Martin Lorenz Max-Planck-Institut fur Mathematik Gottfried-Claren-Str. 26 D-5300 Bonn 3, Fed. Rep. Germany

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1 ON AFFINE ALGEBRAS Marti Lorez Max-Plack-Istitut fur Mathematik Gottfried-Clare-Str. 26 D-5300 Bo 3, Fed. Rep. Germay These otes cotai a uified approach, via bimodules, to a umber of results of Arti-Tate type. Throughout we will keep the followig otatio: k is a commutative rig (with I), ad R ad S are k-algebras. As is customary ad coveiet, (R,S)-bimodules V are assumed to have idetical k-operatios o both sides: ~v = v~ (v V, ~ 6 k) I. BIMODULES AND AFFINE ALGEBRAS LEMMA I. Let 0 --> U --> V --> W --> 0 be a short exact sequece of (R,S)-bimodules ad assume that V S ad R W are fiitely geerated, say V = Rv1+'''+RVm+U for suitable vl 6 V. If S is affie over k, the there exists a affie k-subalgebra R' ~R ad a fiitely geerated (R',S)-subbimodule U' EU such that V = R'v I R'v m + U' PROOF. Write V = wis WS k-algebra geerators for S The ad let Xl,...,x t 6 S be m w i = [ rih v h + u i v.x. = [ rij hv h + h=1 ' 1 ] h=1 uij m for suitable rih, rij h6 R ad u i, uij 6 U. Let R' ER be the k-subalgebra geerated by the rih ' s ad r ijh ' s, ad let U' cu _ be the (R',S)-subbimodule geerated by the u.'s ad u. 's. The i 13 V' = R'Vl + "'" + R'Vm + U' cotais w. l ad VhX j for all i,j,h.

2 122 Hece m = ~ V'x3 h=1 R'VhX j + U'xj ~ R'V' + U' = V' Sice V'k = kv' _ c V' it follows that I V' = V. w.s c V'S c V' i=i 1 --, whece COROLLARY I. Let R ~ S be k-algebras such that S is affie over k ad fiitely geerated as a left module over R. The S is also fiitely geerated as a left module over some affie subalgebra R' c R, with the same module geerators. PROOF. Take U = 0 ad V = RSs i the lemma. defied Recall that, for a left R-module V, the trace of V i R is by TrR(V) = [{Imf If6 HomR(V,R)} TrR(V) is a two-sided ideal of R, ad V is a ~eerator for R-mod, the category of left R-modules, if ad oly if TrR(V) = R. LEMMA 2. Let V be a (R,S)-bimodule such that R V ad V S are fiitely geerated, ad assume that S is affie over k. Suppose that R cotais a affie k-subalgebra A c R ad a fiitely geerated left ideal I with I c TrR(V) ad R = <A,I>k_algebra (=A + IA). The R is affie over k. This happes i particular if R V is a geerator for R-mod. PROOF. By assumptio o I, there exist fiitely may fi 6 HOmR(V,R) with I ~ Jim fi. After elargig I if ecessary we may therefore assume that a fiite direct sum of copies of R V maps oto I. By Lemma I, with U = 0, there exists a affie k-subalgebra

3 123 R' ~ R such that R,V is fiitely geerated. Hece R,I is also fiitely geerated, ad A,R', ad the geerators of I over R' together geerate <A,I>k_algebra = R. COROLLARY 2. Let R ~ S be k-algebras with S affie over k. Assume that S ad TrR(S) are fiitely geerated as left modules over R. The R is affie over k if ad oly if R/TrR(S) is affie over k. PROOF. Apply Lemma 2 with V = RSs ad I = TrR(S) 2. SOME APPLICATIONS (A) CORNERS OF RINGS. Assume that S is affie over k ad let 2 e = e 6 S. If SeS is fiitely geerated as left ideal of S, the ese is affie over k. (Motgomery-Small [6]). PROOF. By [6, Lemma I]I, e S is fiitely geerated as left module over ese. Now take V = es ad R = ese i Lemma 2 ad ote that R V maps oto R ~ via es ~--> ese (S 6 S) (B) MORITA EQUIVALENCE. If A ad B are Morita equivalet rigs, the there exists a (A,B)-bimodule P such that A P ad PB are fiitely geerated projective geerators for A-mod, resp. mod-b. I case A ad B are k-algebras, ad the left ad right k-operatios o P agree, we coclude from Lemma 2 that A is affie over k if ad oly if B is affie ow~r k. (Wadsworth, cf.[6,ackowledgemet]). (C) RESULTS OF ARTIN-TATE TYPE. Let R E S be k-algebras with S affie over k ad R S fiitely geerated. The R is affie over k i each of the followig cases:

