Graded Modules of Gelfand Kirillov Dimension One over Three-Dimensional Artin Schelter Regular Algebras

Transcription

1 Ž JOURNAL OF ALGEBRA 196, ARTICLE NO JA Graded Modules of Gelfandrllov Dmenson One over Three-Dmensonal ArtnSchelter Regular Algebras Martne Van Gastel and Mchel Van den Bergh Departement WNI, Lmburgs, Unerstar Centrum, Unerstare Campus, Buldng D, 3590, Depenbeek, Belgum Communcated by J T Stafford Receved December 12, 1996 Let A be a three dmensonal ArtnSchelter regular algebra We gve a descrpton of the category of fntely generated A-modules of Gelfandrllov dmenson one Ž modulo those of fnte dmenson over the ground feld The proof s based upon a result by Gabrel whch says that locally fnte categores can be descrbed by module categores over topologcal rngs 1997 Academc Press CONTENTS 1 Introducton 2 Notatons and conentons 3 Pseudocompact rngs 4 A matrx representaton for pseudocompact rngs 5 Global dmenson 6 A classfcaton problem 7 More classfcaton 8 Proof of Theorem 11 * E-mal address: mvgastel@lucacbe The second author s a senor researcher at the NFWO E-mal address: vdbergh@ lucacbe $2500 Copyrght 1997 by Academc Press All rghts of reproducton n any form reserved

2 252 VAN GASTEL AND VAN DEN BERGH 1 INTRODUCTION Let us start wth our man result Afterwards we wll ndcate the applcaton we have n mnd k s an algebracally closed feld THEOREM 11 Let A be a k-lnear locally noetheran Grothendeck category Žthat s, an abelan category whch satsfes AB5 and has a famly of noetheran generators Let G : A A be an autoequalence and let : G d be a natural transformaton such that Ž F A s surjecte for eery njecte object n A Let B be the full subcategory of A consstng of objects M wth Ž M 0 and let Cf be the full subcategory of A consstng of fnte length objects whose composton factors le n B Assume that eery smple object n B has fnte njecte dmenson n A and furthermore that there s a CohenMacaulay cure Yoer k such that B s equalent to QchŽ Y, the category of quas-coherent OY-modules For x Y denote by Px the smple object of B correspondng to x Then we hae the followng Ž 1 1 There s a bjecton : Y Y such that G Ž P x P x Ž 2 C f zy²: C f, z, where Cf, z s the full subcategory of Cf con- sstng of objects whose JordanHolder quotents are gen by Py wth yo Ž z Ž 3 There s a category equalence F between Cf, z and the category of fnte dmensonal rght modules oer a rng C z Ths rng Cz has the followng form: Ž a If O Ž z then Cz s gen by the lower trangular matrces wth entres n O ˆY, z In ths case z s regular on Y and thus we hae Oˆ k x Y, z Ž b If O Ž z r n then Cz s gen by a rng of n n matrces of the form 0 R RU RU RU R R where R s a complete local rng of the form R k²² x, y:: Ž, wth yx qxy hgher terms Ž 11

3 GRADED MODULES OF GELFANDIRILLOV 253 for some q k, or yx xy x 2 hgher terms Ž 12 U s a regular normalzng element n radž R such that RŽ U O ˆY, z If z s not fxed under then z s regular on Y and also U rad 2 Ž R Ž 4 Let I f O Ž z and I n f O Ž z n In ths way the elements of Cz correspond to I I-matrces For I let e be the correspondng dagonal dempotent Then eery fnte dmenson rght Cz-repre- sentaton W satsfes W We Ž 5 Put S ecradž ec Then FŽ P z z z S Ž 6 Defne the followng normal element N of C z Ž a If O Ž z then N s gen by the matrx whose entres are eerywhere zero except on the lower subdagonal where they are one Ž b If O Ž z then U 1 0 N Let N N 1 Then we hae the followng commutate dagram C G f, z Cf, z F Ž r z r z Mod Ž C Mod Ž C F where Mod Ž C r z denotes the category of rght Cz-modules Ž 7 If M s an object n Cf, z then one has the followng commutate dagram FŽ Ž M FGŽ M FŽ M N FŽ M FŽ M Ž 8 Let C be the pullback of C n QchŽ Y f, z, Y f, z Thus the objects of C are the fnte length object n QchŽ Y f, z, Y whose support s contaned n the -orbt of z Put D C Ž N Ł O ˆ Let Ž ˆ be a shorthand for the z z Y, p z

