojec i e module, and we wan o ge a ee basis o i . The p ocess desc ibed in [7] uses he p ima y decomposi ion o an ideal o R[x1, . . . , xn]. To a oid i , we gi e wo new QS-algo i hms. The p ocedu es a e by induc ion on n, he numbe o a iables. I n= 0 we ha e a p ojec i e module o e an MC-PID, and we can compu e he Smi h no mal o m. Assume ha n≥0 and ha we ha e an algo i hm o ings o polynomials wi h n a iables and coe icien s in an MC-PID. Now conside he polynomial ing R[x1, . . . , xn][y] in n+ 1 a iables. The i s s ep is educing he p oblem o ind a ee basis o he modules PMo e he ings R[x1, . . . , xn]M[y] o a ini e se o maximal ideals Mo R[x1, . . . , xn]. He e we need Algo i hm 1 o compu e a maximal ideal ha con ains an ideal in R[x1, . . . , xn]. These ee bases a e pa ched oge he o ob ain a basis o he module P, as shown in [16], so he p oblem is educed o gi e an algo i hmic p oo o Ho ocks’ heo em ([17, p. 28]). 3.1. Fi s QS-algo i hm in R[x1, . . . , xn]. Theo em 1. Le Pbe a p ojec i e module o e R[x1, . . . , xn][y], de ined as he ke - nel o a unimodula ow = ( 1, . . . , m), and Ma maximal ideal o R[x1, . . . , xn]. Then he e exis s a m×m-in e ible ma ix Uwi h en ies in R[x1, . . . , xn]M[y] such ha ·U= (1,0,...,0). The las m−1columns o U o m a ee basis o PM. P oo . W i e A=R[x1, . . . , xn]M[y]. Le Sbe he mul iplica i e se o monic polynomials o A, and S0⊂R[y] he se o monic polynomials. As is a unimodula ow, we can compu e a column gsuch ha ·g= 1, and M=I−g· is a ma ix whose columns o m a se o gene a o s o he R[x1, . . . , xn][y]-module Syz( ). F om he commu a i e diag am R[x1, . . . , xn][y]→(S−1 0R[y])[x1, . . . , xn] ↓ ↓ R[x1, . . . , xn]M[y]−→ AS we see ha he module S−1PMis ex ended om S−1 0P. By Co olla y 1, S−1 0R[y] is an MC-PID, and by he induc ion hypo hesis and ex ension we compu e a ma ix US∈GL(m, S−1R[y]) such ha ·US= (1,0,...,0). Le 1, . . . , m−1∈PM be he las m−1 columns o US. These ec o s o m a ee basis o S−1PMin AS. Le k=R[x1, . . . , xn]/M, A =A/MA=k[y] and AS=k(y). Compu e a ma ix U∈GL(m, k[y]) such ha ·U= (1,0,...,0) and le e1, . . . , em−1be he las m−1 columns o U. This se is a ee basis o PM. Take a1, . . . , am−1∈A such ha ai=ei, i = 1, . . . , m −1. Then ei=ai−g· ·ai, i = 1, . . . , m −1, CONSTRUCTIONS IN R[x1,...,xn]. APPLICATIONS TO K-THEORY 5 a e elemen s o PM ha go o e e1, . . . , em−1. By sol ing a linea sys em, we ge W∈GL((m−1), k(y)) such ha ( 1, . . . , m−1)W= (e1, . . . , em−1) because 1, . . . , m−1and e1, . . . , em−1a e bases o he ec o space PSo e he ield k(y). As poin ed in [3, 14], we can ake W∈GL(m−1, AS) ha li s o W. Change he basis 1, . . . , m−1o S−1PMby he basis ( 1, . . . , m−1)·W. Then ei= i+hi, hi∈ MS−1PM, i = 1, . . . , m −1. Following [12, 3], i Cis he sub ing o S−1R[y] o med by /g, wi h g∈Sand deg( )≤deg(g), hen MS−1PM=MPM+MQ, whe e Q=L iy−1C. By he di ision algo i hm, decompose hi=gi+g0 i, whe e gi∈Am, and he