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On Two-Dimensional Regular Local Rings and a Lifting Problem Annali Della Scuola Normale Superiore Di Pisa, Classe Di Scienze 3E Série, Tome 27, No 4 (1973), P

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On Two-Dimensional Regular Local Rings and a Lifting Problem Annali Della Scuola Normale Superiore Di Pisa, Classe Di Scienze 3E Série, Tome 27, No 4 (1973), P ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA Classe di Scienze PAOLO VALABREGA On two-dimensional regular local rings and a lifting problem Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 3e série, tome 27, no 4 (1973), p. 787-807 <http://www.numdam.org/item?id=ASNSP_1973_3_27_4_787_0> © Scuola Normale Superiore, Pisa, 1973, tous droits réservés. L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ ON TWO-DIMENSIONAL REGULAR LOCAL RINGS AND A LIFTING PROBZE1VI (1) by PAOLO VILABREGA (2) Introduction. In [10] and [11], in order to prove the existence of nonecellent regu- lar local rings of characteristic 0, we showed that two classes of nonexcel- lent one-dimensional regular local rings of positive characteristic p can be lifted to characteristic 0, i.e. these DVB’8 are isomorphic with a quotient ring R/pR, where R is a two-dimensional regular local ring of characteristic 0. In the present paper we want to investigate the lifting problem for 0 from a of in characteristic p &#x3E; general point view. , Precisely we consider a one dimensional regular local ring B of positive characteristic p and prove that, if B contains a coefficient field .g and its fraction field is separably generated over (where X is a suitable parameter), then there is an isomorphism : B = where R is a two- dimensional regular local ring of characteristic 0. Moreover R can be chosen faithfully flat as an algebra over Zpz (ring of integers localized at the prime p). The translation of our lifting result into the language of Spectra shows that it is strictly related with a lifting theorem for smooth schemes (see [7], proposition 18.1.1.). We remark also that our result gives a partial answer to a lifting problem modulo p" for formally gmooth algebras, extending to the non-com. plete situation results which are well known when the rings are supposed Pervcnuto alla Redazione il 26 Gennaio 1973. (i) The present paper was written while the author was supported by the CNR an a member of the GNSAGA. (2) AuthOr’8 permanent address : Istitnto matematico delFUnrversita - via Carlo Alberto, 10 - 10123 TORINO (Italy). 788 to be noetherian complete local rings (see [5], theorem 19.7.2 and remark 19.7.3). To deal with our main preblem we need a few general results on formal series In we have to consider the power’ rings. particular following problem : let C be a discrete valutation ring, C its completion and Y an indeterminate; if t is an arbitrary formal power series in C[[Y]] does t belong to some non complete regular local ring R, with completion and containing Our positive answer, with an explicit construction of the ring R, has applications both when (where R = field and X = indeter- minate) and when C = complete discrete valutation ring with parameter the prime number p. It must be observed that this last case is really a key-point for our lifting. The lifting result given in the present paper generalizes the main result of [11]. Also the main theorem of [10] can be slightly improved by means of properties that we prove here. We are also able to produce a class of regular local rings of charac- teristic 0 and equal characteristic, which are not pseudogeometric, genera- lizing an example of Nagata (see [8], Appendix, example 7). We wish to thank Michael Artin for some useful conversations on the subject of this paper. n. 1. All the rings considered are commutative with 1. A ring A, through the present paper, is a ring with a unique maximal ideal, but not necessarily noeth,erian. We start with a few general properties of local domains and LEMMA 1 : Let A be local dontain with fraction field L, M any subfield of B = An M. Then the following properties .are true : a) B is a local d01nain domitzated by A ; i-e a principal ideal of B then PROOF: a) Let m be the maximal