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TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 171, September 1972 THE FINITENESS OF / WHEN R[x]/I IS FLAT BY JACK OHM(7 AND DAVID E. RUSH ABSTRACT. Let J? be a commutative ring with identity, let X be an indeterminate, and let / be an ideal of the polynomial ring /?[X]. Let min / denote the set of elements of I of minimal degree and assume henceforth that min / contains a regular element. Then /?[Xj// is a flat /?-module implies 7 is a finitely generated ideal. Under the additional hypothesis that R is quasi-local integrally closed, the stronger conclusion that / is principal holds. (An example shows that the first statement is no longer valid when min / does not contain a regular element.) Let c(7) denote the content ideal of 7, i.e. c(7) is the ideal of R generated by the coefficients of the elements of /. A corollary to the above theorem asserts that 7?[Xj// is a flat R-module if and only if / is an invertible ideal of R[X] and c(/) b R. Moreover, if R is quasi-local integrally closed, then the following are equivalent: (i) /?[X|/7 is a flat Tî-module; (ii) R[X]/I is a torsion free R-module and c(/) = R; (iii) / is principal and c(/) = R. Let zf denote the equivalence class of X in R[X]/I, and let (1, f, •••, £ > denote the Tî-module generated by 1, £, •••, £ . The following statements are also equivalent: (i) (1, <f, ---, <?') is flat forall t > 0; (ii) (1, <f, ••-, <?') is flat for some t > 0 for which 1, £, ,-••, ¿s are linearly dependent over R; (iii) / = (ft,'", fn>, fi e mm /> and c(/) = R; (iv) c(min /) = R. Moreover, if R is integrally closed, these are equivalent to R[X\/1 being a flat 7?-module. A certain symmetry enters in when <f is regular in R[^ \, and in this case (i)—(iv) are also equivalent to the assertion that R[¿¡] and 7?[l/f] are flat Ti-modules. The main results ties of these reader might of this paper sections already benefit by first occur Additional technical difficulties ring and 7 is subject element. §4 condition on min /. In particular, rings then modules that is easily R[X]/I Presented answered arise to the Society, Research supported R is an integral to this The implies class that this as to the class generated R with the property generated is exactly of rings by the editors of is R-flat, for finitely I is finitely R a regular when one removes / of R[X], if R[X]/I question and the to the case min / contain a conjecture March 27, 1971; received Polynomial ideal, finitely domain, case. that happens there The corresponding R-module 3- Many of the difficul- when one proceeds of what we make AMS 1970 subject classifications. (7 and to the restriction as follows: flat Key words and phrases. module, content, invertible the ring that for every ideal generated. is a finite in §§2 his attention to a discussion R with the property / is finitely when confining is an arbitrary is devoted are found May 10,1971. Primary 13A15, 13B25, 13C10; Secondary 13B20. ring, flat module, generated ideal, torsion-free module, principal ideal. projective in part by NSF Grant GP-8972. Copyright © 1972, American 377 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use Mathematical Society 378 JACK OHM AND D. E. RUSH the class cludes of rings section in more than R denotes R[X]. this fact with maximal the integral closure If R C R (RAS, If R and allow When ring with A non-zero-divisor will always \ denotes of R. cp: R —> R with defining us to avoid / an ideal be as- to signify quotient ideal that R is quasi-local ring of R, and that (R')p set-complement.) mention denotes We use is a ring homomorphism, explicit R denotes R is integrally of R, then homomorphism For any subsets emphasize cp. The then A that the element regard R'[X] phism cpx obtained for an ideal also that zf of R as an R[X]-algebra, generates we shall by applying R have R is isomorphic as an R -algebra as if AA = R' = R[<f] Whenever in mind the defining of elements / of R[X], /R'[X] = the ideal of R'[X] generated R'[X] may nota- of R , we define as an R-algebra. cp to the coefficients fot R following of cp and to treat A of R and the ](R) ¡2 cp(a,)a'.\ a. £ A, a'. £ A'\; similarly, A' n R is defined to be 0-1U'). signifies of of the polynomial we sometimes R = R, we say ideals and of R. and is a domain, P is a prime (Here ate rings an overring question. mean commutative homomorphisms the total T(R)). and an R-algebra conventions it were of R (in S = R\P. R of a polynomial work on this zero-divisors. R. R, m is used denotes of the maximal be considered the case Nagata's indeterminate, discussion than 222. T(R) ate rings where the intersection tional X a single D rather ideal briefly Ring in- into the identity. When the ring under by using closed. ring a ring, element. (and hence rings). in particular, a ring without a regular are projective Ring will always will mean to map the identity ring discusses and, and terminology. and domain R-modules and noetherian of the paper a ring will be called sumed flat rings, one variable 0. Notation identity, finite quasi-local The final ring for which domains, [September we homomor- of R[X]. Then by <px(/). Note to R' ®„ R[X] via the map r ® f(X) -* r ■f\X). By "module" called we shall torsion-free if of R. We sometimes etc.) to emphasize generators the rank Let whenever say that mean a unitary module. An R-module rm = 0, r £ R, m / 0 £ M, then that M is R-torsion-free we are regarding of M, we write M= M as an (222,,•••, of M. If S is a multiplicative Rç-module clear) always r is a zero-divisor (or R-flat, R-projective, R-module. If 222., • • • , 222 are 222). When system (m.s.) M is R-free, of R, then R^ ®R M; when M is an ideal of R, we shall identify H be a nonempty denotes More generally, the ideal subset of R[Xl. of R generated if U is a subring generated by the coefficients a 1X + ... + a n X" with c „(H) of R, c^H) a ft 4 0, then (or merely by the coefficients of the elements denotes of H. Let a fa is called M is rkR M denotes zM„ denotes the if the JJ-submodule / /= 0 £ R{X]. ° the M~ with MRS. of the elements the leading License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use c(H) R-free, coefficient R is of H. of R If / = aQ + of '/, and 19721 THE FINITENESSOF / WHEN R[x]/I IS FLAT we write deg,, subscript j = n. When there X. Similarly, More generally, is no possibility of confusion, if an = a. = -..=a._. if H is a nonempty, = 0 and nonzero 379 subset we shall a ./ omit the 0 we write subd /= i. of R[X], we define deg 7/ = min {deg f\ f / 0 £ H\ and subd H = min [subd f\ f / 0 £ H\. For any ideal / of R[X], we define min / to be {f / 0 £ l\ deg / < deg g for all g ^ 0 e /!. Note that min / U [Oi is then an ß-module where coefficient a . = leading We shall call an ideal and that, for any f., A of R locally trivial ly principal P (resp., locally of R[X], IR[X]pi Our primary lar, C denotes it is clear 1.1 Content. and content of R[X] finitely P of R, local- if for every prime generated). will be Zariski-Samuel containment. sense; ideal Also, [ZS]. (A : B)r the subscript C will In particu- denotes be dropped and flatness. If l = 1,a.b.£L, and when has makes on content c ¡¡(I), is defined (/-module; < strict this at primes (resp., prime / of R[X] will be called Let L be a free R-module with basis of R. denoted = a f., from the context. 1. Preliminaries a subring generated) for terminology containment {c £ C\ cB C A !, whenever when finitely is principal reference £ min /, ö./, if for every either AR p = 0 or AR p = Rp. Similarly, an ideal ideal /, /.. a . £ R, the (/-content (/-module X a . U. c to be the U = R, c the following (I) is independent B = {b \. ., and let U he of 7 with respect (I) is a finitely of the choice to B, generated of basis B. The properties: (a) For any I £ L and any ideal A of fi, I £ AL «$^ cy(0 c A. (b) c u(rl) = 7 - cAl) for r £ R, I £ L. If H is any nonempty tained subset in R and generated indeterminates, this basis. a free R-module (c) Content [No]. large With one exception, content formula content formula We also R that For any in the proof that is that some is an ß-algebra and con- then of power-products function defined multiplicative with C of the respect to property: /, g £ C, c^/g^^g)" of 2.17, our further an element need (/-module = cö(f)c v(g)n + l 72. confine r / 0 £ R such to be the of indeterminates), consisting a (/-content has the following formula R = (/; so we shall R (in any number with basis C has function for all sufficiently exists ring over and hence This c ..(H) by {c (h)\ h £ H\. If C is a polynomial is, in particular, of L, we define deal to this g £ C is a zero-divisor r • c cR(gh) we only discussion (g) = 0. Another = cR(h) properties L is a free if c 2?-module the case that It follows of C if and only immediate from the if there consequence of the (g) = R. of the content R' ®„ L is a free -R'-module with basis with case. formula under localization. with basis {b.\. ., then L = il ® è.}.£. [L, p. 418, Proposition License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use If 8]. 380 JACK OHMAND D. E. RUSH For any' I £ L, I = 2 r zb z'., r.efi; z c Al)R = cd'(1 submodule cular, and hence ' ® /). More generally, of L' generated [September 1 ® / = 2 r z.(1 ® b z.). Therefore, if K is a submodule of L and K is the by \l ® l\ I £ K\, then c AK) • R = cR'(K if S is a multiplicative system of R, then cR(K)Rc = c „ (KA ). In parti(= c AK)<. by notation). We also Let need, in the proofs G be a multiplicative of 1.5, system 1.6, and 2.17, of the polynomial the following ring observation. C with the property that g £ G =^-cR(g) = R, and suppose in CQ, f /g x = (h/g 2)(l/g ?), f, h, I £ C, g{ £ G. Then cR(f) = cR(hl); and if cR(h) = R, then cR(f) = cR(l). For, there exist g, g £ G such formula. free that (The R-module initions. gf = g hi, and then awkwardness here and hence does One can get around not have such free R-algebra (b), (c) above. [OR] for a detailed 1.2 Flatness R-modules with criteria. Let L free. Then a content problems of a not necessarily See the asssertion follows is due to the fact that which function by working has study Cr in the sense function satisfying setting (a), matters.) 