4 254 VAN GASTEL AND VAN DEN BERGH ˆ z product of the complecton functors Ž Then the followng dagram s commutate C C f, z, Y f, z Ž ˆ z F Mod Ž D Mod Ž C r z r z From ths theorem we can extract the followng corollary COROLLARY 12 If z s not a fxed pont for then z s regular on Y If we thnk of the curve Y as beng embedded n a knd of non-commutatve space A then Theorem 11 gves us some nsght nto the structure of A n a neghborhood of Y Now for the applcaton Recall that an Ž ArtnSchelter regular algebra s a graded algebra A k A1 wth the followng propertes A has polynomal growth A has fnte global dmenson A s Gorensten That s, there s a number such that k f Ext AŽ k A, A ½ 0 otherwse If A s generated n degree one and has global dmenson three then accordng to 3 there are two possbltes: A has 2 generators n degree one and 2 relatons n degree three A has 3 generators n degree one and 3 relatons n degree two Assume that A s a regular algebra of dmenson three generated n degree one In order to understand the geometrcal propertes of A, people have studed varous classes of graded A-modules 5, 1 We wll now show that Theorem 11 apples to A and hence can be used to gve a descrpton of the category of fntely generated A-modules of Gelfandrllov dmenson one Žmodulo those of fnte dmenson over k We frst ntroduce a few notons from the theory of graded rngs Assume that A k A1 s a rght noetheran graded rng Let GrŽ A be the category of graded rght A-modules TorsŽ A GrŽ A s the category of graded A-modules whch are drected unons of rght bounded ones Followng 2 we put QGrŽ A GrŽ A TorsŽ A and we let :GrŽ A QGrŽ A be the quotent map We denote by s the automorphsm of GrŽ A whch sends M to MŽ 1 snduces an automorphsm on

5 GRADED MODULES OF GELFANDIRILLOV 255 QGrŽ A whch we denote by the same letter By Proj A we denote the couple ŽQGrŽ A, A Let Ž X,, L be a trple where X s a noetheran scheme, AutŽ X, Ž Ž and L Pc X Defne B n Bn wth Bn X, L L n1 L B s a grade rng wth multplcaton a b ab m, a B m, b B n we call B the twsted homogeneous coordnate rng assocated to the trple Ž X,, L 6 Let QchŽ X be the category of quas-coherent OX-modules We say that Ž Ž 1 L s ample f the functor s : Qch X Qch X : M M L 1 1 Ž n has the property that for every coherent M one has H X, s M 0 and s n M s generated by global sectons for large n The key result of 6 s that we have functors Ž GrŽ B QchŽ X Ž 13 QchŽ X GrŽ B whch, f L s -ample, factor though to gve nverse equvalences between QGrŽ B and QchŽ X Now let A be agan a regular algebra of dmenson three, generated n degree one Accordng to 4, A possesses a regular normalzng element g s degree three or four Ždependng on whether A has three or two generators such that B AŽ g s a twsted homogeneous coordnate rng assocated to a trple Ž Y,, L wth Y a plane curve of arthmetc genus one and L O Ž Y 1 Basc objects n the geometrcal study of A are the so-called pont modules These are graded A-modules, generated n degree zero, whch are one-dmensonal over k n every degree All such pont modules are annhlated by g 4, 5 and hence t follows easly from the category equvalence Ž 13 that they are of the form P ŽkŽ x x for x Y Put PxP x Let G be the autoequvalence of Gr Ž A gven by A ga and denote by the same letter the nduced autoequvalence on QGr Ž A The natural transformaton Ž M s the obvous map GŽ M M obtaned form the ncluson ga A It s now clear that the hypothess for Theorem 11 are satsfed, whence we can apply that theorem n order to gve a descrpton of the category of A-modules of Gelfandrlov dmenson one modulo those of fnte dmenson Theorem 11 has also some sgnfcance for the study of non-commutatve surfaces 7 These are graded rngs of Gelfandrllov dmenson three satsfyng some sutable regularty propertes At present t s unclear why, but noncommutatve surfaces whch are suffcently generc seem to possess a regular normalzng element of postve degree, such that factor-

6 256 VAN GASTEL AND VAN DEN BERGH ng by t yelds a twsted homogeneous coordnate rng of a Cohen Macaulay curve Ž at least n hgh degree The proof of Theorem 11 s based upon a result by Gabrel 8 statng that locally fnte categores are dual to pseudocompact rngs Pseudocompact rngs are rngs equpped wth an especally nce topology Sectons 35 wll be devoted to some generaltes concernng pseudocompact rngs We are especally nterested n the relatonshp between topologcal and non-topologcal propertes of such rngs Ths should remove the anxety some readers mght experence when confronted wth topologcal rngs In Sectons 6, 7 we gve some classfcaton theorems whch are slghtly more general than what we need for the proof of Theorem 11 Fnally n Secton 8 we gve the proof of Theorem 11 2 NOTATIONS AND CONVENTIONS If C s a rng then ModŽ C refers to the category of left modules over C The category of rght modules s denoted by Mod Ž C r Note that n the Introducton GrŽ A was used to denote the category of graded rght-modules over a graded rng A An unspecfed module wll always be a left module If M s a left module over a rng C and s an automorphsm of C then M s the left C-module whch s equal to M as a set, but whch has ts multplcaton twsted by : c m Ž cm A smlar notaton s used for rght modules 3 PSEUDOCOMPACT RINGS Recall that a left topologcal module M over a topologcal rng A s pseudocompact f t s Hausdorf, complete and ts topology s generated by left submodules of fnte colength Ž not necessarly by all such submodules A s sad to be a pseudocompact rng f A s pseudocompact as a left A-module In the rest of ths secton A wll be a pseudocompact rng The category of pseudocompact modules over A s denoted by PCŽ A It s an abelan category satsfyng AB5 and AB3 8 Its dual category s a locally fnte category, that s, a Grothendeck category possessng a set of generators of fnte length Conversely assume that C s a locally fnte category If M, N C then the natural topology on Hom Ž M, N s the lnear topology generated by C

