F-Rational Rings and the Integral Closures of Ideals

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1 Mchgan Math. J. 49 (2001) F-Ratonal Rngs and the Integral Closures of Ideals Ian M. Aberbach & Crag Huneke 1. Introducton The hstory of the Brançon Skoda theorem and ts ensung avatars n commutatve algebra have been well documented n many papers (see e.g. [AH1; LS]). We wll therefore only brefly revew the relevant concepts and theorems. Frst recall the defntons of the ntegral closure of an deal. Defnton 1.1. Let R be a rng and let I be an deal of R. An element x R s ntegral over I f x satsfes an equaton of the form x n + a 1 x n 1 + +a n = 0, where a j I j for 1 j n. The ntegral closure of I, denoted by I,s the set of all elements ntegral over I. Ths set s an deal. Let R o be the set of all elements of R not n a mnmal prme. An equvalent though less standard (but for our purposes a more useful) defnton of ntegral closure s the followng. Equvalent Defnton 1.1. Let R be a Noetheran rng and let I be an deal of R. An element x R s ntegral over I f there exsts an element c R o such that cx n I n for all n 0. A theorem proved by Brançon and Skoda [BS] for convergent power seres over the complex numbers and generalzed to arbtrary regular local rngs by Lpman and Sathaye states as follows. Theorem 1.2 [BS; LS]. Let R be a regular local rng and let I be an deal generated by l elements. Then, for all n l, I n I n l+1. Ths was partally extended to the class of pseudo-ratonal rngs by Lpman and Tesser [LT]. However, they were unable to recover the full strength of Theorem 1.2. Theorem 1.3 [LT, (2.2)]. Let R be a Noetheran local rng and assume that the localzaton R P s pseudo-ratonal for every prme deal P n R. Suppose that I has a reducton J such that dm R P δ for every assocated prme P of J n. Then Receved February 22, Revson receved October 10, Both authors were partally supported by the NSF. 3

2 4 Ian M. Aberbach & Crag Huneke I n+δ 1 J n. In partcular, f J can be generated by a regular sequence of length δ, then the above contanment holds for all n 1. The present two authors, as well as Lpman, have pushed the orgnal theorem further by ntroducng coeffcents ; see [AH1; AH2; AHT; L]. The methods used by the present authors have reled on the theory of tght closure. These mprovements, however, have been vald only n regular rngs, and the queston of whether the statement of Theorem 1.2 remans vald n arbtrary pseudo-ratonal rngs has remaned open snce Recent progress was made by Hyry and Vllamayor [HyV], who proved (among other thngs) that f R s local Gorensten and essentally of fnte type over a feld of characterstc 0, then I n+l 1 I n for an arbtrary deal I wth l generators. In ths paper we wll use tght closure methods to prove that Theorem 1.2 s vald for F-ratonal rngs (the defnton s n Secton 2). In characterstc p, Smth [Sm] proved that F-ratonal mples pseudo-ratonal, but t can be stronger n general. However, for affne algebras n equcharacterstc 0, the concepts of ratonal sngularty, pseudo-ratonal sngularty, and F-ratonal type all agree, owng to work of Lpman and Tesser [LT] for the equvalence of ratonal sngularty and pseudo-ratonal sngularty, and of Smth [Sm] and Hara [Ha] and ndependently Mehta and Srnvas [MS] for the equvalence of ratonal sngularty and F-ratonal type (Smth proved that ratonal mples F-ratonal type and the other authors have just recently proved the converse). It follows from these equvalences that, n equcharacterstc 0, we are able to prove Theorem 1.2 for ratonal sngulartes. The basc dea of ths paper s nspred by the proof of a cancellaton theorem (see [Hu1]). The key dea s to relate an arbtrary deal I to a system of parameters n a manner that closely approxmates the structure of the powers of I. We do ths by usng frst a basc constructon and then a theorem that relates the ntegral closure of powers of I wth the tght closure of the system of parameters. In the next secton we brefly dscuss tght closure; see [HH1; Hu2] for more references and nformaton. We begn wth the defnton. 2. Tght Closure Defnton 2.1. Let R be a Noetheran rng of characterstc p>0. Let I be an deal of R. An element x R s sad to be n the tght closure of I f there exsts an element c R o such that cx q I [q] for all large q = p e, where I [q] s the deal generated by the qth powers of all elements of I. Every deal n a regular rng s tghtly closed. We say that elements x 1,...,x n n R are parameters f the heght of the deal generated by them s at least n (we allow them to be the whole rng, n whch case the heght s sad to be ). If the deal they generate s proper, then the Krull heght theorem says that the heght s exactly n.