0 —►K —> L —» M —> 0 be an exact the following a of our def- in the more general a content of such from the content is not necessarily sequence of are equivalent: (a) M is flat. (b) For every ideal A of R and every r £ R, (A : r)M Z) AzM : r (where AM : r = \m £ M \ rm £ AM\). (c) For every finitely (d) For every generated ideal A of R, K DAL C AK. x £ K, x £ c(x)K. (a) <=£► (c) follows from [B(a), p. 33, Corollary], (a) <#> (b) from [B(a), p. 65, Exercise (a) «#$►(d) is easy [B(a), 22]; and p. 65, Exercise 23]. Our basic reference for flat- ness will be [B(a)]. We shall an exact then also sequence 0 —» R R'-modules Corollary use the following of R-modules, ®R K —> R' [B(a), p. 30, Proposition 1.3 Corollary. Let By localizing local with maximal ideal following and let ®„ L —» R' R 0 —> K —>L —►/Vl—>0 be be an R-algebra. If M is M —►0 is an exact sequence 4] and R' ®r M is R'-flat 0 —>K —>L —»M —►0 be an exact R-flat, of [B(a), p. 34, sequence of R-modules Then c(K) is locally trivial. at P, it suffices to prove the corollary P. If c(K) / R, then K C PL; hence when R is quasi- c(K) = 0 by the lemma: 1.4 Lemma [B(a), p. 66, Exercise Jacobson ®r Let 2]. with L free and M flat. Proof. observations: radical 23-d]. Under the hypothesis of R and K C JL, then K = 0. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use of 1.3, if ] = 19721 THE FINITENESS OF / WHEN R[X]/l IS FLAT Proof. For any flatness. x £ K, x £ c(x)L Therefore x £ c(x)JL, c(x) = 0, which implies Note assertion c(Kp), that when that either if Kp/0 Let Proof. the conclusion of C. l] ring over ) and assume (c(x) ■ I) R is quasi-local If c(I) = 0, then to prove s £ C\ M'. C\m' Therefore with maximal formula, by the content exist formula that l)Mt. The following Theorem result finitely primes finitely of R[X] Proof. Let we may assume (R/P)[X] Then c(x) = c(g). by first of R[X] such Thus x/l M, we may Therefore that with maximal R = c(l) C c(f) c(l)M = at this. is immediate. £ C\ M' and is principal localizing M in proving assume = (f/l)(g/s), g £ C, s .x = fgs 2. But ideal implies = (f/l)(g/s) with Corollary Let I bean c(l) If /,,' £ (c(x) ■ I)M<. Note that = (fg/s) £ ideal of R[X] suchthat at primes is locally a converse ring in one variable and ideal to / is of C. of R[X],(2) then R[X]/I is R-flat. I is locally principal If I at trivial. R is quasi-local is a principal by = R. It follows P' be a prime of R[X], and let P = P' Cl R. By localizing that s . £ M. Therefore c(f) 1.3 will provide C is a polynomial at primes generated and of indeter- at primes Q.E.D. that generated 1.6 Proposition. is locally s ., s together 1.5 in the case locally ideal R is quasi-local c(x) = c(fg). (c(g)- DM- =(c(x)- principal for some / £ I. For any x £ I, (x/l) there =^- c(s .) = R since the content R (in any number the following: ■ I ) i; and thus, / = 0 and the assertion c(l) = R. IM> = (//1)CM- = M' of C, and let M = M Cl R. By 1.2(d) it suffices < = (c(x) c(K)p is a flat R-module. and e(l)M = 0 or RM, then for any x £ I, x/1 c(l to the R, since c(K) = R. If I is locally then C/l Fix a maximal ideal p. 112, Corollary from the lemma, of 1.3 is equivalent for arbitrary P of R, then C be a polynomial I be an ideal trivial, following By Nakayama's Q.E.D. for every prime and let the equality c(x) = c(x)j. K = 0 or c(K) = R. Also, and c(l) is locally [B(a), x = 0. R is a domain 1.5 Theorem. minates), Cl K = c(x)K, and hence 381 domain; with maximal and hence ideal there P. exists at P, R[X]/PR[X] = / £ I such that / = fR[X] + (PR[X] n /). But R[X]/l is K-flat implies PR[X] n / = PR[X] ■I. Therefore I pi =fR[X]pi + PR[X]pi • I pi. Since by Nakayama's lemma that /„' = fR[X]pi. then c(l) = R. Q.E.D. /„' is finitely generated, Moreover, by Corollary we conclude 1.3, if I/O, (2) Heinzer and Ohm, The finiteness of I when R[x]/¡ is R-flat. II, Proc. Amer. Math. Soc. (to appear), have recently proved that R[x]/l is /î-flat implies / is locally finitely generated at primes of ä[x1. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 382 JACK OHM AND D. E. RUSH 1.7 zero The condition ideal degree. leading dition is we have min / contains and let proof R{X]/I follows. First closed with prove assumption That 2.13, There (ii) that R{X]/I these is that R is inte- closed c(l) = R, rather R-flat. This assump- the quasi-local R uses than is explained are equivalent statements be discussed thoroughly lemma. Let I be an ideal of R[X]. exists There will the following / = /R[X], (iii) that asserts field an integrally free and as integrally residue next remove The proceeds the assumption concerns need Lemma. (i) f £ min / such the leading exists the by for an inte- that coefficient g £ min / such Then c(f) the following For the are equivalent: = R. of f is regular that in §3- c(g) = aR, in R, and a regular c(l) = R. in R, and R{X]/l R-torsion-free. Moreover, c(g) hypothesis 2.7); R-torsion element. ideal R is quasi-local the infinite remove which is for / to be principal we only 2.1 R{X]/I a regular indeterminate, R. Conditions present, (Theorem and finally, c(l) = R. generated that then remove of the proof that which grally closed in the case if the so if in ad- X be a single min / contains I is a finitely field; 2.16); part stronger Corollary that an indeterminate the hypothesis apparently such let = 0 for of / is regular. trivial; that for a non- if and only coefficient is locally it follows R be a ring, implies residue (Corollary closed. Let that of / of least / is regular c(I) then Recall elements if and only if rc(f) if and only if some in 1.3 that the theorem infinite is regular element, of R[X] R-flat element. if / £ min /, then seen generated. is tion by adjoining grally / £ R[X] a regular / be an ideal that a regular the set of nonzero of / is regular R-flat, 2. I is finitely is formula, r = 0. In particular, coefficient If R{x]/I min / contain min / denotes By the content r £ R implies only that / of R[X], [September when is a principal Proof, these d then R{X]/l is R-flat and, for any g £ min /, ideal. (i) =#►(ii): of /. Then hold, is regular Let / be as in (i), and let in R. For d be the leading bd = 0 for some b £ R implies coefficient bf = 0 or deg bf < deg /. Since bf £ I and / 6 min /, it follows that bf = 0. But c(f) = R, so b = 0. Now let g £ L By the division algorithm tot some integer c(h) = dnc(g) 7z and some by the content h £ R[X]. formula, dng and hence in R[Xl, so then g £ c(g)fR[X]. (ii) =^ (iii): R{X]/I is free [B(a), p. 29, Proposition R-flat [ZS(a), p. 30, Theorem 9], d"g = hf Therefore, dng £ c(h) £ d"c(g)fR{X]. Therefore by Theorem But License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use and since d is regular in R / = /R[X]. 1.5 and hence 3l. ■ /R[X]; is also R-torsion- 19721 free THE FINITENESS OF / WHEN R[X]/I IS FLAT (iii) =^ and g £ I implies (i): c(g) = aR implies c(g) = aR implies c(f) = R. We have seen R-flat. (ii) since already also g = af fot some / £ I. Therefore, implies / £ R[Xl. g £ min / implies in the course of the proof for any g £ min /, g = af, 383 R[X]/I is R-torsion- / £ min /. Also, that (ii) implies a £ R; and then ac(f) R[X]/I = is c(g) = aR c(f) = R. 2.2 cipal) Remark. If D is a Bezout domain (i.e. and D[Xl/7 is D-flat, then 2.1(iii) principal. This discussed in §5. then D[X]/I Theorem observation is D-flat 2.15. includes The case domain / is finitely / need generated case are prin- of Nagata's D is disposed generated, not be principal ideals and hence, by 2.1(ii), the one variable of a Prüfer implies However, finitely is satisfied, which in this / is theorem of almost as easily; will follow from our case, as can be seen from Example 2.14(3). We impose regular element. is extended that on / in much of what T(R) 2.3 This condition to a torsion-free denotes Let insures overring the total Lemma. follows the condition that that deg min / does not decrease of R, as the following quotient lemma a when shows. / Recall ring of R. I be an ideal of R[X], let a regular R' be a torsion-free and let I = IR [X]. If min / contains regular element deg min / = deg min / . // R C R' C T(R) and min / contain element, then R-algebra, min / contains and there a exists f £ min / such that c (f)R' = R', then f is regular in R[X] and /' = fR'[X]. Proof. the leading For the first coefficient ficient algorithm = hg fot some of g is regular the image h £ R [X]. in R and c „(f) contains cients of / is zero assertion, a regular element. Note contain by Lemma 2.1. that it can easily a regular element. contains a nonnilpotent putation shows is is R-torsion-free. 2.4 Lemma. R of min /. Then r. £ R , 9], ang . £ gR[X] > deg g since It then C T(R) that / is regular for some the leading also follows coefthat / £ min /'. and c Af) the annihilator in R[X]. But c „'(f) • R = R implies of the coeffi- Therefore, = c „(f) deg / = • R' = R', so Q.E.D. happen that R[Xl// For an example But then element of min /'. that It follows zero-divisor / / R[X]. deg g R-torsion-free. element that deg min / = deg min /', and hence p. 30, Theorem Thus R observe and hence g is a regular in R. If g '£ I , g =%r.g., [ZS(a), of g in R [X] is a regular For the second /' =/R'[X] suppose a of g is regular g . £ I. By the division n; so a"g assertion, By Theorem (with a, and let 1.5, is and yet / / 0), let / = (1 + aX). R[X]// / Cl R = min / U [0| R-flat is R-flat consists Let I be an ideal of R[Xl, and let R min / does R be any ring which Then a simple and hence com- a fortiori only of zero-divisors. be a ring such that License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use not 384 JACK OHMAND D. E. RUSH fiCfi'C [September T(R). If R[X]/I is R-torsion-free, then IR'{X] n R[X] = /. Proof. Let '/ £ IR'[X] n R[X]. Then '/= 2 (a iAs i" )f.i with 'iii f. £ I, a., s . £ R,' and s . regular But since for every R{X]/l 2.5 is Lemma. i. Then sf = 2 a. f. £ I tot some regular R-torsion-free, Let then b £ R, and let element s of R. / £ I. cp denote the R-automorphism of R[X] defined by <p(X)= X + b. Then c(f) = c(rp(/)) for each f £ R[Xl. Proof. Let anX" + • • • + aQ = f(X) £ R[X]. Then cpifiX)) = aniX + b)n + an_xiX = a\Y\ + b)n~l )xn-ib')+a + ia n b" + a n— 1 h"-1 + ■■■+ aQ ,(y[ )X"-1~i*!l + ■■■+aA.u 2.6 Lemma. Lez: R' be an R-algebra, = h(0) = 1. // the coefficients g and h are integral over + ■■■+aAX + b) + an and let g, h £ R'[X] be such that g(0) of gh are integral over R, then the coefficients of R. Proof. Let g(X) = a Xn + • • • + a XX+ 1, and let h(X) = bjA71 + ■■■ + b XX+ 1. Let g = a n + a n- ,X + • • • + a ,Xn~ ' + X" and similarly J h = b mm—1 + b ° 1 1 b ,Xm~ cients + Xm. integral over The proof that Then of g h ate g h integral has over the same R. coefficients But then R by [B(c), p. 17, Proposition of the following R/222 is finite requires 2.7 Let theorem Lemmas 2.9 as 2.10 of g the coeffi- and h ate Q.E.D. will be broken and gh, and hence the coefficients ll]. .X + • • • + into two parts. and will be given The case after Lemma 2.10. Theorem. that fiCfi'C such that T(R) and R is integrally R[Xl/7 g £ min /, then Proof R, m be a quasi-local is R-torsion-free I is principal (in the case that and ring, and let R be a ring such closed in R'. If I is an ideal of R[X] c(l) = R, and if c Ag)R' and is generated R/222 is infinite). by a regular Let element / e / be such License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use = R' for some that of min /. c(f) = R. 1972] THE FINITENESS OF / WHEN R[X]/I IS FLAT Since R/ttz /(a) there is a unit of R. X + a. Then same further some a £ R such cp he the h and for each exists h £ R[X] that assume = 1. By Lemma /(0) (g/g(0)) • (g(0) over and hence Thus, g/g(0) f(a) cp(f)(0) 2.3, /R'[Xl by Lemma = /(a), and the we may, after replacing = gR [X]. / by /(0), in R', c „(g/g(0)) / = gh tot we have / = of g/g(0) it follows being we may In particular, in R'[X] 2.6, the coefficients closed by cp(X) = 2.5) dividing £ R[X] Cl IR [X] = /, the equality £ min / and defined (by Lemma is a unit of R'; and hence R is integrally g/g(0) / 0 (mod m); and hence of R[X] content is a unit of R. After " ^)- Therefore, R. Since But then g(0) the same and since /(0) that R-automorphism cf>(b) have </>(/), assume h £ R [X]. tegral Let since degree / with 2.4. is infinite 385 that a consequence = R, so the result are in- g/g(0) £ R[X]; of Lemma follows from Lemma 2.1. 