Solving Algebraic Equations in Commutative Rings Hideyuki Matsumura Nagoya University

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1 s 0 Itroductio Solvig Algebraic Equatios i Commutative Rigs Hideyuki Matsumura Nagoya Uiversity Let A be a commutative Rig, ad let f(x) = ( f 1,..., f,) be a vector of polyomials i X 1,, X with coefficiets i A: f; (x) E A[x 1,, x] (1 <. i <. r). The problem is to fid solutios of f(x) = 0 i A, i.e. to fid a,,.... a E A such that f 1 (a) =... = fr(a) = 0. Whe A is a algebraically closed field this problem leads to Algebraic Geometry, while whe A = Z this is the problem of Diophatie Equatios. Let..P : A-+ B be a homomorphism of commutative rigs, ad let tf(x) deote the polyomial obtaied by applyig cf> to the coefficiets of (x). The f(x) = 0 has a solutio i A ~ t'(x) = 0 has a solutio i B. This is trivial (but useful to prove the usolvability of f(x) = 0 i A. For istace if A = Z ad B == Z/() the the solvability of fp(x) = 0 i Z/() ca be checked by a fiite umber pf trials, ad if f 10 (x). = o has o solutio the f(x) = 0 has o solutio.) The opposite implicatio is obviously false i geeral, but there are some importat cases i which it is true. The mai purpose of my talk is to explai this aspect of the problem i two cases. The first is the case of (simultaeous)liear equatios, ad the secod is that of the Arti Approximatio Theorems. Our discussio will show, hopefully, the importace of the operatios of Localizatio, Completio ad Heselizatio. I the followig all rigs are assumed to be commutative, to have a uit elemet 1 ad to be differet from {OJ s 1. Liear Equatios. Local Rigs A rig A which has oly oe maximal ideal m is called a local rig; we say that (A, m) is a local rig, meaig that A is a local rig ad m is its maximal ideal. As a example, lf?t 0 be the rig of holomorphic fuctios defied i eighbourhoods of the origi i Cfl, where the domai of defiitio may vary from fuctio to fuctio, ad two fuctios which coicide i a eighbourhood of the origi are cosidered equal. The the set m cosistig of fuctios vaishig at the origi is the oly maximal ideal of 0, because if f E 0, f Et: m the 1 /f E 9. Therefore 0 is a local rig. (This rig ca be idetified with the rig of coverget power series i variables x 1,..., X with coefficiets i C, ad is deoted by C{ x 1,, x } ) We ca costruct may local rigs from a arbitrary rig by the followig method. *Text of a lecture delivered to the Sigapore Mathematical Society o 23 February Professor H. Matsumura is professor of mathematics at Nagoya Uiversity, Japa, sice. 1968, ad has taught at Kyoto Uiversity, Columbia Uiversity, Bradeis Uiversity ad Uiversity of Pesylvaia. He was a VJSitig professor at Muster, Germay, ad Torio, Italy. He has made importat cot"butios to algebraic geometry ad commutative algebra. He is the author of the classic text, "Commutative algebra". 18

$Here, two fractios sads' are defied to be equal iff there is s" E A - P such that s"(s'a -sa') = 0; similarly i Mp.$

2 Localizatio Let A be a rig ad P be a prime ideal (i.e. a ideal such that x, yea- P implies xy $ P). Put ad if M is a A-module put A P = { : I. a E A; s E A - P } Mp =i; / x EM, sea- P}. a a'. Here, two fractios sads' are defied to be equal iff there is s" E A - P such that s"(s'a -sa') = 0; similarly i Mp. The AP is a rig (by the usual formulas of sum ad product of fractios) ad Mp is a AP - module. Moreover, AP is a local rig. I fact, m ={ : I a E P, s E A - P } is a ideal of AP ad the elemets of AP - m are uits (i.e. have their iverses) i AP. Therefore m is the maximal ideal of AP. X There is a atural map M ~ Mp which seds x EM to1 E Mp. We will write X Xp fort. The Xp = 0 ~ 3 SEA- P such that SX = 0 ~ P ~a \X), where a(x) = {a E A 1 ax = o}. We will use the followig two properties of localizatio: (I) Localizatio preserv es exactess. Namely, if... ~L~M~N~... is a exact sequece of A-modules, the the correspodig sequece of AP - modules.... ~ Lp ~ Mp ~ Np ~... is exact. (II) If x E M ad if XP = 0 for all maximal ideals P the x = 0. Proof It x =I= 0 the a(x) :I= A (because 1 x :j: 0), hece there exists a maximal ideal P cotaiig a(x). The Xp =to. THEOREM 1. If a system of liear equatios (*).r. 8;- X.= b., i = 1,... ; m (a;-, bi E A) 1=1 I J l J has a solutio i AP for every maximal ideal P of A, the it has a solutio i A. PROOF. Let f : A ~ Am be the A-liear mappig give by f(x 1,, x) 19