3 F-Ratonal Rngs and the Integral Closures of Ideals 5 Defnton 2.2. A Noetheran rng R of characterstc p>0 s sad to be F- ratonal f the deals generated by parameters are tghtly closed. Ths defnton arose from the work of Fedder and Watanabe [FW] because of the apparent connecton to the concept of ratonal sngulartes. The concept of pseudo-ratonalty was ntroduced n [LT], partly as a substtute for the noton of ratonal sngulartes n postve and mxed characterstc, where desngularzatons are not known to exst n general. Ther defnton s as follows (see [LT, Sec. 2]). Defnton 2.3. Let (R, m) be a d-dmensonal local Noetheran rng. The rng R s sad to be pseudo-ratonal f t s normal, Cohen Macaulay, and analytcally unramfed and f, for every proper bratonal map π : W X = Spec(R) wth W normal and closed fber E = π 1 (m), the canoncal map Hm d (π (O W )) = Hm d (R) H E d (O W) s njectve. In [LT] t s proved that, for a local rng essentally of fnte type over a feld of characterstc 0, the notons of pseudo-ratonal and ratonal sngularty agree. In [Sm] t s shown that, n postve characterstc, F-ratonal mples pseudo-ratonal. Smth uses ths to prove that rngs of fnte type over a feld of characterstc 0 that are F-ratonal type have ratonal sngulartes. Here F-ratonal type essentally means that characterstc-p models of the varety are F-ratonal. More precsely, we next ntroduce the dea of a model. Let R be a rng that s fntely generated over a feld of characterstc 0, say R = k[x 1,...,X n ]/I. Then we can choose generators for the deal I and, by collectng coeffcents of those generators, fnd a fntely generated Z-algebra A k such that defnng R A = A[X 1,...,X n ]/(I A[X 1,...,X n ]) yelds R = k A R A. We call the map A R A a famly of models of R. We sometmes nsst that the map A R A be flat, whch one can always obtan by expandng A by localzng at a sngle element. A typcal closed fber of R A over A s a characterstc-p model of R. Defnton 2.5. Let R be a fntely generated algebra over a feld of characterstc 0. Then R s sad to have F-ratonal type f R admts a famly of models A R A n whch a Zarsk dense set of closed fbers are F-ratonal. (Ths does not depend on the choce of models.) The theorem n [Sm] states that, f X s a scheme of fnte type over a feld of characterstc 0, then f X has F-ratonal type t has only ratonal sngulartes. Recently, the converse has been proved by Hara [Ha] and ndependently by Mehta and Srnvas [MS]. 3. F-Ratonal Rngs and Tght Closure In ths secton we frst dscuss a basc constructon that wll play a crucal role n the paper. Gven an deal I n a Noetheran local rng (R, m), a mnmal reducton