2.8 Definition infinite, we pass of R(Y). In removing to a certain overring from 2.7 R(Y) the restriction of R which that is defined R/m be as follows [Na3, p. 18]. Let Y be an indeterminate, and let S = {f £ R[Y]| c(f) = R\. S is a multiplicative system of regular is customarily denoted (i) An ideal ideal of R[Y] R(Y). consisting We list ztz of R(Y) elements, a few properties is maximal and the ring of this RfY],. ring. if and only if there exists a maximal m of R such that m = 77zR(V) [Na3, p. 18]. (ii) For each ideal A of R, R(Y)/AR(Y) ^ (R/A)(Y) [Na (iii) R(Y) is R-flat [Na3, p. 64, Exercise p. 18]. 2]; and hence by (i) R(Y) isa faithfully flat R-module. (iv) R(Y)[X] is a faithfully 2.9 Lemma. Let flat R[X]-module R, m be a quasi-local [B(a), p. 48, Proposition ring, and let I be an ideal If IR(Y)[X] = fR(Y)[X] for some regular element f £ min(/R(Y)[X]), principal and is generated Proof. Since element R(Y)[X] is a faithfully extends and contracts to show that there by a regular to itself exists [B(a), g £ I such flat R[Xl-module, deg., We may assume and hence degx /> degx /.. But / £ min(/R(Y)[X]) / < degx /¿. Thus • • • + ö(Y'. Hence degx atYl)f; /. = degx /. Since tion that 2.14(3) and since a . g mR(Y) will Therefore / and show that 2.9 = + f( Yl, j. £ I; and p. £ 7R(Y)[X], so the leading / is regular for some it suffices IR(Y)[X] coefficient of / is we have /. = a ./ with a. £ R(Y). Thus ;*. Then is the unique maximal ideal of R(Y). Therefore Example every ideal of R[X] / £ ¡R[Y , X]. Then / = /n + f {Y + ... regular in R(Y) and IR(Y)[X] = /R(Y)[X], / = (a0 + ajY+--.+ 9]. g = degx gR(Y)[X]. degx of R[X], then I is of min /. p. 51, Proposition that 5]. in R(Y)[X], l = ar] + a]Y a . is a unit of R(Y), since f.R(Y)[X] =/R(Y)[X]. is no longer valid R is quasi-local. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use without + 77zR(Y) Q.E.D. the assump- 386 JACK OHMAND D. E. RUSH 2.10 Lemma. Let I be an ideal of R[X] such element of R[X]. // R{X]/I is R-torsion-free that [September min / contains then R(Y){X]/IR(Y){X] a regular is Ro- tors ion-free. Proof. aAsy c(s) Suppose f = f'/s2, = R\. af £ 1R(Y){X], a regular in R(Y) and / £ R(Y){X]. Then a = where a' is regular in R{Y], f There exists show R{Y, X]/IR{Y, Let s £ S such sa f' £ R{Y, X], s . £ S = \s £ R{Y]\ £ IR{Y, X], and thus it suffices to X] is R[Y]-torsion-free. T be the total min /. Since that quotient the leading ring coefficient of R, and let g be a regular of g is regular element in R, we have of c Ag) = T and IT{X] = gT{X] by Lemma 2.3. Hence by Theorem 1.5, T{X]/IT{X] is T-flat. Therefore, T{Y, X]/IT{Y, X] ( =S T[X]//T[X] ®_ T{Y]) is T[Y]-flat by [B(a), p. 34, Corollary 2l and a fortiori T[Y]-torsion-free. lar, let / £ R{Y, X], and suppose T[y]; so / £ IT{Y, X]. rf £ IR{Y, X]. c R\x](rf) When Let contains is A contains a regular have r • cd("v-i(/) R-torsion-free, = and hence element, then that, of 2.10 element remains valid statement can be to of omitting this assump- (0 : c Aa)) R = 0; and af as a polynomial r Af) C / for some that Thus, ideal a domain or a noetherian in Y 22 > 1. If cDry~|(/) / £ IR{Y, X]. generated without without interest so that we can conclude (e.g. valid of some Now regard and hence for any finitely a regular (a related of R[Y], c Aa)nc R-torsion-free), remains perhaps the possibility / £ R{Y, X]. to obtain 2.10 element It is thus element By [Kaj, example shows p. 63, Exercise (i) R is its own total pA has zero quotient that (p., such that 25,722. = 0, p ym2 = 0 and this *- ' if R is a A of R, (0 : A)R = 0 the assumption assumption exists ring, annihilator, that (iii) / = gR[X]. a is a regular element of R{Y, X]; and by (i) R[Xl/7 cannot a ring (ii) there there 222 2 <¿ p XR. Let and let Then By (ii), that 7], there such 722jX. is that ring), then min / contains element. The following erties: r of R such Y, we then of Lemma to consider formula with the property the conclusion eral. in order a regular R{X]/I implies element R{X]/I a regular 18.12]). a be a regular the content (assuming in the conclusion af £ IR{Y, X] fot some c Aa) a regular ^ ' slnce min / contains for a moment and apply exists Q.E.D. that from 2.10. suppose ring c R\x~\if) in [Na ,, p. 63, Theorem digress a £ R{Y] be regu- af £ IR{Y, X] C IT{Y, X]. Then a is regular in there / as a polynomial R is noetherian, the assumption tion Regarding C /. Therefore, / £ IR{Y, X]. found Thus, Now let element is trivially exist exist m2¿ pxR. Thus, nonzero nonzero in gen- p., R-torsion-free. X]. p2£ R g = p l + p 2X, g is a regular Let /= However, / ¿ R{Y, X]/IR{Y, X] is not R[Y]-torsion-free. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use prop- 272,, 222 £ R a = p , + p 2Y, let of R{Y] and af = p ,2222 + p 2m xYX = (2222 + 222xY)g £ IR{y, gR{Y, X] since be omitted R with the following 272 + 1972] THE FINITENESS OF / WHEN R.[X]/I IS FLAT We shall Proof now complete of Theorem the previous proof. the proof 2.7 in the general Otherwise R(Y), which is quasi-local R(Y)[X]/IR(Y)[X] R(Y). By Lemma regular element R(Y), contained closed in R of Theorem we must with infinite 2.9 it suffices in the total and such is infinite case ring cß(y)(g)l?" apply to the ring field by 2.8. By Lemma 2.10, /R(y)[X] For this we merely by passing and c(l) = R implies to show quotient that If R/m to this residue is R(Y)-torsion-free; of min (/R(Y)[xl). 2.7. case. reduce 387 cR(y)(/R(Y)[X]) is principal it suffices of R(Y), generated by a to find an overring such that = R"• We claim R(Y) = R of is integrally R = R [Ylr is such a ring, where S = {f £ R[Y]\ cR(f) = R\. Since cR(g)R' = R', it follows that cR(y)(g)R'[Y]s = R'[Y]S. Also, R[Y] is integrally and hence 16]. R(Y) = RMs Finally, R[Y] and is integrally any element cR(g) total quotient =sR, with corollary closed of R'[Y]„ to Theorem 2.7 Let R, m be a quasi-local such that of R[X] and R'[Y]S is the main result 2.11 torsion-free p. 22, Proposition in the form f/g in R; and hence and integrally be an ideal [B(c), 13], with /, g £ is contained in the Q.E.D. R is quasi-local Corollary. in R'[V]S can be written s regular ring of R[Y]. The next closed in R'[Y] [B(c), p. 19, Proposition closed (in its total a regular I is principal for the case that ring). integrally min / contains c(l) = R, then in §2 quotient closed ring, element. and is generated and let If R[X]/I is by a regular I R- ele- ment of min /. Proof. Take min /, then T(R) to be the the leading coefficient R of Theorem 2.7. of g is regular If g is a regular in R and hence element is a unit of of T(R). Q.E.D. Another consequence relationship between following lemma 2.12 Then torsion-free Tp there is T = T(R), and R[X]/l the corollary of R[X] such 2.13, which being R-torsion-free. in a global that illuminates the The form. min / contains a regular (if and) only if Rp[X]/lRp[X] is Rp- P of R. let a £ R, s, s' and hence exists therefore R-flat is R-torsion-free prime is Corollary P be a prime £ r\p, f £ R[X], of R, and suppose and a/s (a/s)(f/s is regular ) £ in R p. Then £ ¡Rp[X] C 7Tp[X]. But T[X]//T[X] is T-flat by Lemma 2.3 and Lem- Rp-flat, element being 2.7 us to give I be an ideal R[x]/7 Let where (a/s)(f/s') ma 2.1, Let for every Proof. }Rp[X] R[X]/I will enable Lemma. element. of Theorem Tp[X]/ITp[X] so a/s s" £ R\P r £ R such //l that is regular such that is Tp-flat in Tp, s"f £ IT[X], rs"f £ I. But since and a fortiori and hence f/s and hence R[X]/I is T „-torsion-free. £ ITp[X]. there exists R-torsion-free, £ IR p[X]. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use Also Therefore a regular then s"f £ I; 388 JACK OHMAND D. E. RUSH 2.13 element Corollary. Let and consider I be an ideal the following (i) For every prime ideal ment of min(/R„[X]); (ii) For every of R{X] such [September that min / contains a regular properties: P of R, }R p{X] is principal generated by an ele- and c(l) = R. prime ideal P of R, IR p{X] is principal; and c(l) = R. closed, then (iii) R{X]/I is R-flat. (iv) R{X]/I is R-torsion-free, Then (ii) (i) =^ =^» (iii) and c(l) = R. => (iv); and if R is integrally (iv) =£» (i). Proof, (i) =#►(ii): Trivial. (ii) =$> (iii): Apply Theorem 1.5. (iii) =^> (iv): c(l) = R by Corollary 1.3 and the fact that c(I) contains a regu- lar element. (iv) =^ (i): Let P be a prime ideal of R and let g be a regular min /. Then Rp C T(R) „ C T(R A, and R p is integrally [B(c), p. 22, Proposition 16]. Also, in Rp{X] is in min(/Rp[X]); cR (g/l)T(R)p. Since Rp[X]//Rp[X] lows from Theorem 2.7. Note Lemma that To see this ting of zero-divisors that min / contains it suffices and such to exhibit that torsion-free. existence R 0 be a noetherian prime ideals exists ting R = RAA, of a ring ring of Krull and P, A of R. whose we get a ring = P AA. a regular 2.14 integral tains To see prime proceeding can happen further, if one alters Examples. closure a regular this, g/l of g R with such dimension a prime > 3. Then is needed a prime element but Rp{X]/PRp{X] associated R whose Thus be an associated Before element R having P2 primes in P consis- of Rp. For would not be R p- can be seen as follows. there a chain are exactly associated consists primes exists R. we shall the hypothesis (1) Let Then element choose of (0) in R [B(b), exists and such that and let R[X] —>R[<f]. R[rf] is R-torsion-free 2.13(iv) of P,. Let- P 2 C P,; P x but _be an P2 5]. to illustrate what 2.13. R is different / of R[X] is satisfied / be the kernel since and P~R~¿ 2 P 2 would 8] and hence examples of Corollary an ideal since p. 134, Proposition some R be any ring for which there <f £ R\R give P. of (0) are exactly of zero-divisors P 2,R-t;P 2 contains a regular element of R--, & P 2 , for otherwise associated prime of (0) in Rp [B(b), p. 137, Proposition would image T(R)p = P Q < P x < P 2 < P} of R Q, and by [ZS(a), p. 230, Theorem 21] there an ideal = P j/A a regular a ring PR p contains would be R-torsion-free The the