3 (~a 1.x. ~a 2 x.,..., ~a x.), ad let {3 = (b 1,..., b ). Deote the cokerel off J I' I I m1 I m by N. Thus N = Am /f(am ). Let~ be the image of {3 i N. The (*) is solvable i A iff p = 0. (*) is solvable i Ap iff ~P = 0, because oe ca idetify Np with the cokerel of fp : Ap -+ A; iduced by f. Thus the theorem follows from the remarks (II) above. Determiatal Ideals Let M = (u.. ), u.. E A, be a r x s matrix. The rak of M is defied as usual, amely as the siz~ of l~rgest o-vaishig miors. Let t be a iteger betwee 1 ad Mi (r, s). The ideal geerated by the txt miors of M is deoted by lt(m). Cosider the system of liear equatios (*) ad put M = (aii), M' = (M, ~') ~m m x m x ( + 1) If (*) has a solutio i A the the last colum of M' is a liear combiatio of the colums of M. Hece: THEOREM 2. If(*) is solvable is A, the Remark: I geeral this coditio is ot sufficiet. A itegral domai A is called a Dedekid domai if (1) it is oetheria ad (2) Ap is a discrete valuatio rig for every o-zero prime ideal P. (A discrete valuatio rig is a pricipal ideal domai which has a uique maximal ideal (11"), so that the o-zero ideals are (1f' ), = 1, 2,... ) The rig of algebraic itegers i a umber field (i.e. a fiite extesio field of Q) K is a typical example of a Dedekid domai. THEOREM 3 If A is a Dedekid domai, the the system (*) of liear equatios is solvable i A iff (1) rak M = rak M', ad (2) I r (M) = I r (M') where r =rak M. 20

4 Example 3x + 4y + 5z = 2 { x- 2y +z = 2 Therefore it is solvable i Z. A=Z r=2 I (M) = I (M') = 22. r r Proof of Th. 3. By localizatio we may assume that A is a discrete valuatio rig. The by elemetary trasformatios o the variables ad o the equatios we ca brig (*) to the followig form: e. 1 7r Xi=b/, i=1,...,r; where 1r is a prime elemet of A. (This is the so-called elemetary divisor theorem.) e 1 + +e e. The I (M) = 1r r A, ad I (M) =I (M') implies b.' E 1r 1 A as oe ca imr r r 1 mediately check. Whe A = Z this theorem was foud i Sice the elemetary divisor theorem is valid oly i pricipal ideal domais, localizatio (i.e. Th. 1) is essetial i the geeralizatio to Dedekid domais. We also like to poit out that if there is oly oe equatio the Th. 3 is just a tautology. The value of Th. 3 is to reduce the solvability of a system to that of a sigle equatio. The Dedekid domais are, rig-theoretically, characterized as the oe-dimesioal, itegrally closed oetheria domais. For more geeral rigs (e.g. for a polyomial rig k [x, y] over a field k, which is a two-dimesioal itegrally closed oetheria domai) o useful ecessary ad sufficiet coditios seem to be kow. i 2. Arti Approximatio Theorems. From ow o, we will mea by a local rig a oetheria local rig. A local rig (A, m) has the m-adic topology, i which a fudametal system of eighbourhood of a elemet x of A is give by { x + mv lv = 1, 2,... }. A complete local rig is a local rig i which every Cauchy sequece (i11radic topology) coverges. Every local rig has its completio. The rig of p-adic itegers (iveted by HeseJ i the 19th cetury) is the completio of Zpz. The completio of the coverget power series rigc/x 1,, x} is th~ formal power series rig C[x 1,, x ]. The classical lemma of Hesel is stated as follows: Hesel's Lemma. Let A be a complete local rig ad let k = Aim be its residue field. Let f(x) = x +a 1 x -t +.. +a EA[x] ad letf(x) =x +8 1 x a Ek[x] be its image i k [x]. If f(x) has a simple root c ik, the f(x) has a root a E A such that Zi =C. 21

This is a algebraic aalogy of the implicit fuctio theorem i aalysis [if f(x, y) is coo i a eighbourhood of (o, c) ad f(o, c) = 0, a f (o,c) =1= 0, the there ay exists a C 00 fuctio y = y(x) such that