7 GRADED MODULES OF GELFANDIRILLOV 257 the subgroups of the form Ž S f : M N fž S 0 4, where S runs through the objects of fnte length n C The followng result s proved n 8 THEOREM 31 If E s an njecte cogenerator for C then A End Ž E C, equpped wth the natural topology, s a pseudocompact rng, and the functor whch sends M C to Hom Ž M, E Ž wth the natural topology C s an equalence of categores between C and PCŽ A One easy property of a lnear topology wll be used repeatedly below LEMMA 32 Assume that M s a topologcal group wth a topology generated by subgroups and L M s an open subgroup Then L s also closed and the quotent topology on ML s dscrete Proof L s the complement of the unon of cosets of L n M whch are not equal to L Snce ths unon s a unon of open sets, t s tself open Thus L s closed Let e be the unt of M L s the nverse mage of e n ML and hence e4 s open and closed n the quotent topology The followng proposton records for further reference some of the propertes of the forgetful functor PCŽ A ModŽ A 8 PROPOSITION 33 The forgetful functor PCŽ A ModŽ A s fathful and commutes wth kernels, cokernels, and products In partcular t reflects somorphsm and exactness If M PCŽ A then the subobjects of M n PCŽ A are n one-one correspondence wth the subobjects of M n ModŽ A whch are closed Let FnŽ A be the full subcategory of ModŽ A consstng of objects whch are of fnte length and let PCFnŽ A be ts pullback n PCŽ A A module of fnte length carryng a lnear topology can only be separated f ts topology s dscrete So we conclude mmedately that the forgetful functor PCFnŽ A FnŽ A s fully fathful The followng lemma gves us more nformaton on Ž PCFn A LEMMA 34 Ž 1 An object n PCŽ A s smple n PCŽ A f and only f t s smple n ModŽ A Ž 2 The objects n PCFnŽ A are precsely the fnte length objects n PCŽ A

8 258 VAN GASTEL AND VAN DEN BERGH Proof Ž 1 On drecton s clear For the other drecton assume that 0SPCŽ A s smple n PCŽ A We want to show that S s smple n ModŽ A Take 0 x S Snce S s Hausdorf there exsts an open submodule L S, not contanng x L s also closed Ž Lemma 32 and thus t s pseudocompact f we gve t the nduced topology Snce S s smple n PCŽ A we obtan L 0 Ths mples that S SL carres the dscrete topology But then every submodule of S s closed and thus s a subobject of S n PCŽ A Snce S s smple n PCŽ A there can be no non-trval subobjects and thus S s smple n ModŽ A Ž 2 Ths follows from Ž 1 Let us say that M PCŽ A s fntely generated n PCŽ A f there s a k surjectve map A M n PCŽ A for some k PROPOSITION 35 Proof Assume that M, N PCŽ A, M fntely generated Then Hom PCŽ A Ž M, N Hom ModŽ A Ž M, N If m M then the map a am s contnuous whch yelds Hom PCŽ A Ž A, M Hom ModŽ A Ž A, M Ths proves the proposton for M A and hence also for M A k Ž Now assume M general There s an exact sequence n PC A 0 M F M 0 wth F A k Ths yelds a commutatve dagram wth exact rows PCŽ A Ž PCŽ A Ž PCŽ A Ž 0 Hom M, N Hom F, N Hom M, N ModŽ A Ž ModŽ A Ž ModŽ A Ž 0 Hom M, N Hom F, N Hom M, N The vertcal maps are njectve and the mddle one s an somorphsm It follows that the left map must be an somorphsm COROLLARY 36 An object n PCŽ A s fntely generated n PCŽ A f and only f t s fntely generated n ModŽ A COROLLARY 37 A drect summand n ModŽ A of a fntely generated object n PCŽ A s a drect summand n PCŽ A In partcular a fntely generated object n PCŽ A s projecte npcž A f and only f t s projecte n ModŽ A Proof If M s a fntely generated object n PCŽ A then a drect summand of M s the mage of an dempotent n End Ž M ModŽ A The result now follows from Propostons 35, 33