canonical = T(R) then is R p-tors ion-free by Lemma 2.12, (i) fol- then R[X]/PR[Xl Let 2.3 cR(g)T(R) of by Q.E.D. the assumption 2.12. by Lemma and since element closed in T(R)p such but of the that from its min / con- 2.13(iii) is not. R-homomorphism R[£] C T(R), and c(l) = R since License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use <f 1972] THE FINITENESS OF / WHEN R[X]/l IS FLAT is integral regular over R. Moreover, element since ç= of min /. However, a/b, R[çJ 389 a, b £ R, b regular, is not R-flat since bX — a is a R < R[<f] C R [A, p. 803, Corollary 2]. (2) [Na2, p. 446, Example which is not integrally 2]. closed, fied but not 2.13(a). Let This is an example and an ideal k he a field of a quasi-local / of D[X] and such that t he an indeterminate, k[t2, z*3] 2 , , and let / be the kernel of the D-homomorphism r D[l/t\ / i does is the quotient field not have principal (3) This local, (Corollary exists domain an ideal the quotient osition shows closed.) (iii) Let that D[X] —» D[b/a], of D, D[b/a] implies class domain domain D, which but group is not quasi- R is quasi-local is not Bezout Since of the 7]). D[b/a] and hence 1.3- £ min / and / is not princi- [C, p. 222, Theorem if / is the kernel c(l) = D by Corollary when which is not principal. D = by 2.1. is satisfied / is principal is D-torsion-free Therefore, then closed y but z°X - t be principal 2.13(iii) D be a Prüfer with nontrivial D-flat. / cannot such (a, b) of D which field 3] is so let D, is satis- D[X] —>D[l/t]. a is D-flat; of an integrally / of D[Xl 2.13 and integrally Dedekind content, is an example and an ideal pal. of D and hence domain 2.13(iii) (e.g. any Then there is contained by [B(a), in p. 29, Prop- D-homomorphism Moreover, / is not principal. For, if / = /D[X], then c(l) = D implies c(f) = D. But aX - b £ I implies aX - b = df, d £ D; and then by equating (a, b). [Na2, (Nagata Note that in Lemma Example 2.9- For D(Y)[X]/ID(Y)[X] contents, p. 446, 3 also D[X]/l (a, b) = (d), a contradiction Example shows is D-flat is D(y)-flat; l] gives that a similar the quasi-local implies to the choice of example.) assumption (by tensoring; see and D(Y) is the Kronecker is needed 1.2) that function ring of D [G, p. 384, Theorem 27.4], which is Bezout [Kr, p. 559, Satz 14], so ID(Y)[X] is principal by 2.2. The next 2.13(i) theorem gives some and is the tool needed 2.15 element, Theorem. and assume (i) For every element Let conditions which are equivalent to pass from the local to the global I be an ideal of R[X] such that c(l) = R. prime further ideal Then the following P of R, ¡R Ax] to case. min / contains a regular are equivalent: is principal generated by an of min(/Rp[X]). (ii) c(min /) = R. (iii) / = (f y • • •, fn), f. £ min /. Proof, prime (i) =^ (ii): P of R, it follows that /p = fRp[X]. Since that by Lemma for each 2.3, deg min / = deg min(/p) prime P of R there exists for each / £ min / such Then c(f) p = R p, and hence c(f) t P. It follows that c(min 7) = R. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 390 JACK OHM AND D. E. RUSH (ii) ==> (iii): R. Therefore By (ii) there for any prime /. in R p{X] is in min(/p), shows that such /,,•••,/ it follows that c(f x, • • •, / ) = i; and since the image prime P of R, and hence of This / = if x, • • •, / ) Theorem l]. =^ that £ min / such j£ P for some from Lemma 2.1 that Ip =/.Rp[X]. I p = (f y • • • , f ) p fot each [B(a), p. Ill, (iii) exist P of R, c(/.) [September (i): c(/.) Since \t P. c(/,, • • •, / ) = R, for any prime But since the image P of R there of /. in R p[X] exists is in min(/p) /. (by Lemma 2.3), it follows from 2.1 that lp = /.Rp[X]. 2.16 Corollary. such that ated by finitely Proof. R-flat, ated. give the assumption Note also eses 2.16, of Corollary additional condition dimension zero, lar element principal that then rem l], and hence 2.15(iii). not by Lemma 2.1, R to R, where R denotes 2.17 Let 2.14(3) that shows c(g) R{X]/I gener- a regular that under as follows: Let 2.1, it follows is principal ele- the hypoth- if one imposes of R be a noetherian and Lemma But then that / is not finitely however, spectrum that is satisfied, R such min / contains to be principal 2.13 P of R. that be principal; 2.15(i) closed but such and 2.16 that Q.E.D. of an integrally the maximal The final part of the proof Lemma. is just / may be seen prime c(l) = R. shows that Example of min /. By Corollary for every I is gener- 1.3 and the assumption is satisfied / need then 2.13 in 2.15 ment is essential. R-flat, from Corollary an example c(l) = R, and 2.15(i) Thus, is Corollary conclusion we shall If R{X]/I of min / and element. the remaining closed, and let I be an ideal of R[X] element. c(/) = R follows a regular In 2.21 is a regular many elements That min / contains and then Let R be integrally min / contains the space of g be a regu- that c(g)p by [S, p. 318, is Theo- / is principal. of the main theorem the integral closure R, m be a quasi-local of §2 consists of R in its total ring, let in going quotient I be an ideal from ring T(R). of R{X], let g £ I, and let P' be a prime ideal of R[X] lying over m. If c(l) = R and IR{X]p" =gR{X]p" for every prime P" of R[X] lying over m, then IR{X]p> = gR{X]p>. Proof. it suffices First we show /R[X]p- = gR[X]pz, and by [B(a), p. Ill, to check the integralness tracts to P' this locally of R[X]p< in R[X]. over Therefore R[X]p' R[X]p", in R[X] and hence to 222in R. Since we have that ideals P /R[X]p" ideal of R[X]p' is a prime l] of R[X] p>. It follows any maximal the localizations are all of the form hypothesis, where at the maximal Theorem of R[X] = gR{X]p" of R[X]p< from con- at its maximal which contracts tot all such ideals to P' P" by ¡R{X]pi = gR{X]p<. Now we show /R[X]pz = gR[X]p-. Since IR{X] p< = gk~{X]p<, for any / £ I License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 1972] THE FINITENESS OF / WHEN R[x]/¡ IS FLAT there exist h £ R[X] R = 772implies there exists yields and s s £ R[X\\P' cR(s¿) = R. Therefore, /. £ / such that c Af such cR(j) that s f = s 2gh. = cR(gh) by the lying-over Moreover, C c~^(g)- Since ) = R; so the above R = c (/ ) C c~jï(g)- Hence 391 reasoning theorem, c(/) = R, applied cR(g) to /. = R. Therefore, cR(/) = cR(gh) = cR(/>), so cR(h) CR and h £ R[X]. Thus, / £ gR[X]p-. 2.18 Theorem. element. Then Let I be an ideal the following of R[X] such that P' Cl min / contains Q.E.D. a regular are equivalent: (i) R[X]/IJs R-flat. (ii) R[X]//R[X] is R-flat. (iii) I pi Proof, is principal (i) =£>(ii): for every prime Since R[X]/I P of R[X], and c(l) = R. is R-flat, the sequence 0-> 7 ®R R -♦ R[X] ®R R -+ (R[X]/l) ®R R -> 0 is exact; and hence (R[X]/7) ®R R S R[X]//R[X] (as R-algebras). But (R[Xj//) ®R R is R-flat by [B(a), p. 34, Corollary 2l. (ii) =£> (iii): Let P' he a prime ideal of R[x], and let P = P' n R. By localizing at P, we may assume (R/P)[X] is a principal R is quasi-local ideal domain, with maximal so there exists ideal P. g £ I such Then that / = gR[X] + (PR[X] n /). Therefore, IR~[X]= g~R[X]+ (PR~[X]Cl 1R~[X]).Since R[X]//R[X] is R-flat, PR[X] n /R[X] = PR[X] ■ /R[X] [B(a), p. 33, Corollary]. Therefore, 1R~[X]= gR[X] + PR[X] so 7R[X] lemma, • IR[X]. is a finitely IR[X]p" But min(/R[X]) generated = gR[X]p« ideal for every contains a regular by Corollary 2.16. prime P" of R[X] apply Lemma 2.17 to get IR[X]pi = gR[X]p' R[X]//R[X] is R-flat implies element Thus, lying provided by Lemma 2.3, by Nakayama's over P. We can now we first note that c(IR[X]) = R by Corollary 1.3 and hence that c(l) = R. (iii) =#»(i): Apply Theorem 1.5. Note prime that (ii) =$> (iii) P of R there R[X] lying primes over exists P. Thus, P of R which ideals of the above g £ I such if (ii) [Kaj, argument that is valid are contractions of R by Kaplansky Q.E.D. actually proves Ipi = gR[X]pi and that fot every a is the cardinality of maximal p. 17, Theorem ideals 27]), of R[X] then for each prime P of of the set (called of the G- / can be generated by a. elements. 2.19. Main theorem of §2. regular element. If R[X]/l Let I be an ideal of R.[x] such that min / contains is a flat R-module, then I is a finitely a generated idea I. Proof. flat; ma 2.3- exists As observed and min / contains Therefore, /R[X] a finitely generated previously, a regular is finitely R[X]/l element is R-flat implies implies generated min(/R[X]) by Corollary ideal A of R[X] such that AC/ License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use R[X]//R[X] does 2.16. also Hence is Rby Lemthere and AR[X] = /R[X]. 392 JACK OHMAND D. E. RUSH Claim. A = I. It suffices localizing Let at (R/P){X] to show A p< = Ip< tot every prime P = P fl R, we may assume P be a prime ideal of R lying R is quasi-local over P, let and let <p be the homomorphism (or applies [B(a), p. 48, Proposition (R/P){X] extends to itself P' of R[X]. with maximal since R/P in (R/P)[X]. —» easily every Therefore, P. R[X] One checks is a field, By ideal <p be the homomorphism R[X] -♦ (R/P)[X]. 5]) that and contracts c/>(/R[X]) implies [September ideal of cp~(AR[X]) = <p(A) = <p(/). Thus, / = A + (PR{X] n /). Since R[X]// is R-flat, PR{X] n / = PR{X] ■ I [B(a), p. 33, Corollary]; so I = A + PR{X] ■ I. Then I p> = A p< + PR{X]pi conclude and since by Nakayama's Note eses ■ Ipi; that the last /p' is finitely lemma that part generated Ip< = A p>. of the above proof by Theorem portion of the proof could also we Q.E.D. actually shows of 2.19) for any ideal A of R[X] such that AC/, A = /. This 2.18, be done that (under the hypoth- AR{X] = IR{X] implies using (ii) <$> (i) of 2.18 and Lemma 2.4. 2.20 Corollary. lar element. Let Then I be an ideal R{X]/I is R-flat of R[X] such if and only that min / contains if I is an invertible a regu- ideal and c(l) = R. Proof. (==H every prime ment, / is invertible Corollary P / is finitely of R[X] generated by Theorem by Theorem by [B(a), 1.3 and the fact c(I) then contains element of degree for a moment 1. Then give a simple lai. By 1.2, (yR : x)R{x/y] fore there R[X]// construction exist R[X]// the special = R[<f] = (yR[x/y] since 4]. case ele- from element. at primes that of R[X] min / contains <f £ T(R). of /. Let f=l+a. ' for a regular c(l) = R follows is R-flat by Theorem 1.5. : x) and hence that Ip> is principal / contains That principal where for the generators a z. £ (yR : x) such w and a regular / is locally [B(a), p. 148, Theorem 4]. Therefore Let us consider Thus, p. 148, Theorem that (<^=) If / is invertible, 2.18. 2.19, In this <f = x/y, by Q.E.D. a regular case we can x, y £ R, y régu- 1 £ (yR : x)R{x/y]. + a,X+...