deg ee o he denomina o s o g0 ia e g ea e han he deg ee o nume a o s. Compu e zi he no mal o m o giwi h espec he module MPMo e he ing A. Then, by [12, 3], he elemen s 0 i= i+zi+g0 i, i = 1, . . . ,m −1 o m a basis o PM. Rema k 3.The algo i hm desc ibed in [14, algo i hm 4] is incomple e, because o ex ac he componen in MPMwe need no mal o ms, and no only quo ien s. An analogous ema k is applied o [14, p. 418]. Example 2.Conside he polynomial ing Z[x], he unimodula ow = (13, x2− 1,2x−3) and P he p ojec i e module de ined by ke ( ). We can compu e g= (2,−20,10x+ 15) wi h ·g= 1. A basis o S−1Po e S−1Z[x] is o med by he ec o s 1=1,−13 x2−1,0 , 2=0,−2x−3 x2−1,1 . Fo e e y maximal ideal Min Z, a basis o he module S−1PMis ob ained by ex ension. Le M=h2imaximal ideal o Z, and A= (Z/M)[x]. By Euclidean algo i hm in A, we ge a basis o PMwi h elemen s e1=−x2+ 1,1,0 , e2= (1,0,1) . Then W=−x2+ 1 1 0 1 ∈GL(2, AS) is a ma ix wi h ( 1| 2)W= (e1|e2). Li o W=−x2+ 1 1 0 1 ∈GL(2, AS) and a new basis o S−1PMis 1=−x2+ 1,13,0 , 2=1,−2(5 + x) x2−1,1 . We can compu e elemen s e1=e1−g· ·e1, e2=e2−g· ·e2∈PMsuch ha hey apply o e e1, e2. Le h1=e1− 1=g1+g0 1, h2=e2− 2=g2+g0 2, whe e g1=h1, g0 1= 0, and g2=−20 −4x, 200 + 40x, −150 −130x−20x2 , g0 2=0,2x+5 x2−1,0 . The espec i e no mal o ms o g1, g2wi h espec o MPMa e z1= (0,0,0) , z2=−4x2+x3+ 4x−1 x3−2x2−1,−(x−1)x2(x2−3x+ 1) x3−2x2−1,0 . 6 JES´ US GAGO-VARGAS Then 0 1= 1, 0 2=0,2x−3,−x2+ 1 o m a ee basis o PM. I U= (g| 0 1| 0 2), hen de (U) = −13 is a uni in ZM. To ob ain a ma ix wi h de e minan 1, we conside U1=g|− 1 13 0 1| 0 2. Le 1= 13. We epea he p ocess o M2=h13i, and ob ain he ma ix U2=  2 1 0 −20 −52 52x−3 10x+ 15 26 5x+39 5−x2+ 1  . In his case, 2= 5, and h 1, 2i=Z. By pa ching oge he he solu ions as desc ibed in [16], we ge V= −128x2+ 60x3+ 60x1 + 1144x2−780x3−144x2+ 100x3−4x −1−30x13 + 390x−50x−3 270x−375x2−130x+ 4875x21−625x2  wi h de (V) = 1 and ·V= (1 0 0). 3.2. Second QS-algo i hm in R[x1, . . . , xn].The algo i hm desc ibed in he p e- ious sec ion uses he no mal o m o a ec o wi h espec o a module. We gi e ano he me hod, based on [19, 17], whe e is no needed. We begin wi h an easy lemma. Lemma 2. ([17, Lemma 3.2.5].) Le Rbe an MC-PID and Ma ee R-module. Le be a nonze o elemen o M. Then Mhas a basis 1, . . . , such ha =α 1 o some α∈R. The ollowing algo i hm sol es he local s ep, i.e., compu e a basis o he R[x1, . . . , xn]M- module PM. Ou s a ing poin is a se o gene a o s o PMas a submodule o a ee module, and p oceed by induc ion o e ank(P) = m. We build a se o gene - a o s o a p ojec i e module P0wi h ank m−1. Remembe ha i P0is p ojec i e hen i is o sion ee, so i is isomo phic o a submodule o a ee module o ini e ank ([10, P op. 10.11]). This isomo phism can be compu ed, because he ela ions be ween he gene a o s o P0can be ound by sol ing a linea sys em in he ield Q(R). Then we apply he induc ion hypo hesis. Theo em 2. ([17, Thm. 