9 GRADED MODULES OF GELFANDIRILLOV 259 COROLLARY 38 If M PCŽ A s fntely generated then a submodule LM s open f and only f ML PCFnŽ A Proof If L s open then t s closed and of fnte colength Hence MLPCFnŽ A Conversely assume ML PCFnŽ A By Proposton 35 the quotent map M ML s contnuous Snce ML carres the dscrete topology, 0 ML s open and thus so s ts nverse mage L Snce PCŽ A s the dual of a locally fnte category, t has projectve covers The projectve covers of the pseudocompact smples are the ndecomposable projectves Furthermore every projectve n PCŽ A s a product of such ndecomposable projectves By 8 the ndecomposable projectves are of the form Ae, where e s a prmtve dempotent n A Recall also from 8 that f Ž e I s a summable set of prmtve, parwse orthogonal dempotents wth sum 1 then A Ł I Ae and every ndecomposable projectve n PCŽ A s somorphc to at least one Ae LEMMA 39 The Ž Ae are the projecte coers n ModŽ A I of the smple A-modules whch are pseudocompact Proof Ths follows from the fact that End Ž Ae End Ž Ae ModŽ A PCŽ A s local snce the Ae are projectve covers of smple modules n PCŽ A From Proposton 33 t follows that M s noetheran n PCŽ A f t satsfes the ascendng chan condton on closed subobjects Thus f M s noetheran n ModŽ A then t s noetheran n PCŽ A We now show that the converse holds For the purpose of the proof we ntroduce the followng noton We say that M PCŽ A s strongly fntely generated f M s a quotent of a fnte drect sum of Ž Ae Then we have the followng lemma LEMMA 310 Assume that M PCŽ A s strongly fntely generated and N s a submodule of M such that N M Then N M Proof Assume N M Consder the partally ordered set P N N M N ModŽ A 4 Snce M s fntely generated, P has a maxmal element by Zorn s lemma Agan wthout loss of generalty we may replace N by ths maxmal element In that case MN s smple However, MN s not pseudocompact snce otherwse by Proposton 35, N kerž M MN would be pseudocompact and hence closed whch s mpossble because N M N 1 Let : Ae M be a non-zero map Then ether Ž N Ae or 1 Ae Ž N s smple but not pseudocompact The last case s mpossble snce by Lemma 39, Ae has only one smple quotent, and ths smple quotent s pseudocompact Thus N contans the mage of every and therefore N M

10 260 VAN GASTEL AND VAN DEN BERGH PROPOSITION 311 Eery subobject n ModŽ A of a noetheran object n PCŽ A s closed and hence les n PCŽ A Proof Let M be a noetheran object n PCŽ A and let N M be a subobject n ModŽ A Then N s a noetheran subject of M n PCŽ A Snce the Ž Ae form a set of generators for PCŽ A, N s a quotent of a drect sum of a fnte number of such Ae, and n partcular s strongly fntely generated Hence by the prevous lemma N N COROLLARY 312 An object n PCŽ A s noetheran n PCŽ A f and only f t s noetheran n ModŽ A Ž COROLLARY 313 Assume that M PC A s noetheran Then the topology on M s the cofnte topology That s, a submodule L M s open f and only f ML has fnte length Proof One drecton s clear For the other drecton, let L be a submodule of M of fnte colength By Proposton 311, L s closed n M Hence ML s pseudocompact and snce t s of fnte length t carres the dscrete topology Thus 0 ML s open, and so s ts nverse mage L Let R, I, M be respectvely a rng, an deal n R, and an R-module Then the I-adc topology on R s the lnear topology generated by the submodules of M of the form I n M In a pseudocompact rng A the Jacobson radcal radž A s the common annhlator of the smple pseudocompact A-modules 8, dual of Proposton IV12 The followng s a reformulaton of the prevous corollary COROLLARY 314 Assume that M PCŽ A s noetheran Then the topology on M s gen by the radž A -adc topology Proof It suffces to show that MradŽ AMs a fnte sum of smples To prove ths we may replace M by MradŽ AMand A by AradŽ A Then by 8, dual of Proposton IV12, A s a product of endomorphsm rngs of vectorspaces over dvson rngs and M s stll a noetheran A-module Now one shows by drect verfcaton that M must be a fnte drect sum of smples Ž DEFINITION 315 We say that A s locally noetheran f the Ae I Ž are noetheran n PC A We say that A s noetheran f A s noetheran n PCŽ A Ž PROPOSITION 316 Let A be locally noetheran and M, N PC A Assume that M s noetheran Then Ext M, N Ext M, N Ž Ž PCŽ A ModŽ A