+a 0 1 There- X" £ I. Let zz b I. £ R be such that a.x = Jyb l.. Then it follows that / = (/, a.X - bn, • • •,' a zz X —b 72). l ' 0 0' For, if P' then / £ P' is a prime ideal implies a . i P' /Rp[X] = (a.X - b^RpiX] of R[X] containing fot some (/, a X - b., • • • , a X - b ), i. Therefore, if P = P'nR, by Lemma 2.1, and then a fortiori then /R[X]p< = (/, a0X-b^,...,anX-bn)R{X]pl. We conclude of our other lar element. results Recall this section that a ring is flat [B(a), p. 64, Exercises 2.21 Example. with an example are not true without This R is said which shows the hypothesis that to be absolutely 16—17; p. 168, Exercise is an example of a ring that 2.19 flat if every a regu- R-module 9; p. 173, Exercise R and an ideal License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use and many min / contains / of R[X] 17]. such 1972] that THE FINITENESS OF 7 WHEN R[X]/I IS FLAT R[X]/I Let example, but yet / is not a finitely R to be a countable sum ideal.) R-flat chosen is R-flat take the direct is is A be any non-finitely-generated because to be Then every direct / = AR[X] R-module R-projective A slight modification with is. Added of this the additional c(l) = R. In this case January R[X]/7 of copies note the direct that (For A to be and R[X]/I if sum ideal), R. and generated, reference, with flat ring of a field not finitely For later to the recent platitude et de projectivité, paper is the a domain, «-variable A is then 3. / principal. an ideal AR[X] being in this that that is dropped. R[Xl// Thus, being principal. that min / contains when R-flat, R[zf] l/f is in T((p(R)[cf]), when to denote no confusion We remind generators 3.1 ç denote element. 2.1 closed. what ring that to / / is principal in Corollary and 2.11 Example closure 2.14(2) assumption conditions in addition to conclude solution R and are equivalent However, in order a satisfying R is The problem which when the integral on R[Xl/7 that of the theo- 2. and we have proved class to this so of X in R[X]/I. is the canonical the ring reader that to / is problem (when <Jj(R)[f]. </6(R)[l/f] If f is a regular makes Then R[X]/l = c/>(R)[f ], homomorphism. R —> cf>(R), we can that R(a •-., • . . , a . When the ring ) for this Lemma. in the case sense; Regarding more </>(R)[rf] as simply element and we again use the nota- of </>(R)[(f], then write R[l/f] can arise. the a (ctj, ...,a a regular is to determine via the homomorphism tion de element). tp: R[X] —> R[X]/l R-algebra our Critères of a quasi-local on R[X]// valid provides zf will denote the equivalence where the case R is integrally be imposed 3.6 a regular has brought generalization R. We know by Lemma is the main problem The Summary 2.19 and Ohm in footnote is no longer must and R-module. and L. Gruson, ?z-variable min / contains R[X]/l R-flat the for the moment and of content is true flat gives element Vasconcelos of our Theorem contains is to find conditions the converse a cyclic with work of M. Raynaud to Heinzer such R implies the converse shows 3.4.2 section principal of content is even a regular Invent. Math. 13(1971), 1—89- Corollary 3-4.7 of that Consider / of R[X] treated R/A / = AR[X] + XR[x], / contains A conversation generalization attributed namely that important and Corollary rem previously example, property = 28, 1972. attention let product is also (as is the case ideal. of an absolutely R[X] projective. an example an generated ideal 393 Let R is a ) denotes understood, the R-module we shall also having merely use of R[X], and R-module. R, m be a quasi-local the equivalence class + • • • + a X" £ / with aQ, • • • , a ._ ring, let of X in R[X]/l. I be an ideal If there exists f = a . + a .X £ m and a . = 1 (/ > 0), then ( 1, f, - • • , f "+ A = (1, f, •" - , f7" \ fl+1 +\ - - - , e+i > for all i > 0. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 394 JACK OHM AND D. E. RUSH Proof. Let [September E = (1, {;, • ■ • , <f" *); and for e £ E, let e denote its image in E/272E.ThenJ' + Vi?'+_! + ' ' ' Aa J" " ° i^lies ?' £ ^ <f'*V Therefore E = (1, £, • • • , ' ' .f"'1) tf _z, <f lemma. !, • • • , cf" £ (£'+»♦', ■•-, t"+i). *\, and hence the lemma follows l.' " ' ' ?!>« ?'+1 £ from Nakayama's Q.E.D. Before known ,_:• • , ^ proceeding facts which be a finitely generating generated set generating for sets module). to the proof will be used R-module. M contains contain Moreover, of Theorem there. Let It follows a minimal the same 3.2, we shall from Nakayama's generating number mention R, 222be a quasi-local and let lemma that M any set and that any two minimal of elements if zM is free of rk 72, then a few well- ring, ( = dim M/mM as an any minimal generating set R/m- is free [B(a), p. 109, Corollary]. Theorem 3.2 shows that ( 1, tf, • • - , <f ) is equivalent equivalent regular false, conditions element, as we have as Example 2.14(2) be integrally closed, that a finitely free [B(a), 3.2 for "free" Theorem. £222rilet to the equivalent conditions R{X]/I already by Corollary module over 3-e], Let is 2.13 of all the modules of Lemma R-flat in Lemma If one imposes and These 2.1; but the converse the converse a quasi-local 2.1. min / contains the additional is flat "flat" that Recall R also if and only could a is condition is valid. ring so that the word in Theorem rf denote ( = R[<f]) noted shows. then generated R, flatness that p. 167, Exercise substituted R[X], imply for a quasi-local if it is equally well be 3.2. R, m be a quasi-local ring, the equivalence of X in R{X]/I. class let I be a nonzero Then ideal of the following are equivalent: (i) (1, zf,• • • , tf ' ) is a free R-module for all t > 0. (ii) (1, ff, • • • , <f') is a free R-module f or some t > 0 for which 1, <f,• • • , (* are linearly (iii) c(l) - R. dependent / = /R[X] (Note: Proof, some there exists elements for some this (iii): First / £ I such of / having over, ther that f(a) consider that content the case as a linear c(f) leading coefficient assertions that = R. Choose is regular, of Lemma term 1. It follows the R-automorphism we have the residue field of / to have 72 = deg /. of / are in 222and a = 0. Thus, that combination R, and let = u é m; and by replacing by applying assume f £ R{X] whose is one of the equivalent ef7 can be written all of the coefficients such R. and 2.1.) (i) =£> (ii): Trivial. (ii) =^ Since over minimal Claim, R/222 is infinite, / by f/u of R[X] reduced R/222 is infinite. 1, zf, • • • , f7, • • • , ç degree there exists we may assume given among f £ min /. Since not a £ R u - 1. More- by X —►X + a, we may fur- to the case that / has constant that rf is a unit of R[f ]. By Lemma 3.1, (1, <f,• • •, ¿jn l) = License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 1972] THE FINITENESS OF / WHEN R[X]/¡ IS FLAT (fm,---, f"+!) for all i> 0. In particular, for i = t-n, zfz (1, f, • • • , f "~ 1) is free; that (1, ff, • • •, <f"_1) pendent over R, for otherwise quasi-local), and hence, since is free. Therefore, contradicting some (1, f, •••, f*+7 = f is a unit of R[f ], it follows 1, f, • • • , f" proper the choice 395 subset forms of /. Thus must be linearly a basis (because / £ min /. Therefore indeR is (iii) holds by Lemma 2.1. It remains device used by R(Y), have to remove in §2 mR(Y). the restriction of adjoining There a commutative that R/m be infinite. an indeterminate are then a few things Y, thus that must We do this replacing by the the ring be checked. R, 772 For one, we diagram 0 — / ®R R(Y) -, R[X] ®r R(Y) — (R[X]/I) ®R R(Y) -> 0 ] 1R(Y)[X] 0— The top row is exact isomorphisms. implies since Let is there exists and hence, Also the ! — R(Y)[X]//R(Y)[X]->0. R-flat, and the vertical maps R-module (1, <f, • • • , f). Then and since R(Y) we have a regular element by Lemma =^ (i): 2.9, Let R-flat, argument / = a. + ajX are R(Y)-algebra E is R-free an injection for the ring R(Y) and Therefore by the above argument, such 7R(Y)[X] / £ min (IR(Y)[X]) 7 is principal generated that by a regular = /R(Y)[X], element of min /. as above. + ...+a flQ, • • ■, a,_ j £ ??z and a, = 1. Since each t = n + i fot some t < n; so assume is ®R R(Y). Thus, (ii) is satisfied (1 ® 1, <f ® 1, • • • , (f ® 1)'). c(l) = R by the same (iii) is R(Y)-free; E ®R R(Y)—. (R[X]/7) the R(Y)-module R(Y) E denote E ®R R(Y) i R(Y)[X] -> X". / £ min /, Since c(f ) = R, we may assume (1, <f, • • • , <f') i > 0. By Lemma is certainly 3.1, (1, f, •••, free for f"+z) = <1,f, - • • , f '" », f/+1+¿,• • • , f "+¿>.Clatm. I, £ ..., f'-\ f/+1+¿,• • • , f " +¿ are linearly ,rf ~ independent b,Z+ ,l+Z^.Çl+l+i+... over R. Suppose èn + è ,rf + • • • + è,_ + b72 +2.<fn+¿=0, b.£R, = J + and let g° = b n0 + b ,X + • • • + b72+2.Xn+i. 1 Then g £ 1; and since / = /R[X], g = hf fot some z? £ R[X]. Let 7>= c0 + CjX + ••• of h must be + c. X1 (the degree lar element of R). Equating z since coefficients the leading coefficient of / is a regu- of X , X + , • • • , X +1 in g = hf, we get a set of 7 + 1 equations: 0 = c„a,+ c,a, u / any c .a, z I—j 0=c0a/ +1 +c1«r+e^/_1 + ... +<7«,_i+i 0= c a, + c.a,. . + c a, 0 Z+2 (We define ,+■■■+ 1 /— 1 a . having subscript . .+••• 1 l+i—1 less than 2 z zero or greater License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use than tz in these 396 [September JACK OHM AND D. E. RUSH equations to be zero.) unknowns cQ, • • •, c ., the coefficient elements If we regard of 272above fore by Cramer's these the diagonal, rule equations matrix and hence [L, p. 330], cQ = c as has z' + 1 equations in the a, = 1 down the diagonal z+ 1 and its determinant is a unit of R. There- = • • • = c . = 0. Thus h = 0, and hence g = 0. Q.E.D. If one imposes (= R[X]//), then sense to speak then be replaced the mild restriction l/cf is in the of the ring 3.2(i) implies limits by symmetric R[<f] and and R[l/<f] shall next demonstrate then following a suitably 3.3 R' (ii) AR Proof. exists mal ideal P. czR'. Since Therefore a prime the equivalent let modules conditions when in the case that Consider this of Theorem r.1 £ R such and that homomorphism, R' = R{b/a\ We element. R is a domain, since of R. ideal Then of R which there exists an Then that R{a/b] has that R. lemma. ç= the lemma when satisfies (iii) be satisfied since + ■■• + xjt(b/a), properties. Let taking use / X -» l/<f. ^ R'. f. £ lemma, (a, b)1 ■• • , b*). Therefore It follows by lying that over, b/a is a PR{a/b] ^ Q.E.D. <p: R[X] —> R{X]/I be the canon- qS(X) and that we are denoting for the kernel = PR i > 1. There- b)1; so by Nakayama's But then, maxi- (a, b)R x. £ P, b)1 tot some a2-2b2, A = (a, b) with if we had <p(R)[<f] by merely R[g]. If f is regular in R[f], then <p(R)[l/£] and we shall set for A R is quasi-local + ••• + r.