3.2.1] Le Pbe a p ojec i e R[x1, . . . , xn][y]-module, gen- e a ed by a se o ec o s o R[x1, . . . , xn][y]s, and Ma maximal ideal o R[x1, . . . , xn]. Then we can ind a ee basis o PM. P oo . Le Mbe a ma ix whose columns a e he gene a o s o P, and m= ank(P). (1) I m= 1 hen Pis isomo phic o an ideal o R[x1, . . . , xn][y]. Then S−1 0P is a p ojec i e ideal o (S−1 0R[y])[x1, . . . , xn], so i is ee, hence p incipal. Using a G ¨obne basis we can ind i s gene a o , ha is a basis. (2) I m≥2, le 1, . . . , ma basis o S−1 0PM, ha we can compu e because S−1 0(R[x1, . . . , xn][y]) = (S−1 0R[y])[x1, . . . , xn]. Choose i∈PM aking o denomina o s. (3) Le e1, . . . , embe a basis o PMo e k[y], wi h k=R[x1, . . . , xn]/M. (4) Compu e a basis q1, . . . , qmo PMwi h 1=αq2(Lemma 2). Le Vbe a change basis ma ix and Va li ing wi h en ies in A. (5) Li q1 o PM h ough M·V. (6) By sol ing a linea sys em, le q1=Pm i=1 a0 i iin S−1A, so we can ind s∈Asuch ha sq1=Pm i=1 ai i, ai∈A. CONSTRUCTIONS IN R[x1,...,xn]. APPLICATIONS TO K-THEORY 7 (7) Take ksuch ha a1+sykis a monic polynomial in he a iable y. (8) Le p=q1+yk 1, and P0=P/pA. Then P0is p ojec i e and ank(P0) = m−1 ([17, 19]), so is o sion ee, and we can compu e a se o gene a o s. Se P←P0, and go o s ep 1.  Example 3.Conside Example 2, and le M=h2i ⊂ Z. We wan o compu e a ee basis o he A=ZM[x]-module PM. A se o gene a o s o Pis o med by he columns s1, s2, s3o M=I−g· . I is easy o see ha ank(P) = 2. As S−1 0Z[x] is an MC-PID, we can ind he Smi h no mal o m o he module S−1 0PM. Then   10 1 0 10x+ 15 0 −2 13 x2−1 2x−3  M  0 1 2 1−10 −20 0 5x+ 7 10x+ 15  =V1MV2=  100 010 000  . The nonze o columns { 1, 2}o M·V2 o m a basis o S−1 0P. Now i is easy o see ha he ec o s e1= (−1,0,−1) , e2=0,−1,−x2+ 1 a e a basis o he module PM. As 1=e2, we ake q1=e1, q2=e2, and q1= (−25,260,−130x−195) ∈ PMgoes o e q1. Then sq1=a1 1+a2 2wi h a1= 10, s = 1 and a1+sx is monic in x, so p=q1+x 1= −25 −2x3+ 2x, 260 −19x+ 20x3,−115x−195 −10x4+ 10x2−15x3 . We know ha P0=PM/pA is a p ojec i e A-module wi h ank equal o 1. Now we ha e o compu e a ee basis w+hpio P0, which is gene a ed by s1+hpi, s2+ hpi, s3+hpi. The i s s ep is o ind d2, d3∈Asuch ha d2(s2+hpi) = λ2(s1+ hpi), d3(s3+hpi) = λ3(s1+hpi),so we sol e he sys em s2|s3=p|s1a11 a12 a21 a22  in he ield o ac ions o A. Le d= 5x(2x+3), λ2=−5(2x+3), λ3=−2(10 +x), and conside he mo phism be ween A-modules ϕ:P0→(s1+hpi)Ade ined by ϕ( ) = d· . Then ϕis injec i e, and P0≃ϕ(P0)⊂(s1+hpi)A. Since ϕ(P0) is gene a ed by only one elemen , i mus be a mul iple o s1+hpi. Then conside he ideal J=hd, λ2, λ3iA. By compu ing a G ¨obne basis in Awe ob ain u= 85 = 0·d+λ2−5λ3, a uni in A, so P0is gene a ed by ϕ−1(s1+hpi) = u−1(s2−5s3)+hpi. Le w=u−1(s2−5s3) = 1 85 −2x2+ 20x−28,20x2−200x+ 281,−10x3+ 85x2+ 10x−215 . Then {p, w}is a ee basis o PM. Rema k 4.These algo i hms allow us o ex end he esul s in [14] o ind bases o p ojec i e modules o e a monoid ing R[M], because all we need a e he con- s uc ions in S−1R[x] desc ibed in Sec ion 2 and he Quillen-Suslin algo i hm in R[x1, . . . , xn] ([8]). In he same way, we ha e a QS-algo i hm o quo ien s o he o m R[x1, . . . , xn]/I, wi h Ia monomial ideal, ex ending [13]. 