11 GRADED MODULES OF GELFANDIRILLOV 261 Proof The case 0 follows from Propostons 35 For 0 we use an exact sequence 0 M P M 0 n PCŽ A wth P a fnte drect sum of Ae Snce A s locally noetheran, M s also noetheran The proposton now follows by degree shftng Proposton 316 yelds the followng corollary COROLLARY 317 Let pcž A resp modž A be the full subcategores of PCŽ A resp ModŽ A consstng of noetheran objects Then the functor pcž A modž A s fully fathful and ts essental mage s closed under extensons In partcular Ž Ž PCFn A s closed under extensons nsde Fn A Ž PROPOSITION 318 If A s locally noetheran and M PC A s noetheran then proj dm M proj dm M ModŽ A PCŽ A Now we dscuss brefly automorphsms of pseudocompact rngs LEMMA 319 Assume that A s a pseudocompact rng If AutŽ A then s contnuous f and only f for eery pseudocompact A-module S of fnte length, we hae that S s pseudocompact Proof Ž One needs that the left multplcaton on S Ž wth the dscrete topology s contnuous Snce s assumed to be contnuous, ths s clear Ž Assume that L A s an open deal Then AL s pseudocom- Ž 1 pact of fnte length and hence AL A Ž L s pseudocompact of 1 fnte length Thus Ž L s open n A COROLLARY 320 homeomorphsm If A s locally noetheran and AutŽ A then s a Proof It suffces to show that s contnuous By Lemma 319 and Corollary 317 we must show that f S s pseudocompact smple, then so s Ž S Ths s clear snce S Aerad Ae for some prmtve dempotent e 1Ž Ž 1 and thus S A e rad A Ž e To close ths secton we dscuss noetheran pseudocompact rngs

12 262 VAN GASTEL AND VAN DEN BERGH PROPOSITION 321 forgetful functor s an equalence of categores Let A be a noetheran pseudocompact rng Then the pcž A modž A Ž 31 Proof By Corollary 317 we only have to show that Ž 31 s essentally surjectve Let M modž A Then M has a resoluton F 1 0 F M0, where the F are fntely generated free A-modules By Proposton 35, Hom Ž F, F Therefore M coker PCŽ A PCŽ A 1 0 PROPOSITION 322 Put J radž A Then Assume that A s a noetheran pseudocompact rng Ž 1 AJ s semsmple Ž 2 A s complete for J-adc topology Ž 3 The topology on A concdes wth the J-adc topology ŽŽ Conersely f A s a left noetheran rng satsfyng 1, 2 then A s pseudocompact when equpped wth the J-adc topology Ž n Proof Snce A s semsmple and J s fntely generated, all J n J n1 are fnte drect sums of smples Hence AJ n has fnte length for all n Therefore Ž 2 mples that A s pseudocompact Ž By 8, AJ s a product of endomorphsm rngs of vectorspaces Snce AJ s also noetheran, t must be semsmple Ths proves Ž 1 Property Ž 2 follows from Ž 3 and Ž 3 s precsely Corollary 314 An object n PCŽ A s sad to be cosemsmple f t s a drect product of smple modules Ths s equvalent wth beng semsmple n the dual category PCŽ A If MPCŽ A then we defne MradŽ M as the quotent of M whch s the socle of M n PCŽ A By constructon MradŽ M s the largest cosemsmple object n PCŽ A whch s a quotent of M From the fact that takng socles s left exact t follows that the functor MMradŽ M s rght exact Ž Ž Ž LEMMA 323 Nakayama s lemma If M PC A then M rad M f and only f M 0 Proof Ths s clear f we look at the dual statement Ž PROPOSITION 324 If Mrad M s fntely generated then so s M A smlar statement holds for strongly fntely generated

13 GRADED MODULES OF GELFANDIRILLOV 263 Proof The proofs of the two statements are dentcal, so let us prove the frst one By lftng the generators of MradŽ M we can construct a map : A k M whch becomes surjectve after applyng the functor TTradŽ T Gven the rght exactness of ths functor we obtan that CradŽ C for C coker By Nakayama t follows that C 0 Ths fnshes the proof From ths we deduce the followng PROPOSITION 325 Let A be a pseudocompact rng and N radž A a regular normalzng element nducng a homeomorphsm on A Assume that MPCŽ A s such that MNM s noetheran Then M s noetheran Proof Ths s a standard proof Frst let S be a subobject of M n Ž p Mod A Let p : M MN M be the quotent map Ž p By Proposton 311 t follows that ps s a closed subobject of MN M 1 Ž Ž p Hence p p S SN M s a closed subobject of M We deduce Ž Ž p p S SN M SN M S, p p where the last equalty follows from 8, Proposton IV11 We deduce Ž Ž p S S N M 32 p Ž Let us call S saturated f MS s N-torson free From 32 t follows that f S saturated then so s S Let T M be an arbtrary submodule We want to show that T s fntely generated We defne frst S t M k : N k t T 4 Obvously S and hence S s saturated Snce SNS MNM s strongly fntely generated, t follows from Proposton 324 that the same holds for S But then by Lemma 310 have S S Thus S s fntely generated It now follows from the defnton of S that there exsts a k such that N k S T Snce N k S f fntely generated, t now suffces to show that TN k S s fntely generated Ths follows from the fact that the latter s k subobject of the noetheran object MN M 4 A MATRIX REPRESENTATION FOR PSEUDOCOMPACT RINGS If A s an arbtrary rng, M a left A-module, and Ž e 1,,n a fnte set of parwse orthogonal dempotents wth sum 1 then t s classcal that A s