ideal of A. We want to see that tn n B is the unique maximal ideal of B. In fact, if x is in B - m n B), then x is invertible both in ~. and in M, hence in B. q) Let x = ay, with y E A, be in JPA n B. Then y E A i.e. 789 LEMMA 2 : Let C be any D VR with fraction field M and parameter a. If E is a 8ubfield of’ M containing a, then R = L fl c is a DVR with para- meter a. PROOF: By lemma 1, a), R is a local domain dominated by 0. By lemma 1, b), ~ce have: Therefore the maximal ideal 00 00 of R is Moreover R an C = since C is a Zariski principal. C- (0), 1 f1 1 ring ([2], chap. III, § 3, n. 3). This implies that R is a D VB with para- meter a ( [3], chap. VI, § 3, n. 6, proposition 9). REMA.R]K: Lemma 2 can be applied in the following situation : 0 = formal power series ring in the indeterminate X over the field K. Therefore, if F is any set of formal power series, then (F) n nK [[ X ]] is a With parameter X and, of course, with completion .g [[X]]. We now give a few useful properties of two-dimensional regular local rings. PROPOSITION 3 : Let C be any DVB and Y an indeterminate. If L is a sub,field of the fraction field of C [[Y]], such that then R = = Z t1 regular local ring with c01npletion 0 [[Y]I. PROOF: By lemma 1, a), R is a local domain dominated by C[[Y]]. Let now a be a parameter of C. We want to show that the maximal ideal of R is generated by a and Y. Let x be an element in R but not in 00 x If = a (a, Y) R and suppose that = I bn Yn. bo aco ,1 for suitable co E 0, n says that x E (a, Y) B. Therefore ba must be invertible in 0 and x is inver- tible in R. Now we want to show that B is a noetherian local ring. For this pnrpose, let’s first consider the. quotient field K = ClaO and the ring C [~ Y ]]/a C [[ Y J] = g [[ Y J J. We want to see that BlaR is a DVR. But we have: by definition of R. Therefore BlaB is a subring of K[[Y]], with maximal ideal generated by Y and separated in its Y-topo- logy. This says that is a D VR ( [3], chap. VI, § 3, n. 6, proposition 9). Let’s now go back to R. To see noetherian property it is enough to show that every prime ideal is finitely generated ([8], chap. 1, theorem 3.4). Of course every prime ideal containing a is finitely generated; therefore we can assume that P is a prime ideal such that a f 1. 790 The image of 1P modulo aR is a principal ideal Yr (BlaR), where r is a suitable integer ~&#x3E; 1 ; sue we can select b in 1P such that its image modulo aR is Yr . Then we obtain the following inclusions : If x E f) then x = bc~ + ad~ , where c, and d, are suitable elements which implies: It is now clear that, for each n, we can choose en and dn such that Therefore we deduce the following inclusions : Extending the ideals to 0 [[Y]] we obtain: the last equality depending on the fact that 0 ([h]~ is a Zariski ring for the aC [[Y]]-topology ([2], chap. III, § 3, n. 3). Hence we have : Therefore lemma 1, b) says that fl5 = bR. So R is a local ring satisfying the following properties: (i) R is noetherian; (ii) the maximal ideal of R is generated by a and Y; Therefore we deduce that R has completion 0[[YI] ([2], chap. III, § 3, n. 5, proposition 11). Hence R itself is regular ( [9J, vol. II, chap. VIII, ~. 11). A useful application of proposition 3 is the following key-result to deal with our lifting problem : THEOREM 4 : Let 0 be any DVR, Y an indeterminate over it8 completion an arbitrary fort)ial power series in 791 Then there is a local ring R satisfying the following condition8: + (ii) R is regular with co?npletion 0 Y]];J (iii) R is the smallest ring satisfying (i) and (ii) (I.6. if 8 satisfiø8 (i) and (ii), then B r- S). Fitrthermore such a ring R is not C0filp16t6 and its fraction field is gene- rated over the fraction field of 0 [Y] by t, ao, .,. ~ I.... PROOF: Let’s put: L = fraction field of By lemma 2, B is a D VR with completion C. Now we consider the fraction field of B [Y, t], say K, and put : .R = By proposition 3, R is regular local with completion C [[Y]J ; hence it satisfies requirements (i) and (ii). Property (iii) depends on the fact that a regular local ring R with "- fraction field 2T and completion R satisfies always the following~ equality : ([8], chap.
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