4 6 Ian M. Aberbach & Crag Huneke J of I say, J = (a 1,...,a l ) and an nteger N, we wsh to construct an deal A generated by parameters such that J A modulo m N and such that A s closely related to I and ts powers. For example, one would lke I A, but ths s n general not possble snce I may not be contaned n any deal generated by parameters. We record what we need n Proposton 3.2. We need the followng lemma from [AHT]. Lemma 3.1. Let (R, m) be a local rng wth nfnte resdue feld and let I R be an deal of analytc spread l. Let J I be a mnmal reducton of I. Then there exsts a basc generatng set a 1,...,a l for J such that (1) f P s a prme deal contanng I and ht P = l then (a 1,...,a ) P s a reducton of I P, and (2) ht((a 1,...,a )I n : I n+1 + I) + 1 for all n 0. (3) If c a modulo I 2, then (1) and (2) hold wth c replacng a. Proof. The frst two statements are found n [AHT, Lemma 7.2]. The last statement follows from the proof of Lemma 7.2 n [AHT]. The choce of a basc generatng set depends only on the mages of the a n the assocated graded rng G = R/I I/I 2. In partcular, snce c and a have the same leadng forms n G, (3) follows. Proposton 3.2. Let (R, m) be an equdmensonal and catenary local rng wth nfnte resdue feld and let I R be an deal of analytc spread l. Let J I be a mnmal reducton of I. We assume that ht I = g and J = (a 1,...,a l ), a basc generatng set for J as n Lemma 3.1. Let N and w be fxed ntegers, and suppose that for g + 1 l we are gven fnte sets of prmes ={Q j } all contanng I and of heght. Then there exst elements a 1,...,a l and t g+1,...,t l such that the followng hold (we set t = 0 for g for convenence): (1) a a modulo I 2 ; (2) for g + 1 l, t m N ; (3) b 1,...,b g,b g+1,...,b l are parameters, where b = a + t ; (4) f R/I s equdmensonal then the mages of t g+1,...,t l n R/I are parameters; (5) there s an nteger M such that t +1 (J ti M : I M+t ) for all 0 t w + l, where J = (a 1,...,a ); (6) t +1 / j Q j, where the unon s over the prmes n. Proof. We choose the a and t nductvely. We frst modfy a 1,...,a g to a 1,...,a g n such a way that these elements form parameters. We can do ths wth a a modulo I 2 for 1 g. Suppose we have chosen a 1,...,a and t 1,...,t so that all sx of the lsted statements are true for these elements. Fx the mnmal prmes P 1,...,P k (all necessarly of heght ) above B = (b 1,...,b ). Dvde them nto two sets: let P 1,...,P n be the ones that contan I, and let P n+1,...,p k be those that don t contan I. We frst change a +1 to an element a +1 a +1 modulo J 2 such that a +1 / k j=n+1 P j. Ths choce s possble because the nlradcal of J s

5 F-Ratonal Rngs and the Integral Closures of Ideals 7 the same as the nlradcal of I. Next choose M such that the heght of I +(J I M : I M+1 ) s at least + 1, and choose M to be the maxmum of the M. (Ths s possble by Lemma 3.1.) Ths choce forces all (J ti M : I M+t ) + I to be of heght at least + 1 for all t 0. For suppose that (J ti M : I M+t ) + I Q, where Q s a prme of heght at most. Snce I Q, ths forces (J I M : I M+1 ) Q, and after localzaton at Q we have (I M+1 ) Q = (J I M ) Q. But ths forces (I M+t ) Q = (J ti M ) Q for all ntegers t, and so (J ti M : I M+t ) Q, a contradcton. Usng prme avodance, choose t +1 w+l t=0 (J ti M : I M+t ) m N ( k j=n+1 P ) j and t +1 / ( n j=1 P ) ( j j Q j). Ths s possble because I s contaned n each of the prmes n the second lne and all these prmes have heght, whle the heght of I +(J ti M : I M+t ) s at least +1. We set b +1 = a +1 + t +1. We clam ths choce proves (1) (6) for these new elements. Our choce of a +1 and t +1 make statements (1), (2), (5), and (6) trval. To prove (3) we need only prove b +1 / k j=1 P j. If j n, then a +1 I P j whle t +1 / P j. Hence b +1 / P j. If j n + 1, then a +1 / P j whle t +1 P j. Agan b +1 / P j, provng (3). Statement (4) follows from (3). Clearly the heght of (I, b g+1,...,b +1 ) s at least that of b 1,...,b +1 and hence at least + 1. But (I, b g+1,...,b +1 ) = (I, t g+1,...,t +1 ). Snce R s equdmensonal and catenary, t follows that the mages of the t j n R/I form parameters. Theorem 3.3. Let (R, m) be an equdmensonal and catenary local rng of characterstc p havng nfnte resdue feld. Let I be an deal of analytc spread l and postve heght g. Let J be a mnmal reducton of I. Fx w, N 0. Choose a and t as n Proposton 3.2. Set A = B l = (b 1,...,b g,...,b l ). Then I l+w (A w+1 ). Proof. Our choce of elements means that a 1,...,a g,a g+1 + t g+1,...,a +1 + t +1 s part of a system of parameters. Fx the notaton as n Proposton 3.2. By our choce of the t j we have that t j I M+k Jj 1 k I M for all 1 k w + l. We frst clam that ths mples tj n I M+nk Jj 1 nk I M for all n 1. Assume ths s true for a fxed n, and multply by t j I k. We obtan that (t j I k )(tj ni M+nk ) (t j I k )Jj 1 nk I M. Snce t j I M+k Jj 1 k I M, we now have t n+1 j I M+(n+1)k Jj 1 nk J j 1 k I M as requred. Fx c I M R o. Note that the above contanment shows that, for all n 1, ct n+1 j I (n+1)k J (n+1)k j 1. (3.4) Set B = (b 1,...,b ). Let g l and w r 0. We show by nducton that c g J (+r)q (B r+1 ) [q]. The base case s when = g and r w s arbtrary.