14 264 VAN GASTEL AND VAN DEN BERGH somorphc to the matrx rng 0 A A A A A A A A A n n n1 n2 nn wth Aj eae j and M s somorphc to the set of column vectors Ž M 1,,Mn t wth Mj em j An element a n A s sent to the matrx Ž eae j j and an element m of M s sent to Ž em j j It s clear how to extend ths result to the pseudocom- pact stuaton LEMMA 41 Assume that Ž e Is a summable set of orthogonal dem- potents n a pseudocompact rng A such that Ýe 1 Let M be a pseudo- compact A-module and put Aj eae,memthen j A s somorphc to the rng of doubly nfnte matrces Ž a Ž A j j j j wth summable columns and M s somorphc to the set of summable column ectors Ž m Ž M The somorphsms are gen by the maps Ž aj Ýa j j j Ž m Ým Note that ths lemma only says somethng about the rng structure on A and the module structure on M, but nothng on the topology Below we gve the Aj the topology nduced from A and M the topology nduced from M Snce Aj eae j s clearly closed n A, ts complete A smlar result s true for M Furthermore the topology s lnear Ž gven by abelan subgroups We also have multplcaton mappngs Aj A jk Ak A M M j j and snce there are nduced from the multplcaton on A and M they are contnuous Ths makes A nto a topologcal rng, Aj nto a topologcal A Ajj-bmodule, and M nto a left topologcal A-module LEMMA 42 Ž 1 A s a pseudocompact rng and Aj s a pseudocompact A-module Ž 2 M s a pseudocompact A -module

15 GRADED MODULES OF GELFANDIRILLOV 265 Proof It suffces to prove Ž 2 Indeed f we take M Ae Ž j and after- wards j then we obtan part Ž 1 Let L M be an open submodule If T s an A submodule of em contanng L emelthen T AT L s an A-submodule of M contanng L, and furthermore T emt Ths yelds that the length of ememl Ž s bounded by that of ML Snce the topology on ems nduced from that on M we deduce that ems pseudocompact Unfortunately t s not n general true that A and M carry the nduced topology from the product topologes on Ł Ž A j j on Ł M A counter example s gven by the endomorphsm rng of an nfnte dmensonal vectorspace Under some mld extra hypotheses ths defect can be repared Note that the Ae are pseudocompact projectves Hence they are products of ndecomposable pseudocompact projectves PROPOSITION 43 Let Ž e Ibe as n the preous lemma Assume that eery ndecomposable pseudocompact projecte s a summand of at most a fnte number of Ae Then as topologcal spaces Ł j, j A A Ž 41 Ł M M Ž 42 Proof We certanly have A Ł Ae Hence t suffces to prove Ž 42 j j We have an ncluson Ł M M Ž 43 whch s gven by the product of the maps M M : m em These maps are contnuous and hence the ncluson s also contnuous We now show that the topology on M s courser than the nduced topology for the ncluson Ž 43 Let L M be an open submodule Snce ML has fnte length the hypothess mply that Hom Ž Ae, ML A s non-zero for at most a fnte number of Snce HomŽ Ae, ML e Ž ML we deduce that for almost all, m eml Hence Ł Ž M L s open n Ł M Observng that M Ł Ž M L L fnshes the proof The proof we have just gven also shows that f Ž m Ł M then Ž m s summable n M Sendng Ž m to Ým defnes an nverse to the ncluson Ž 43 A smlar result holds for A