(b/a)1. z the desired technical and recall over P O R = P, and in a generating a2 £ P(a, a1' £(ai~lb, 1 = r,(b/a) 1 is integral a rather R 1 = x xf x(b/a) , • • • , b') + P(a, that that a £ A. to prove of R, (ii) would b1), and hence a/b such element of elements we conclude a2~2b and the ring rp(R)[l/ff] 3.2. a regular generated ideal P, we may assume ideal = R'. ■-., of R it suffices at the ring PR' P R -regular element, equation, a2~2b2, exist ideal By localizing (a, b)1 = (a2~lb, T(R[<f]); are then properties. A be a finitely P be a prime that conversely min / contains ring has the required on the number a regular We next need ical of whether for some assume = t«*'"**, R{a/b] these and let P is a maximal From unit of R since can R-modules It follows arises is the case makes 3.2 the finite thus is well known By induction contains there since R-flat R be a ring, = aR zz regular. fore, l/rf it thus of Theorem is R-torsion-free, there (iii) R[X]. involving and (ii) of R[<f] R such that (i) with (i) and are both this element, ring of R[tf], The question valuation Let element \ • " ' > 1} are isomorphic. implies that lemma a regular R-algebra R-flat chosen Lemma. contains which are quotient conditions R[l/tf] <f be a regular Conditions (l/£)'~ of flat modules. R[zf] The total R[l/<f]. (1, £• •♦, <f'} and ((!/£)', direct that of the homomorphism the ring is a subring of R[X] —» Then a QXn + a ,X"- 1 + . • • + an, a . £ R, is in /* if License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 1972] THE FINITENESS OF / WHEN R[X]/l IS FLAT and only if aQ + aX subset + -'-+a H of R[X], degrees) deg 77 (subd of the nonzero 3.4 Lemma. element, Xn £ I. Recall Let H) denotes elements also that, 397 for any nonempty the minimum of the degrees which I be an ideal reduces (sub- of 77. of R[X] such that min / contains a regular let P be a prime ideal of R, and let if/: R[X] —► (R/P)[X] morphism nonzero coefficients mod P. // zf is regular be the homo- in R[f] and R[<f] z's R-flat, then deg / - deg xf/(l)< subd xfAf). Proof. Let / be a regular R-algebra R such cR(f)R that which is element of min 7. By Lemma R-torsion-free, = aR for some which has R -regular a prime element 3.3, there P lying a £ cR(f). exists over an P, and Also, R'[X]//R'[X] ^(R[X]/7) ®r R' is R'-flat by [B(a), p. 34, Corollary 2], and / • 1 £ min(/R 2.3 [X]) by Lemma of R'[X]). /jR Therefore coefficient 2.3, deg / = deg(/R'[X]). that P over of R lying of /. will denote fl £ R'[X] is regular. g = b0+b.X g • 1 = hf + -'' Then P, the homomorphism IR'[X] = that, by Lemma a. £ R', a^ 4 0, let /* = + • • • + b m £ I*,I the + b Xm £ I. Therefore g* • 1=7) the identity suchthat Note also . For any7 g* = bñXm + è.X"2-1 72 ° 0 1 polynomial h £ R [X] such a subscript If fl = aQ + a yX + • ■• + aXn, + ••• +a corresponding 1 without by Lemma 2.1 there exists [X] and the leading anX" + a,X"-1 01 (where / .. Since xfj "extends" there there exists exists a prime to a homomorphism if/ of R'[X]; and it follows that subd xfj'(I*R'[X]) > subd xfj'(f*¿ = n - deg xfj'(f,). Similarly, deg xfj'(¡RAX]) > deg </>'(/j) since IR'[X] = /jR'[X]. Therefore, subd xfj (I R [X]) > deg / - deg xfj (IR [X]). i/-'(/V[X]) tify R[X], with an appropriate Theorem. and let element conclusion follows since = </>(/*)(R7P')[X] and xfj'(¡R'[X]) = xfj(¡)(R'/P')[X] provided we iden- (R/P)[X] 3.5 The desired Let f denote of R[f], then subring of (rVp')[X]. R, m be a quasi-local ring, the equivalence of X in R[X]/I. the following class is equivalent let I be a proper ideal of If f is a regular to conditions (i)—(iii) of Theo- rem 3.2: (iv) min / contains Proof. a regular element, (=^0 (iii) of Theorem while (i) implies that R[f] direct limits (<#=) so a of the modules Let principal ideal. a . of generators z7 is as large this / = '01 is a regular leads a_+a,X element Since and 3.2 implies R[l/rf] are in (i) [B(a), + ... + a 72X", for c(f) from among and R[rf] and min / contains R-flat, since are R-flat. a regular these element, modules are 2]. a 72 í 0,' he a regular ° 2.1, it suffices we can choose the R[l/f] p. 28, Proposition of R. By Lemma R is quasi-local, as possible and element to show a minimal a ., and we do this i'. < ¿- < • • • < z,. We shall to a contradiction. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use set in such assume of min /, c(f) is a a . , •••, a way that t > 1 and show 398 JACK OHMAND D. E. RUSH There choose [September exist b,. £ R such that a, = 2l. , b,.a . , 1 = 0, 1, • • • , n; and we can z; z ,=ll]ij' = b. = •. • = b . = I. Moreover , since z. was chosen as large as b. 21 1 possible, 222 ¿z. itt ' 1 b £ 722for /> z'.. Then /- ZVi. + IV,]* +■•■+ 2X«,-.)*" z= l = «f/fcoi By applying + *„X + • ■• + ¿7nlX") + . • • + fl;^0i the flatness of R[<f] c . £ (a . , • • • , a .) : a . Cm z z2' If such ij (1) via criterion a . by applying 1.2(b), we conclude that there exist that lH + *»i+Similarly, + bXtX + ■■■+ bntX"). + *,lf'*Efif' the flatness of R[l/<f], there exist d. £ (a . ,••• , ) : a . C m such that zz-1 2t (2) b0td/on + bxtd/ct)n-1 +.. • + *wf = £ zi.d/f )«-. 1=0 If i/z denotes the homomorphism mod 222,we have from R[X] —►(R/t2z)[X] (1) that deg <//(/) < i, since which b. reduces coefficients = 1 and b. £ m tot I > i.. Similarly, it follows from (2) and the fact that b . = 1 that subd xp(l ) < n - i (where is the kernel deg / ip(l) > 22- i a contradiction 3.6 R{X]/I. inequality to the assumption We shall contains of the homomorphism the first now globalize Summary. Let a regular element, Then that t > 1. and summarize R be a ring, and let the following (i) (1, I, • •• , £') R[X] —►R{l/¿;]). following let from Lemma n - i > n - Therefore i < z,, Q.E.D. the preceding / be an ideal <f denote Then 3.4. local results. of R[X] the equivalence such class that min / of X in are equivalent: is a flat R-module for all t > 0. (ii) (1, <f,• • • , rf') is a flat R-module for some t > 0 for which 1, zf, • • • , <f' are linearly (iii) dependent over R. c(l) = R; and for every the leading coefficient prime P of R, there of / is regular in R p and exists / £ Rp[X] such that IR p[X] = /Rp[X]. (iv) / = (fx, • • •, fn), f. £ min /, and c(l) = R. (v) c(min /) = R. (vi) every R[zf] regular is R-torsion-free and c(g) is invertible for some g £ min /). If <f is regular in R[f], then (i)—(vi) are equivalent (vii) R[f] and R[l/<f] are R-flat. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use to g £ min / (or for 1972] THE FINITENESS OF 7 WHEN R[X]/¡ IS FLAT If R is integrally closed, then (i)—(vi) are also 399 equivalent to (viii) R[f] is R-flat. Proof, localizes (i) «^ (ii) <#> (iii): Apply Theorem 3.2 and the fact that flatness well [B(a), p. 116, Proposition 15]. (iii) <$> (iv) <#£► (v): Apply Theorem 2.1.5(vi) <^> (iii): Suppose is Rp-torsion-free by Lemma if it is finitely [B(a), generated, p. 148, Theorem and is generated Therefore cR (g/l) 4]. follows 2.12. Recall Then that a regular Therefore c(g) element. from Lemma g £ min /, g/l is principal therefore is satisfied. contains by a regular (iii) any regular (vi) by Lemma an ideal element, 2.1. is invertible implies g/l if and only Conversely, principal c(g)p £ min(Ip) is principal by Lemma suppose fot any prime 2.1, and hence P of R, R[f]p and is locally is invertible Moreover, £ min (/p) for any prime (iii) is valid. P of R, by Lemma c(g) is locally 2.3For 2.3- principal Then and invertible. (vii) <^- (iii): Apply Theorem 3.5. (viii) ^ (iii): Apply Corollary 2.13- The simplest case deg min / = 1, i.e. to which the above min / contains a regular Q.E.D. results apply element f is in the total quotient ring of R. Then rf = a/b, is isomorphic as an R-module (1, f, • • • , f') Theorem grally (viii) domain We have shown min / contains pal. We conclude 3.7 §3 Theorem. Let Proof. Let that field that [B(a), p. 148, if D is an inte- of D, then D[f] as a very special if R is quasi-local then generalizations a regular element, is flat case of our R[X]/I is of this ring, integer is I bean R-flat, such If t = n this that X" be a regular c(f) is immediate, = (a , a element ,,•••, closed / is princi- ideal then of R[X] IR[X] is 72 )r..a.,'j ¿^ j z = 0, 1, ■■■ , t — 1, i=t and License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use of min /, and let a ). We shall so we may assume r z;.. £ R such that a.= 1 implies result. and let ¡f R[X]/I integrally R-flat ](R) C R, then i is principal. / = a0 + fljX + ...+a the largest = a(R. 2.13 R, m be a quasi-local If in addition c(f)R may be regarded element, with some principal. that 3] has proved is in the quotient which in Corollary min / contains exist f = a/b a regular that / denote , • • • , a ) = (a, b)1. Moreover, (vi). and such and (a, b) is invertible, =^ a, b £ R; and (1, tf, • • • , rf') (b , ab [R, p. 797, Proposition where 1, or equivalently, is flat if and only if the ideal (a, b)' is invertible 4]. Richman closed implies to the ideal is the case of degree prove t < n. Then there 400 JACK OHM AND D. E. RUSH [September zz /-¿a/(ro/- + 'i,Jf+"- + ri-iyx<"l+x/)- 7=2 If tf denotes the equivalence class r0z + rlz^+--- of X in R[X]//R[X], + rz-lz^_1 = Hat +X, •••,«„): the equality being and Criterion a consequence 1.2(b).) Thus +? 6{at+V then /(<f) = 0 implies ■■■,an)R{c;]:at atÍR{cf]C «R[f ], of the R-flatness of R[f]. (See g = r Q( + r X[X + • • • + rt_XtXt~l Theorem 2.18 + X1 £ mR{X] + IR{X]. Let 272be anyJ maximal R'[X]//R'[X] by Lemma ideal is R'-flatby 2.3- Since p. 22, Proposition R of R, and let R = R—, 222'= 222R— Then mm [B(a), p. 116, Proposition 15], and / • 1 £min(/R'[X]) is quasi-local l6l and since and integrally closed R' C T(R)— C T(R-), IR [X] is generated element h' £ min(/R [X]) with cR'(h ) = R' by Theorem content formula element of the Claim, that q = t. Let coefficients c(f)R is principal form a . 1, where cp: R [X] —►(R /m'){X] mod 222'. Then cp(b')(R'/m'){X]. and hence t < q < 22 and m of R, c(f)R = cz^R. Therefore c(f)R = a R ; and since + • • • + a 72Xn and For the second d.l £ R such zzZ. = a.a„ Z assertion that a.l = d.1 mod J/, from our previous coefficients mod /. this regular of R . which reduces cp(g • 1) £ Z that, d. £ R. Let since for an arbitrary maxi- by Lemma 2.1, /R[X] = /R~[X], where it will suffice to show ideal only¿_ to show i/z: ß[X] r. that a,Z+F'tz ,,■■•, for any maximal q' = t. It remains _ Then is true z = 0, • • • , 22. That observation a. = d. mod / for some reduces by some the identity a.Z £ R. of the theorem, c(f)R— < q — < n, then ' 772 = a 9 R— 772 for t — have by an from the Since deg cp(g • 1) = t, deg <p(h') < t. But deg (/>(*') > q > t, so mal ideal follows is generated 1 denotes be the homomorphism q = t. Thus, exist 2.7. It follows g • 1 £ m'R [X] + IR [X] implies therefore /'01= a. + a,X in T(R)— by [B(c), —♦ (R//)[X] a there ate in _ / J 272of R, if that for i — < t we be the map which + r. X + • • • + r Xl~ + X1 £ IR{X] + ]R{X] and IR{X] = /r[X], we have z/z(rQi+ 2-X(X+ • • • + x') £ iff(f)(R/J)[X]. But i/KVoz + ''i^X + • • • + X ) and xpif) have if,(r.t) = z/z(a.) for z = 0, • - • , t. Note alent to the rem 2.18. seem; that the hypothesis (apparently Also, for at least in the rather and are both monic, R[X]// hypothesis J(r) is R-flat that in the above R[X]//R[X] C R is not as contrived trivial case of a 1-dimensional D, ?22,it can be seen that ¡(D) C D it (and only if), for every D{X]/I is D-flat implies / is principal. The proof of this remark so Q.E.D. that weaker) the hypothesis the same degree (which is inspired by [E,, License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use theorem is R-flat, as it might quasi-local ideal p. 345]) is equiv- by Theoat first domain / of D[X], requires a few 19721 THE FINITENESS OF / WHEN R[X]/l IS FLAT preliminary observations. the ideal For elements a, b in a ring {r £ R \ ra £ bR \. A criterion overring R we denote of [A, p. 801, Theorem by (bR : a) l] says that an R' of R, R' C T(R), is R-flat iff (bR : a)R' = R' for every a/b £ R' ; and in the case reduces of a simple extension to the following: or equivalently, R = R[x/y], R[x/y] is R-flat if and only if \/(yR it is easily and hence Lemma = R[x/y]. that Now let this (x, y) is not principal = R[x/y], if x/y R[Xl —>R[x/y] criterion is a unit of defined by (x) 4 U, y) / (y); and it follows from / is not principal. D, 772be a 1-dimensional the above the form then that Also, then x/y, y/x I R. Therefore, if R is quasi-local, 2.1 seen if and only if (yR : x)R[x/y] : x)R[x/y] T(R), if / is the kernel of the R-homomorphism X —»x/y, and if y/x £ R\R, prove 401 remark 1 + r (x/y) we need quasi-local only find + ■ • • + r^x/y)" x/y domain £ T(D) = 0, with r.