6 8 Ian M. Aberbach & Crag Huneke In ths case J g (g+r)q (Jg r+1 ) [q] = (Bg r+1 ) [q]. The frst ncluson n the above lne follows at once from [HH1, proof of (5.4)]. Assume now that we are gven r and >gand that the clam s true ether for <(wth r w arbtrary) or for = (wth r <r w). By our choce of c and of the t j, c g J (+r)q c g J [q] J (+r 1)q c g [J g [q] J (+r 1)q + a q g+1 J (+r 1)q + +a q J (+r 1)q ] = c g 1 [cj g [q] J (+r 1)q + ca q g+1 J (+r 1)q + +ca q J (+r 1)q ]. Consder a typcal term n ths sum, ca q j J (+r 1)q, where g + 1 j. Snce b j = a j + t j, we can wrte ths term as ca q j J (+r 1)q = cb q j J (+r 1)q ct q j J (+r 1)q. Usng (3.4) (note + r 1 w + l), we obtan and so c g J (+r)q ca q j J (+r 1)q cb q j J (+r 1)q + J (+r 1)q j 1 c g 1 [cj g [q] J (+r 1)q + (cb q g+1 J (+r 1)q + J g (+r 1)q ) + +(cb q J (+r 1)q whch by the nducton hypothess s contaned n + J (+r 1)q 1 )], J g [q] (B r )[q] + b q g+1 (B r )[q] + (B r+ g b ) [q] + +b q (B r )[q] + (B r+1 1 )[q] (B r+1 ) [q]. In partcular, note that c l g J (l+r)q l (B r+1 l ) [q] (3.5) for all r w. We now prove that I l+w (A w+1 ). Let u I l+w. Choose an element d R o such that du q J (l+w)q. Then c l g du q c l g J (l+w)q (B w+1 l ) [q] by (3.5). It follows that u (B w+1 l ) = (A w+1 ). Remark. Theorem 3.3 s stll vald even f ht(i ) = 0. In ths case, choose c 1 I M and c 2 n the ntersecton of the mnmal prmes of 0 that do not contan I and avodng those that do contan I. Thus c 2 I N = 0 for N 0 and c = c 1 + c 2 R o satsfes equaton (3.4). An almost mmedate consequence s one of our man theorems. Theorem 3.6. Let (R, m) be an F-ratonal local rng of postve characterstc p, and let I R be an deal generated by l elements. Then I l+w I w+1 for all w 0.