16 266 VAN GASTEL AND VAN DEN BERGH 5 GLOBAL DIMENSION In ths secton A s a pseudocompact rng We defne gl dm A sup proj dm M MPCŽ A PCŽ AŽ Note that by Proposton 318 we often have proj dm PCŽ A M proj dm ModŽ A M Therefore, f no confuson can arse, we make no dstncton between those two types of projectve dmenson, and we smply wrte proj dm M LEMMA 51 We also hae gl dmž A sup proj dm PCŽ A S SPCŽ A S smple Proof By Theorem 31 t suffces to prove the dual statement for locally fnte categores So assume that C s such a category and nj dm S n for every smple S n C Hence for every fnte length module F one also has nj dm F n If M C s arbtrary then by defnton M s a drect lmt of fnte length objects By the proof 9, Theorem 1101 monomorphsms nto njectves can be constructed n a functoral way and hence so can njectve resolutons Takng the drect lmt of the njectve resolutons of the subobjects of fnte length of M yelds an njectve resoluton of M of length n ŽC s locally noetheran and hence a drect lmt of njectves s njectve The followng result s very classcal Ž PROPOSITION 52 Let N rad A be a regular normalzng element n A Assume that A s locally noetheran Then gl dm A gl dm AŽ N 1 Ž Proof Let gl dm A N p We have to show that proj dm S p 1 for every pseudocompact smple of what amounts to the same proj dm radž P p, Ž where P runs through the ndecomposable projectves n PC A Ths follows from Lemma 53 below LEMMA 53 Assume that A s locally noetheran and let L be a noetheran pseudocompact A-module whch s N-torson free Then proj dm L proj dm A Ž N LNL

17 GRADED MODULES OF GELFANDIRILLOV 267 Proof By degree shftng one reduces to the case where LNL s projectve over AŽ N In that case the result follows by an approprate verson of Nakayama s lemma The followng type of result seems to be referred to less often PROPOSITION 54 Assume that A s a locally noetheran and N radž A s a regular normalzng element such that for eery ndecomposable pseudocompact projecte one has NP rad 2 Ž P Then gl dm AŽ N 1 gl dm A Ž 51 Proof Ths s an mmedate generalzaton of the proof by Serre that local rngs of fnte global dmenson are regular Let N N 1 By Corollary 320, s a homeomorphsm Let P be an ndecomposable pseudocompact projectve over A wth cosocle S We have an ncluson 1 S NPN radž P radž P N radž P Ž 52 2 Ž Ž Now we also have NP rad P N rad P and thus there s an ncluson NPN radž P radž P rad 2 Ž P Ž 53 Ž 2 Ž rad P rad P s a fnte sum of smples and hence ths ncluson splts From the commutatve dagram NPN radž P radž P N radž P NPN radž P 2 radž P rad Ž P Ž we deduce that 52 s also splt Thus proj dm S 1 proj dm radž P A A proj dm AŽN radž P N radž P Ž Lemma 53 proj dm 1 S A Ž N Ž Takng the supremum over all S yelds 51 6 A CLASSIFICATION PROBLEM Let I be ether or n In ths secton we am to classfy the followng data

18 268 VAN GASTEL AND VAN DEN BERGH Ž A A pseudocompact rng A wth a summable set of prmtve orthogonal dempotents Ž e I such that Ýe 1 and Ae Aej for j Ž B A regular normalzng element N radž A, nducng a homeo- 1 morphsm N N such that Ž e e1 and such that the mage of the Ž e becomes central n B AŽ N The soluton to ths classfcaton problem s the followng PROPOSITION 61 Ž 1 If I then A s somorphc to the rng T Ž R I of lower trangular I I-matrces wth entres n a local pseudocompact rng R The topology on T Ž R I s the product topology Under the somorphsm the e correspond to the dagonal dempotents and N corresponds to the matrx n whch eery entry s zero except those on the lower subdagonal, whch are one Ž 2 If I n then A s somorphc to a rng of n n-matrces of the form 0 R UR UR R UR UR, Ž 61 UR R R R where R s a local pseudocompact rng and U s a normalzng element n Ž 1 rad R nducng a homeomorphsm U U The topology on Ž 61 s the product topology Under the somorphsm of A wth Ž 61 the Ž e correspond to the dagonal dempotents and N corresponds to the matrx U Ž 62 Proof It s clear that the rngs exhbted n ŽŽ 1, 2 and the correspondng N, Ž e satsfy Ž A, Ž B I, so we only have to be concerned wth the converse To smplfy the notatons we put P Ae and S wll be the unque smple quotent of P

19 GRADED MODULES OF GELFANDIRILLOV 269 Recall that by Proposton 43 we have a matrx form A Ž A j j and we also have N Ýe1 NeÝN 1 Snce N s regular we have njec- tons N j jj1, j1 A A Ž 63 N1 A A Ž 64 j 1, j and furthermore snce ANA s dagonal Ž 63, Ž 64 are somorphsms for j 1 Left and rght multplcaton by N are contnuous Furthermore snce A s pseudocompact t follows that the quotent topology on AN, for rght multplcaton wth N, concdes wth the nduced topology We clam that that s also the case for left multplcaton wth N Indeed left multplca- N ton by N s the composton A A A The fact that s a homeomorphsm shows what we want Snce Ž 63, Ž 64 are restrctons from left and rght multplcaton by N they are contnuous Furthermore the topology on the mage concdes wth the nduced topology Ths means n partcular that f j 1 then they are homeomorphsms The fact that N s normalzng also mples N A A N 1, j 1, j1 j1, j We wll frst consder the case N, 1 N1,0 and maps I We defne for 0,,n1, N : A A : an 1 an j j 00 j Note that AjNj A0 and hence N defnes a homeomorphsm A00 A 0 Thus t makes sense to use N 1 Clearly f a A, b A then Ž a Ž b Ž ab j jk j jk k and hence Ž defnes an ncluson of A nto M Ž A j j n 00 We want to understand ts mage If j then s a homeomorphsm and thus Ž A j j j A 00 Hence we look at the case j We have A N 1 A N A, N A N A U Ž j j j j 0j j 0, n1 n1,0 00