4 124 i. R is a fiitely geerated left module over a commutative subalgebra ad k is Noetheria; ii. S is left Noetheria ad R ~ S is a fiite cetralizig extesio (i.e., S = Z Rx with x r = rs. for all r6r) ; i=i 1 l i iii. R S is projective ad, for each proper two-sided ideal M of R, MS ~ S (e.g., if R S is free or if R S is projective ad maximal ideals of R are localizable); iv. k is Noetheria ad, for some commutative subalqebra C ~ R, the module ( ~ is Noetheria. PROOF. (i). Oe ca clearly assume that R itself is commutative. Choose R' ~ R as i Corollary I. The R' is Noetheria, by the Hilbert basis theorem, ad hece R,R is fiitely geerated, as R,S is. Thus R is affie. (ii). Agai, Corollary I yields R' ~ R affie such that R' c S is a fiite cetralizig extesio. As S is left Noetheria, the Eisebud-Eaki theorem [3] implies that R' is likewise. Now argue as i (i). (iii). Set T = TrR(S). The TS = S, by the dual basis lea, ad so we must have T = R. (Actually, by [2], R S maps oto R R ) The result ow follows from Corollary 2. (iv). Let C c R be co~utative with ~ Noetheria ad set X = {r 6RISr ~ R} = a ~. The X = SX c TrR(RS) ad X = vn rt. ar(v+r), where v rus over a fiite geeratig set for. Therefore, ~ ~ for some. Sice ~ is Noetheria, we coclude that ~ is Noetheria, ad hece is also Noetheria. By Lea I, with V = ssc ad W = (ad with sides iterchaged), we ca fid a affie subalgebra C' c C ad a fiitely geerated (S,C'~-subbimodule X' ~ X with ~, fiitely geerated. Now C' is Noetheria ad so ( ~, is fiitely aeerated too. Moreover, sice R S is fiitely qeerated, X' is also fiitely geerated as (R,C')-bimodule, say

5 125 X' = I RX i C' i=i Now set I = Z R x., so that I is a fiitely geerated left ideal 1 i=1 of R with I _c TrR(S ), ad let A _c R be the subalgebra geerated by C' ad the geerators of (-~i-~) c'. The A is affie ad R = A + X' = A + IA. Thus Lemma 2 yields the result. [] REMARKS. (i) is a mild geeralizatio of the origial Arti-Tate Lemma [I] ad has bee observed by a umber of people. (ii) is cotaied i [6]. Usig a result of Formaek ad Jategaokar [4] istead of the Eisebud-Eaki theorem, the same proof yields versios of (ii) which work for certai fiite ormalizig o i extesios R _c S. For example, if S = Z Rx. with rx. = x.r i=i i 1 1 for certai automorphismus a i of R ad if G = <o1,...,~> acts locally fiitely o R, the the argumet goes through, because we ca the choose R' c R to be affie ad ormalized by x's. Also, -- 1 for ay fiite ormalizig extesio R ~ S, proper right ideals of R exted to proper right ideals of S [5]. Thus (iii) above applies to fiite ormalizig extesios R ~ S with R S projective. The questio as to whether the Arti-Tate lemma holds for geeral fiite ormalizig extesios R ~ S, with S left Noetheria, say, was raised i [6] ad is still ope as far as I kow. For the momet, let T deote the class of fiitely geerated left R-modules V such that TrR(V)V = V. The the assumptios i (iii) could be replaced by: R S 6 [ ad, for each proper two-sided ideal M of R, MS ~ S. Now T cotais all fiitely geerated projective modules over R as well as, clearly, all geerators of R-mod, ad T is closed uder direct sums. But I do't kow of a easy characterizatio of the modules i T. The prototype of (iv) (with C = k ) is due to Lace Small (oral commuicatio).

6 126 ACKNOWLEDGEMENT. Research supported by the Deutsche Forschugsgemeischaft/Heiseberg Programm (Lo 261/2-2). I would like to thak Lace Small for umerous iterestig coversatios about affie algebras ad other thigs. REFERENCES. [1] 1:2] [3] [4] [5] [6] E. Arti ad J.T. Tate: A ote o fiite rig extesios, J. Math. Soc. Japa 3 (1951), B. Cortze, L.W. Small ad J.T. Stafford: Decomposig overrigs, Proc. Amer. Math. Soc. 82 (1981), D. Eisebud: Subrigs of Artiia ad Noetheria rigs, Math. A. 185 (1970), E. Formaek ad A.V. Jategaokar: Subrigs of Noetheria rigs, Proc. Amer. Math. Soc. 46 (1974), M. Lorez: Fiite ormalizig extesios of rigs, Math. Z. 176 (1981), S. Motgomery ad L.W. Small: Fixed rigs of Noetheria rigs, Bull. Lodo Math. Soc. 13 (1981),