8 JES´ US GAGO-VARGAS 4. QS-algo i hm in D[x] 4.1. Ideal ac o iza ion in a Dedekind domain. Le Dbe he ing o in ege s o a numbe ield, and Ian ideal o D. Then Dis a Dedekind domain, and he e is an algo i hm ([5, algo i hm 2.3.22]) o compu e he ac o iza ion o Ias p oduc o p ime ideals o D. We p esen he e ano he algo i hm based in G ¨obne bases. We know ha Dis a ee Z-module o ini e ank, and we can ind ω0= 1, ω1, . . . , ωn a ee basis ([4, algo i hm 6.1.8]). Then ωiωj=Pn k=0 ai,j,kωk, i, j ∈ {0,1, . . . , n} o some ai,j,k ∈Z. Le sij =xixj−Pn k=0 ai,j,kxkbe polynomials in Z[x1, . . . , xn], and call J he ideal gene a ed by hem. Lemma 3. (1) D≃Z[x1, . . . , xn]/J. (2) The e is a p imali y es ing algo i hm o ideals o D. (3) Le Ibe a p ope ideal o D. Then he e exis s an algo i hm o ind a se o gene a o s o a maximal ideal Mo D ha con ains I. (4) Le Mbe a maximal ideal o D. Then i is possible o compu e a se o gene a o s o he D-module M−1. P oo . (1) Le p∈Z[x1, . . . , xn] be a polynomial such ha p(ω1, . . . , ωn)=0. By educing pby he polynomials sij , we ha e ha p≡q(mod J), whe e q(x1, . . . , xn) = a0+a1x1+. . . +anxn, ai∈Z. Since p(ω1, . . . , ωn) = 0, hen q(ω1, . . . , ωn) = 0, so a0=a1=. . . =an= 0, because o linea independence o ωiin Z, and hen p∈J. I Iis an ideal o D, we no e ˜ I he li ed ideal o Z[x1, . . . , xn]. (2) Iis a p ime ideal o Di and only i ˜ Iis a p ime ideal o Z[x1, . . . , xn], and by [9, p op. 4.3] we ha e an algo i hm o es he p imali y o ˜ I. (3) Apply Algo i hm 1 o ˜ I. (4) Follow [4, p. 199]. Obse e ha we can always ind p∈Z∩ M a p ime elemen h ough M∩Z.  P oposi ion 2. Le Ibe a p ope ideal o D. Then we can ind p ime ideals p1,...,p o Dsuch ha I=p1·. . . ·p . P oo . I Iis p ime, we a e done. O he wise, le p1be a maximal ideal ha con ains I. Le I1=p−1 1I. Then I1is an in ege ideal and I(I1([18]). We apply again he p ocess o he ideal I1, and we ob ain an ascending chain o ideals I⊂I1⊂. . . ⊂I ha becomes s a iona y because Dis a noe he ian ing. I I =I +1, we know ha I +1 =p−1I , whe e pis a maximal ideal o D ha con ains I . Then I =p−1I and his would imply ha p=D. So I is a maximal ideal, he algo i hm s ops and we ob ain he exp ession I=p1·. . . ·p . Example 4.Le I=h6ibe ideal o D=Z[ω], wi h ω=√−5. An in eg al basis o Dis {1, ω}. Now conside ˜ I=h6, 2+ 5iideal o Z[ ]. Then ˜ Iis no p ime, because h6iZ=˜ I∩Zis no a p ime ideal o Z. Le p1= 2 be a p ime numbe ha di ides 6, and conside he ideal ˜ I0=hp1, 2+ 5i, ha con ains ˜ I. We compu e M1=h + 1i, a maximal ideal o (Z/p1)[ ] ha con ains he polynomial 2+¯ 5. Then p1=h2,1 + ωiis a maximal ideal ha con ains Iand p−1 1=D+1+ω 2D. Hence we ob ain I1=p−1 1I=h6,3+3ωi. Again, I1is no a p ime ideal, so we apply he p ocess o i . I is con ained in he maximal ideal p2=h2,1 + ωi, so we de ine I2=p−1 2I1=h3i. The ideal I2is no CONSTRUCTIONS IN R[x1,...,xn]. APPLICATIONS TO K-THEORY 9 p ime, because 2+¯ 5 is educible in (Z/3)[ ]. A maximal ideal ha con ains I2is p3=h3,1 + ωi, and p−1 3=D+1−ω 3D. Now I3=p−1 3I2=h3,1−ωi, ha i is p ime. Pu ing p4=I3we ge I=p2 1p3p4. 