£ such that satisfying ](D) t- D. To an equation of m ( = -\J(yR : x)), and such that y/x ¿ D; for then y/x £ d\d and yJ(yR : x)D[x/y] = D[x/y]. If mD > m then 772Dj£ D and £ D and it D. Then (ca/b)" b/ac has there exist + • • • + c"r 1 the required ](D)\D suchthat required properties. properties. (a/b)" 3.8 Proposition. the nil radical regular is another for noetherian rings Let ¡f R/A 72 « let closed case there x/y p. 445, and = a/b £ such that R[X]/l is ' exists = b/a has 2.13. A similar Corollary A he an ideal of R[X] x/y z of Corollary in [Na2, R be quasi-local, is integrally ca/b if mD = m, then generalization I be an ideal that r. £ D; and hence _ n, and in this can be found of R, and let element, c £ m such = 0 fot some Similarly, £ ttz for some The next proposition statement a/b + cr ,(ca/b)n~ the 4.3(iv)]. of R contained in min / contains R-flat, then a I is principal. Proof. Let M = R[X]/L Since M is R-flat, 0 -> I/AI -» (R/A)[X] -*M/AM —»0 is exact and M/AM is R/A-flat. a principal 2.19, ideal of (R/A)[X]. it follows 4. The hypothesis examples from Nakayama's case that that of arbitrary ever, to what rings R the condition difficulties extent of §2 / or R[X]//. Thus, they nonetheless We remind Moreover, free this lemma that a regular are false condition §3 by means although give that, <##> projective of §2 of various of this perspective R-module for R quasi-local; and and indeed this In studying for any ideal by Theorem / is principal. hypothesis. can be weakened, the results an interesting the reader generated element, without can be dropped. and 2.13 and Lemma 2.3, I/AI is / is a finitely /. Most of the theorems min / contains the theorems By Corollary Since §3 involve we have One can and in particular, these questions, finiteness section by ask, how- for what assumptions the on either superficial, problem. M, free =£> projective and if M is finitely License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use shown we avoid are somewhat to the general the =£►flat. generated, 402 JACK OHMAND D. E. RUSH then projective local rings, «#> flat and noetherian As usual, shall begin 4.1 for a large R[X] rings. denotes Let We refer remarks A be an ideal (i) A is generated of rings including the reader the polynomial with a few simple Lemma. class [September domains, to [B(a)] quasi-semi- for these ring in one indeterminate facts. over R. We on idempotents. of R, and consider the following statements: by an idempotent. (ii) A is locally trivial. (iii) A =A2. Then (i) =^ (ii) =^ (iii); and if A is finitely general, (iii) valuation => (ii). Take A to be the maximal ideal then (iii) =^ (i). (/t2 of a nondiscrete rank 1 ring.) Proof, (i) =^ (ii) a quasi-local the same ring, follows either localizations 4.2 Lemma. Let that, e = 0 or e - 1 = 0; (ii) ^> (iii) and (iii) 3]. (See also R R . If c(l) satisfies Supp / is open from the observation are equal; by [B(a), p. 83, Corollary then generated, =4> (i) when for any idempotent since two ideals A is finitely [B(a), p. 172, Exercise be a polynomial ring over (Recall that with generated 15].) R, and let I be an ideal 4.1(ii), then Supp / is open; while if c(l) satisfies and closed. e of Supp / denotes the set of 4.1(i), of primes P' of R' such that lp> 4 0 [B(a), p. 132].) Proof. Let P' be a prime of R', and let P = P' C\ R. If c(I) satisfies (ii), I p. 4 0 <$> lp 4 0 <&>c(lp) = Rp <^ c(l) t P <#►c(I)R' \t P', the nontrivial implications regular all following element. from the fact If in addition c(l) = eR l-eeP^(l-e)fi'CP'. asserts open and closed in the proof 4.3 (i) free ideal that if c(I) = Rp tot an idempotent of R , then is generated set and hence of the next Theorem. c(/p) implies Ip e, then contains a c(I) f_ P ■<$» Q.E.D. If / is a locally ma 4.2 that Let / is a projective rk / = 0 or 1. For such an ideal, by an idempotent, rk / is locally constant. then Lem- rk / = 1 on an We need this observation theorem. I be an ideal of R[X]. Then ideal of RlX], and Ai) the following is generated are equivalent: by an idempotent. (ii) R{X]/I is R-flat, and I is a finitely generated ideal of R[xl. (iii) / is a finitely generated flat ideal of R{X], and c(l) is generated by an idempotent. Proof, (i) ==> (ii): Since / is R[X]-projective, it is locally free at primes R[X] by [Ka2, p. 374, Theorem 2]. Therefore by Theorem 1.5, R[X]// By Lemma 4.2, rk / is locally [V2, p. 431, Proposition constant, and hence / is finitely 1.4]. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use of is R-flat. generated by 1972] THE FINITENESS OF 7 WHEN R[X]/¡ IS FLAT (ii) =^ (i): Lemma By Proposition 4.1, c(I) 1.6, / is locally is generated 4.2, rk / is locally free; by an idempotent. constant, and hence 403 and by Corollary By the remark by [B(a), 1.3 following p. 138, Theorem and Lemma l], / is projec- tive. (i) and (ii) =£» (iii): Projective implies (iii) =^ (ii): If / is R[X]-flat, tion 15; and p. 167, Exercise flat by [B(a), p. 28]. then it is locally 3-e]. Therefore, free [B(a), p. 116, Proposi- R[X]// is R-flat by Theorem 1.5- Q.E.D. Note trarily that the above many indeterminates (i). Also, Example 4.3 are essential. 4.4 R proof such 2.21 except shows Theorem. Let that is a finite R'// would R carry through for a polynomial for the appeal that the various be a finitely to Proposition finiteness generated projective then in arbi- 1.6 in (ii) =^» hypotheses R-algebra. R-module, ring of Theorem If I is an ideal I is a finitely of generated ideal. Proof. Let of x . in R // polynomial ated l/L R = R[x satisfies an R-integral /.(X) £ R[X] suchthat it splits. a finitely gether with Therefore, generated /j(*j), <4-5) (2) (3) mains; and, rings. generated ' finitely are finitely exists a monic the ideal of R' gener- R-module. The sequence and since generated implies l/L in / of the generators form a generating set for /. has shown 0 —► R'// is is of l/L to- Q.E.D. that the following are ideals (4.5) include in keeping flat R-modules generated are projective. projective R-modules generated. Projective of R are finitely noetherian with the terminology We can obtain characterization there the image of R-modules; is finitely 2.1] R-module, R: Locally Rings satisfying A(0) is a finite Pre-images (x ) then (1) Finitely , and hence sequence [V-, p. 432, Theorem for a ring is a finite £ I. If L denotes R /L R-module. •••,/ Vasconcelos equivalent R /L —> R // —> 0 is an exact R-projective, R // equation, fix.) by f Ax A, • • •, f (x ), then —* R /L also ., • ••, x ]. Since as a corollary of such rings, but first The following property rings, generated. quasi-semi-local of [CP], to Theorem we need we shall 4.4 rings, and docall a rather a well-known these rings interesting intermediate char- acterization. 4.6 Lemma. of a ring R is equivalent to the proper- ties of (4.5): (4) Every locally trivial ideal Proof. (1) =$» (4): Let A be locally of R is finitely generated. (0) or R p. Then R/A is locally License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use free 404 JACK OHMAND D. E. RUSH and hence R-flat. Therefore by (1), R/A 0 —»A —»R —»R/A —»0 splits, (4) ==> (1): locally fore, Let (0) or Rp by [V., 4.7 Theorem. flat p. 506] by [V., R-projective. Thus the sequence and it follows that A is principal. M be a finite M is projective is [September R-module. and hence The invariant are finitely p. 506, Proposition The following property factors generated of A4 are by (4). There- 1.3]. of a ring R is equivalent to the proper- ties of (4.5): // / is an ideal of R[x] R-module, Proof. then such that I is a finitely R{x]/l generated is a finite ideal. By Theorem 4.4, (1) =£> (5). It remains to show (5) => (4). Let A be as in (4), and let / = AR[X] + XR[X]. / is finitely generated so we must show (0) or (1), R/A generated. that by "cyclic is locally the proof flat" "finite such theorem that implies 4.7 = R/A; R-flat. if and only if A is, and since Therefore A is locally by (5), / is finitely should 1.6 1.6 shows free at primes tive and ring R-flat then probably and thus that allow R have of R[X]. Under c(I) = R, the corollary R[X] flat" can be replaced can be replaced by a poly- the real problem is to re- the property / is a finitely is "yes" that if / is an ideal generated when ideal? The main R is a domain. The next be that of a quasi-local to ask whether, generated at primes us to delete make it a true converse an affirmative "finite Of course, weakened finitely would that in 4.7. Thus, A(0) can be further question tion shows of (5) and that that the answer / is locally to this position answer the stronger to the next R. for a ring of R[X].(3) R, also hypothesis 1.5. Proposi- / is then R{X]/l shows of Propo- Moreover, imply that that proposition R{X]/I An affirmative part of the hypothesis to Theorem would ring of that is locally R-projec- / is, in fact, generated. 4.8 ideal is shows to investigate finitely is many indeterminates. R{X]/I of §2 answer of Theorem Does every The problem R-flat R{X]/I free and hence flat" by merely "flat" Question. R[X] generated. in the statement ring in finitely place case / is finitely Q.E.D. Note nomial flat Proposition. of R[X]. Then Let R{X]/l R, m be a quasi-local is R-free (if and) ring, and let only if there I be a nonzero exists a monic f £ R[X] such that I = /R[X]. Proof. R{X]/I. c(l) = 0 or R by Corollary 1.3; and since Since M is free and R is quasi-local,rk (3) Heinzer and Ohm have now proved ple involving a quasi-local R which shows / /= 0, c(l) = R. Let M = M = rkp/m M/mM [L, p. 418, this-see footnote 2. They also give an examthat the above question must be modified. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use is 1972] THE FINITENESSOF / WHEN R[X]/I IS FLAT Proposition 8l. Moreover, since M is R-flat, —*VM/mM —♦ 0 of R/?72-modules is exact. hence xfj is not an isomorphism. Then by Theorem ing coefficient polynomial 3.2, and hence 4.9 Corollary. and R[X]/I Also, that M/mM and generated M finitely is generated 0 —►I/ml But c(l) = R implies It follows / is principal is regular. the sequence 405 by a monic is a finite by some generated —►(R/?72)[X] I/ml 4 0, and R/m-module. / £ R[X] implies whose / contains lead- a monic polynomial. Let R be a ring, and let I be an ideal of R[X]. // c(l) = R is R-projective, then R[X]/l is R-finite and I is a finitely generated ideal.(i) Proof. M = R[X]/I is R-projective of R. Since c(Ip) = R _, 7^0 form 1, f/l, • • • , (f/1)', so there ly at P for every P. with prime c(l) The appropriate (ii) 4.3, while [Naj, is a finitely in the questions gives this section the 5.1 Lemma. Let prime c(l) = R imM local- 7 is finitely in this shown I is a projective if R[X]/l is paper. R-flat. of R[X] Moreover, assertion the second let X X .,•••, is that proof (i)«#^> assertion. he indeterminates, X ]/I is R-flat. if R is a valuation theorem than the theorem first ring, aroused then our interest of this theorem states, and we shall actually devote of his proof. of Nakayama's the intersection M be an R-module, ideal generalizes Nagata's theorem: is R-finite. of the first that this and elaboration of R, i.e., is the following lemma. Recall of the maximal let A be an ideal that ideals ](R) of R. of R contained in }(R), and let M' be a submodule of M such that M = M' + AM. If M = 0M , where geneous, esis {M .} is a family then P of the generate and then X ] such that R[X „•'••', 3] has with a generalization /-radical 4.9 and it was more information to an analysis We begin denotes ideal; treated considerably 1, cf, • • • , f then R[X]/I R be a ring, p. 164, Theorem generated if and only Corollary Let Then generalization and let / be an ideal of R[X ,,•■•, Nagata ring. is R-projective, theorem. that of Vasconcelos nonnoetherian of our Theorem 5. Nagata's [V,] by an idempotent if c(l) = R and R[X]/I t such M = (1, <f, • • • , f'\, R be a noetherian generated fot every Q.E.D. of the paper Let exists Therefore, by Theorem 4.4. Theorem. R p-ttee where R[X]// = R[zf]. But the hypothesis rk M i s bounded, The content Mp is and hence, by 4.8, Mp has a free basis plies generated implies of finitely generated submodules of M, and M' is homo- M= M . (4) We are indebted to W. Heinzer for showing from the original version of this corollary. us how to delete License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use an extraneous hypoth- / 406 JACK OHMAND D. E. RUSH Proof. lows that Let /VI.= M n M .. Since M = M + AM and AI is homogeneous, M . = M'. + AM .. But then hence M = Ai'. (R//)-module (where Thus, satisfies lar, if R/J R satisfies noetherian and R[Xj,..., X ]// / = ](R)), then ideal R-module zV is the properties if R// such Nakayama theorem then This includes homogeneous that N/JN R-projective of (4.5), is noetherian. This / is any ideal [Vj, it fol- lemma, and is a projective p. 508, Theorem R does also; observation the statement of R[X ,,•••, is R-flat, then / is finitely 2.1]. and in particu- will be used that if R/J X ] such in the is that generated. Let F = © F., z = 0, 1, 2, •'•'•', be a graded ring which is a generated erated flat theorem. 5.2 Theorem. finitely generated (4.5) of the next geneous M . = M. by the usual Q.E.D. If N is a finitely proof [September algebra over of F such F fí. If F Aj(F that F/M is a flat A is noetherian F „-module, and then M is a homo- M is a finitely gen- ideal of F. Proof. Let / = JÍF A. Then F/JF and therefore of F/JF. clude F/JF Taking that is noetherian pre-images M contains and is a finitely generated M/(JF and using a finitely the fact generated algebra over FQ/J, O M) is a finitely that generated ideal /VI is homogeneous, homogeneous ideal M we con- of F such that M = M' + iJF n M). Since F/M is Fn-flat, JF n M = JM; and hence M = M' + //VI. Let AL = F . DM. Since F ./M. is F.-flat ule by [B(b), for each AI. is a finite Example such in §4, [B(a), p. 28, Proposition p. 10, Corollary]. i > 0, and so each Hence each some 2.21 shows condition, it seems quite The next theorem stronger hypothesis device Thus, shows than ©z>0 (P/^A Thus, each 2l. Also, each F . is a finite as remarked of the sequences that above, F AM ■ is in Theorem to us, that this F/M of homogenizing that 5.2. F .-flat. by introducing Q.E.D. A is noetherian, as We have in the case assumption can be done being F AJ(F However, at least the homogeneous FQ-mod- F„-projective 0 —►Af . —>F . —>F ./Al. —> 0 splits. the assumption conceivable that F/M = FQ-module; so AI = M by Lemma 5.1- is needed in one indeterminate, tomary M is homogeneous, of a polynomial can be deleted provided one imposes The proof or mentioned of this a somewhat involves a new indeterminate, ring from 5.2. the cus- and then ap- plying Theorem 5.2. 5.3 Theorem. finitely generated be the canonical flat F „-module Let F = © F ., i' = 0, 1, 2, • • • , be a graded ring which is a algebra over homomorphism. for each F let M be an ideal If E AJ(F k > 0, then of F, and let A is noetherian M is a finitely generated and 2-_ñ cp: F —►F/M cp(F ■) is a ideal. Proof. Let Z be an indeterminate; let Fh = © FhkZk, where F* = £JLfl F .; and let Eh = © EhkZk, where' E* = 2k{=Q cp(F .). If <pA:Fh -» Eh denotes the License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 19721 THE FINITENESS OF 7 WHEN R[X]/¡ IS FLAT graded F.-algebra kernel M that M of cp der first generated. which We shall the case modules f denotes regular element, are also there then are indispensable). no longer of a flat lows 5.3- noetherian and rem 5.3 that has pointed module (b) submodules closure R[X j, • • • , X ]. Any finite is a ring satisfying direct (a) and [E2, pp. 116—117, is modules R-flat noetherian Note state- ring is ring, over a valuation that theorem fol- (a) R/](R) is it follows from Theo- generated satisfies direct 5 and Corollary]). are neither such then domains R is a finite on min / is a valuation since ideal (b); and sum of Prüfer also a ), k > 0, closed / is a finitely sum of Prüfer ) studied the corresponding is why Nagata's are flat, implies f (1, f, • • • , f F if R is any ring (b), then Theorem which of 5.3, when the min / contains over an integrally and this then and the condition out to us that is flat; of flat X ]// and the modules Consi- If / is an (1, <f, •••, S . _0 cp(F .) are flat, More generally, R[Xj,--., domains implies the Q.E.D. cases. of X in R[X]/I, closed 5.2 under generated. special the modules the integral the modules ring any submodule from Theorem class are just in the notation implies M is finitely of M in one indeterminate. ring in 2 indeterminates However, is flat R[X] then the from Theorem image 5-3 in some if R is integrally that W. Heinzer valid. F/M 5.3 that shown ment for a polynomial Prüfer ring R[f ] is R-flat (and we have that Theorem the equivalence X . _Q cp(F .) of Theorem We have proved (see 1, it follows of a polynomial and of F ; and it follows M is the homomorphic by discussing in §3. then Z to by </>(2 ct.Z1) = 2 cp(a )Zl, ideal Since takes conclude of R[X] defined is a homogeneous is finitely homomorphism ideal homomorphism 407 that if R domains there nor quasi-semi-local of exist and which satisfy (a) (see [EO, p. 348, Example 4.5]). BIBLIOGRAPHY [A] T. Akiba, Remarks (1964), 801-806. LbJ on generalized rings of quotients, Proc. Japan Acad. 40 MR 31 #4807. N. Bourbaki, Elements de mathématique. XXVII et XXX. Algebre commutative. Chaps. 1-6, Actualités Sei. Indust., nos. 1290, 1293, 1308, Hermann, Paris, 1961, 1964. MR 30 #2027; MR 33 #2660; MR 36 #146 [(a) Chaps. 1 and 2, (b) Chaps. 3 and 4, (c) Chaps. 5 and 6], [c] L. Claborn, 219-222. [CP] S. H. Cox, projective, [EjJ 284-291- S. Endo, [EO] [G] group is a class group, Pacific J. Math. 18 (1966), Jr. and R. L. Pendleton, Rings for which On flat modules over commutative rings, certain flat modules are MR 41 #6906J. Math. Soc. Japan 14(1962), MR 31 #3475- —-, (1963), 339-352- 343-362. abelian Trans. Amer. Math. Soc. 150 (1970), 139-156- [E2] -, MR 25 #2107[E3J Every MR 33 #408 5- D. Estes On semi-hereditary rings, Projective over modules J. Math. Soc. Japan 13(1961), polynomial rings, J. Math. Soc. 109-119Japan 15 MR 27 # 5808. and J. Ohm, Stable range in commutative rings, J. Algebra Papers in Pure 7 (1967), MR 36 #147. R. W. Gilmer, Math., no. 12, Queen's Multiplicative University, ideal Kingston, theory, Queen's Ont., 1968. MR 37 #5198. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use and Appl. 408 JACK OHMAND D. E. RUSH [Ka J I. Kaplansky, Commutative rings, Allyn and Bacon, Boston, Mass., 1970. MR 40 #7 234[Ka^j -, Projective modules, Ann. of Math. (2) 68 (1958), 372-377. MR 20 #6453[Ü S. Lang, Algebra, Addison-Wesley, Reading, Mass-, 1965- MR 33 #5416[Kr] W. Krull, Beitrage zur Arithmetik kommutative Integritatsbereiche, Math. Z. 41 (1936), 545-577. [Na.l M. Nagata, Finitely 5 (1966), 163-169LNaJ -, Flatness 9 (1969), 439-448. [Na J [No] rings over of an extension [r] 799- ring, of a commutative Local rings, Interscience , New York, 1962- J. Math. Kyoto Univ. ring, J. Math. Kyoto Univ. D. G. Northcott, Tracts A generalization J. Ohm and D. Rush, F. Richman, in Pure and Appl. Math., no. 13, MR 27 #5790. of a theorem Proc. Cambridge Philos. Soc. 55 (1959), 282-288[OR] a valuation MR 41 #191- -, Interscience generated MR 33 #1309- on the content Content modules and algebras, Generalized quotient rings, of polynomials, MR 22 #1600. Proc. Math. Scand. Amer. Math. Soc. 30 (1972). 16 (1965), 794— MR 31 #5880. [s] R. Swan, The number of generators of a module, Math. Z. 102 (1967), 318-322. MR 36 #1434. [V.J W. V. Vasconcelos, 138 (1969), 505-512. LvJ -, Ozz finitely generated flat modules, Trans. Amer. On projective modules of finite rank, Proc. Amer. Math. 430-433- MR 39 #4134[vj -, Simple flat extensions, J. Algebra 16 (1970), 105-107. [ZSJ O. Zariski Math. Soc. MR 42 #252. and P. Samuel, Commutative algebra. Higher Math., Van Nostrand, Princeton, N. J., 1958, I960. [(a) Vol. 1 (1958), (b) Vol. 2 (I960)]. Vols. Soc. MR 42 #252. 1, 2, University MR 19, 833; 22 (1969), Series in MR 22 #11006 DEPARTMENT OF MATHEMATICS, LOUISIANA STATE UNIVERSITY, BATON ROUGE, LOUISIANA 70803 Current Riverside, address California (David E. Rush): Department of Mathematics, 92502 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use University of California,