5 This is a algebraic aalogy of the implicit fuctio theorem i aalysis [if f(x, y) is coo i a eighbourhood of (o, c) ad f(o, c) = 0, a f (o,c) =1= 0, the there ay exists a C 00 fuctio y = y(x) such that f(x, y(x)) = 0 ad y(o) = c] ad is easily proved by successive approximatio. A local rig for which the coclusio of Hesel's Lemma holds is called a Heselia local rig. G. Azumaya ad M. Nagata studied the properties of Heselia local rigs. I particular, Nagata proved the existece of Heselizatio for every local rig. If A is the local rig of a polyomial rig k [x 1,., x ] (k = a field) at the maximal ideal (x 1,, x ), the its Hesel izatio is the rig of algebraic power series, i.e. the set of th~se elemets of the completio 1.\ = k[x 1,,x ] which are algebraic over A. M. Arti ad some other algebraists (maily i easter Europe) have show that Heselia rigs have very good properties. The coverget power series rig k{!} = k{x 1,, x} (k =Cor R, say) is a Heselia local rig. M. Arti first published the followig aalytic theorem: Theorem (ivetioes Math. 5, 1968) Letf.(x,y)Ek{x 1,,x ;y 1,y <,1~i~r. 1, m 1 LetY'{x) = ~ (x),... ;Y m (x) ), yi E k [~] 'Yi(O) = 0, be a solutio of f(x, y) = 0, ad let c > 0 be a iteger. The there exist }'(x) = (y 1 (x),..., y (x)), y.(x) E k{x}, such that m 1 - /'. where m = l:: x. k [ x ] I - Accordig to this theorem, to solve the system of aalytic equatios oe has oly to fid a formal solutio. Coverget solutios ca be foud without ay additioal effort. Remark: The coditio lr/0) is superfluous if f;'s are polyomials. Applicatio. Let A= k {x 1,,x} ad 151. = k[x 1,,x, ~ fo (x)e m A. If so(x) is irreducible i A the it is irreducible i A (k = R or C), ad let Proof. Cosider the equatio (l:: x; T;) (k x. U.) =fo(x). i'=1 ]=1 I I 22

This is a polyomial equatio i T 1,., T, U 1,., U, with coefficiets i A. If it has a solutio i~ the it has a solutio i':t A by Artif's theorem. We ow tur to the algebraic versio of the theory.

6 This is a polyomial equatio i T 1,., T, U 1,., U, with coefficiets i A. If it has a solutio i~ the it has a solutio i':t A by Artif's theorem. We ow tur to the algebraic versio of the theory. Let A be a local rig, m its maximal ideal ad~ its completio.cosider the followig prop.erti~s:_awap) Let f = (f 1,..., f ), f. E A[Y] = A[Y 1,.., YN].iff= 0 has a solutio m A, the it has a solutio im A. ' (AP) f beig as before, if Y' = (?\,..., yn ) is a solutio of f = 0 i~ ad if P E N is give, the there exists a solutio y = (y 1,..., y N) i A such that y. =P:. mod. ftic, where 111 is the maximal ideal of~. 1 1 (SAP) For a give f as before, there exists a fuctio v : N -7 N with the followig property: if y = (y 1,,.VN) E AN satisfies f(ji) = 0 mod mvfcj, the there is yean such that f(y) = 0 ad y = y mod me. It is easy to see that (SAP)~ (AP) > (WAP). actually all three properties are equivalet. Greeberg (1967) proved that excellet Heselia discrete valuatio rigs have (SAP). Arti (1969) proved that algebraic power series rigs over a field have (SAP). Pfister-Popescu ( 1975) proved that all complete local rigs have (SAP). It follows from this that (SAP) ad (AP) are equivalet. Later a etirely ew proof was give to their theorem; the ew proof is based o ultraproduct costructio, a favorite tool of logicias. Popescu (to appear) proved that excellet heselia local rigs have (AP). It is easy to see that a local rig with (AP) must be hesel ia. It has bee cojectured that such a rig must also be excellet. I this directio, Rotthaus proved that if A ad A[x] have (AP) the A must be excellet. Applicatio 1. Let A= k[x 1,, x] be a formal power series rig over a field k, ad let..p (x) Em A be irreducible. Cosider the equatio (*) (~Xi Ti) (~Xi U.) =..P (x). i:::-1 j=7 I Sice A has (SAP) by Pfister-Popescu, there is a fuctio u: IN -7IN as i (SAP), for this equatio. Put r = v (1 ). The (*)has o solutio i A/mr (otherwise (*)would have a solutio i A, makig..p (x) reducible). But the image of ( *) i (A/mr) [T, U] is the same if we replace..p (x) by 'It (x) such that 'It =..P mod mr. Therefore, all such 'It (x) E A are irreducible. I short, all formal power series sufficietly close to..pare irreducible. 23

7 Applicatio 2 M. Hochster used Arti's result to prove the existece of so-called 'big Cohe-Macaulay modules" for local rigs which cotai fields. We caot explai the meaig of Hochster's result here, but it is cosidered as oe of the most importat achievemets i Commutative Algebra i recet years. He gives a stadard way of costructig of a ifiitely geerated module, ad shows that the module thus costructed satisfies the requiremet if certai systems of algebraic equatios have o solutio i A. The he proves the usolvability directly i the case of characteristic p. The he reduces the case of characteristic zero to the case of characteristic p by meas of Arti approximatio theorem. 24

62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +

62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + 62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of