4.2. P ojec i e modules o e D[x].Le Mbe a ini ely gene a ed D-module. Then Mis p ojec i e i and only i Mis o sion ee. In his case, i ank(M) = , hen M≃D −1⊕awhe e ais an ideal o D.Mis ee i and only i ais p incipal ([5, Thm. 1.2.23]). This decomposi ion can be compu ed when Dis he ing o in ege s o a numbe ield ([5, Thm. 1.2.19]), and he c ucial s ep is he ollowing lemma. Lemma 4. I Iand Ja e ac ional ideals o D hen I⊕J≃D⊕IJ as D-modules. A way o ob ain his isomo phism is h ough he p ime decomposi ion o ideals in D(see [5, P op. 1.3.12]) o applying [5, Algo i hm 1.3.16]. Then, i we ha e de e mined he eeness o a o sion ee module Mwe can compu e a basis using his isomo phism. I Mis a maximal ideal in D hen he local ing DMis a disc e e alua ion ing, so a PID. I P(x) is a p ojec i e module o e D[x], hen o each maximal ideal Mo D, he module P(x)Mis p ojec i e o e DM[x], and by he Quillen-Suslin heo em P(x)Mis ee. Then P(x) is ex ended om P(0) ([21]). When Dis he ing o in ege s o a numbe ield, we ha e an algo i hm o he p e ious esul analogous o [14]. This shows us ha o checking he eeness o P(x) o e D[x] is enough o es P(0) o e D. The p oblem is educed o compu e a ee basis o he module P(x)Mo e DM[x] o a maximal ideal Mo D. Bu DMis an MC-PID, and by sec ions 3.1, 3.2 we ha e wo algo i hms o ge a ee basis. Example 5.In D=Z[ω], ω =√−5 conside (x)=( 1(x) 2(x) 3(x)) he unimodula ow in D[x]3whe e 1(x) = −5x2−2ωx + 2x+ω−2, 2(x) = x2−x, 3(x) = ωx −ω+ 1. Le P(x) be he p ojec i e module de ined by ke ( (x)), whose gene a o s in D[x]3 a e gi en by he columns o he ma ix M(x) = I3−g(x) (x), whe e g(x) = (x−1,5x−2,2x−1) .To check he eeness o P(x) we conside he D-module P(0) gene a ed by he columns o M(0). We can see ha P(0) ≃D⊕J, whe e J=h2−ωi. Then P(0) is ee, so P(x). Le M=h2,1 + ωibe maximal ideal o D. Applying Theo em 2 o PMwe ge he ma ix [x−1,−5x4+ 6x3+ωx2−3x2+ωx −ωx4−ω+ 1,−(x−1)(10086662778x +20424937041ω−6861175910ωx −27394274848x2+ 8528154248ωx3 −5214150542ωx2−45243672650x3+ 28030914183) β 5x−2,−25x4−2ωx3+ 14x3+ 2ωx2−12x2+ 5ωx −x−6ωx4−2ω+ 2 , −(144679169033x−61623674056ω+ 99695086617ωx + 120872168654x2 −36583582778ωx3−17235547982ωx2+ 36521518173ωx4−44295714782x3 −240865770565x4−18225840353) β 2x−1,2−4x2−11x4−ωx3+ 7x3+ωx2+ 2ωx −2ωx4−ω+x5, (−65754728708x+ 28500912245ω−41229027211ωx −52828252348x2 −6119253067x5+ 15325343098ωx3+ 13191112346ωx2−18717821941ωx4 +14918939512x3+ 94101342265x4+ 2929481463x5ω+ 15681952346) β, wi h β= (−37835988013 + 20773799974 ω).