7 F-Ratonal Rngs and the Integral Closures of Ideals 9 Proof. There s no loss of generalty n assumng that R has an nfnte resdue feld. We can replace I by a mnmal reducton of tself; suppose that J s that mnmal reducton. The number of generators of J s at most l, so wthout loss of generalty we may assume l s the number of generators of J. Fx an nteger N. We thnk of w as fxed and choose t g+1,...,t l and a 1,...,a l as n Proposton 3.2. In partcular, t m N for all. By (3.3), I l+w (A w+1 ) = A w+1 J w+1 + (t h+1,...,t l ) J w+1 + m N. The equalty (A w+1 ) = A w+1 above follows from [A, Thm. 1.1]. By the Krull ntersecton theorem we obtan that I l+w N (J w+1 + m N ) = J w+1. Ths characterstc-p theorem allows us to prove the same result n equcharacterstc 0. Theorem 3.7. Let R be an algebra of fnte type over a feld of characterstc 0 and havng only ratonal sngulartes. Let I R be an deal generated by l elements. Then I l+w I w+1 for all w 0. Proof. By the work of both Hara [Ha] and Mehta and Srnvas [MS], R s of F-ratonal type. It s straghtforward to prove n ths case that, f the concluson holds n a dense open set of fbers n some famly of models A R A of R, t also holds n R. Hence we may pass to postve characterstc and assume that R s fntely generated over a feld of characterstc p>0 such that R P s F-ratonal for all prmes P. The concluson wll follow f we prove t locally, snce the number of generators can only drop after localzaton. It follows that we can reduce to the local F-ratonal case and apply Theorem 3.6 to fnsh the proof. 4. F-Ratonal Gorensten Rngs Our next theorem s new, even for R regular, as far as we know. The proof s based on a careful analyss of the proof of Theorem 3.5 together wth the deas behnd the cancellaton theorem of [Hu1] (see also [CP] for further cancellaton results). Our man theorem apples to rngs that are F-ratonal and Gorensten. It s known [HH2, (3.4), (4.7)] that F-ratonal and F-regular are the same when the base rng s Gorensten. A rng R s F-regular f every deal s tghtly closed n every localzaton of R. Of course, all regular rngs are F-regular, but the class of F-regular rngs s consderably broader than that of regular rngs. Theorem 4.1. Let (R, m) be an F-ratonal Gorensten local rng of dmenson d and havng postve characterstc. Suppose that I s an deal of heght g and analytc spread l > gwth R/I Cohen Macaulay. Then, for any reducton J of I, I l 1 J. Proof. There s no loss of generalty n assumng that R has an nfnte resdue feld and that J s a mnmal reducton. Fx an nteger N and set w = 0 n the notaton of Proposton 3.2 and Theorem 3.3. We wll prove that I l 1 J + m N. An applcaton of the Krull ntersecton theorem then fnshes the proof.

8 10 Ian M. Aberbach & Crag Huneke We choose t g+1,...,t l and a 1,...,a l as n Proposton 3.2, wth N fxed as before. Let b = a +t for 1 l. Choose x = x l+1,...,x d so that b g+1,...,b l,x s a regular sequence on R/I and set A = (b 1,...,b l, x). We set D = J g : t g+1 and K = (J g,b g+2,...,b l, x). Let Q = (I, b g+2,...,b l, x) + K : D. We clam that A : t g+1 Q. Suppose that t g+1 u = w + vb g+1, (4.2) where w K. Then t g+1 (u v) (J g+1,b g+2,...,b l, x) and hence u v (J g+1,b g+2,...,b l, x) : b g+1 (I, b g+2,...,b l, x) : b g+1 (I, b g+2,...,b l, x) snce R/I s Cohen Macaulay. Hence u v Q and to prove u Q t suffces to show that v K : D. Let d D and consder dv. Usng (4.2), we obtan that t g+1 du = dw + dvb g+1 and hence dvb g+1 (J g,b g+2,...,b l, x). Thus Dv (J g,b g+2,...,b l, x) : b g+1 = (J g,b g+2,...,b l, x) = K. Ths proves our clam and n partcular proves that A : Q A : (A : t g+1 ). We next clam that I l 1 A : Q. Frst observe that (I, b g+2,...,b l, x) I l 1 I I l 1 + A and, by Theorem 2.6, I I l 1 A (usng that R s F-ratonal). Hence t remans only to prove that I l 1 (K : D) A. We use a lemma. Lemma 4.3. Wth the same notaton as before, t g+1 I l 1 J g. Proof. Let z I l 1 and choose an element d R o such that dz n I n(l 1) for all n. Choose c I M nonzero as n (3.4). Usng (3.4), we then obtan dct q g+1 zq ct q g+1 I q(l 1) t q g+1 I q(l 1)+M J q(l 1) g J [q] g, where the last contanment follows because l 1 g and J g has g generators. Hence t g+1 z (J g ). Snce R s F-ratonal, t g+1 z J g, provng the lemma. Lemma 4.3 proves that I l 1 D. Hence I l 1 ((J g,b g+2,...,b l, x) : D) A. We have proved that I l 1 A : Q. By local dualty, we have I l 1 A : Q A : (A : t g+1 ) (J g+1,t g+1, b g+2,...,b l, x) (J, t g+1,...,t l, x) J + m N. Acknowledgment. The authors thank Renhold Hübl for valuable correctons n our orgnal preprnt. References [A] I. M. Aberbach, Tght closure n F-ratonal rngs, Nagoya Math. J. 135 (1994), [AH1] I. M. Aberbach and C. Huneke, An mproved Brançon Skoda theorem wth applcatons to the Cohen Macaulayness of Rees algebras, Math. Ann. 297 (1993),

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