20 270 VAN GASTEL AND VAN DEN BERGH wth U N0, n1 Nn1, n2 N 1, 0 It s easy to see that UA00 A00U and thus U s a regular normalzng element n A Hence Ž puttng R A 0 R UR UR R UR UR Ž A UR R R R From the defnton of U t s clear that U U 1 One computes s a homeomorphsm 0 f j1 Ž N 1 jn N j N j 1 f j1, j n 1 U f 0, j n 1 If U radž R then t s easly seen that N radž A and thus Ž B would be volated Let us now consder the case I The followng three lemmas are standard 1 LEMMA 62 Ext Ž S, S 0 for j, j 1 Žthe hypothess I A j s not used here Proof Assume Ext 1 Ž S, S A j 0 Then there s a non-trval extenson 0S FS 0 Ž 65 j There are now two possbltes NF 0 In ths case Ž 65 s an extenson as B-modules and thus j NF 0 In ths case multplcaton by N defnes a non-trval map S Sj and snce S, Sj are smple, ths map must be an somorphsm Thus S S j Now Sj s a smple quotent of Aej Ae j1 Thus SjSj1 and we fnd j 1 LEMMA 63 Assume I Let M be a fnte length module n modž A wth composton factors among the Ž S Assume MradŽ M j j S Then the composton factors of M are of the form S k, k Proof We prove ths by nducton on the length of M Let St M be a smple submodule By nducton the subquotents of MSt are of the form S k, k

21 GRADED MODULES OF GELFANDIRILLOV 271 Hence f t, then t follows from Lemma 62 that Ext 1 Ž MS, S t t 0 and thus M MSt S t In partcular St s a smple quotent M, dffer- ent from S, contradctng the hypotheses LEMMA 64 Assume I Then HomŽ P, P 0 for j Proof Assume there s a non-zero map P P j Snce Pj s separated 1 there exst an open submodule L P such that P Ž L j s not zero 1 PL has fnte length and s equal to S modulo ts radcal P Ž L j j s a 1 subobject of P L and snce S s a quotent of P Ž L j t follows that S s a subquotent of PjL Ths mples that j by Lemma 63 and we are done Snce Hom Ž P, P A j eaea j j ths last lemma mples that n the case I the matrx form for A s lower trangular For every Ž, j, j there s a homeomorphsm : A A j j 00 obtaned by composng homeomorphsms of the form Ž 63, Ž 64 One checks that j s unquely determned n ths way By a verfcaton as n the case I Ž but somewhat more complcated one also shows that j s compatble wth multplcaton and Ž N 1, 1, 1 Thus A s somorphc to the rng of lower trangular matrces wth entres n R A00 and N has the requred form Ths fnshes the proof of Proposton 61 We now exhbt when pseudocompact rngs n Proposton 61 are locally noetheran and have fnte global dmenson PROPOSITION 65 Let A be a pseudocompact rng as n Proposton 61 Then A s locally noetheran f and only f R s noetheran Furthermore f R s noetheran then gl dm R 1 f I gl dm A gl dm RŽ U 1 f 2 I Ž 66 gl dm R f I1 j Proof We have ½ I RŽ U f I Ž Ž I A N 67 R f I The condton for A to be noetheran then follows from Proposton 325

22 272 VAN GASTEL AND VAN DEN BERGH The statement about global dmenson s clear n the case that I 1 For I 1 one verfes that N satsfes the hypotheses of Propostons 52, 54 Then Ž 66 follows from Ž 67 7 MORE CLASSIFICATION In ths secton we classfy rngs R satsfyng Ž C R s local, complete and contans an algebracally closed feld k, somorphc to ts resdue feld Ž D Let m be the maxmal deal of R We requre that m contans a regular normalzng element U such that RŽ U s a commutatve noetheran CohenMacaulay local rng of rull dmenson one Ž E proj dm Rm R The soluton to ths classfcaton problem s as follows Ž Ž Ž PROPOSITION 71 Assume that R satsfes C, D, E aboe Then where R k²² x, y:: Ž, Ž 71 yx qxy hgher order terms Ž 72 for some q k or yx xy x 2 hgher order terms Ž 73 Ž Ž Ž Conersely eery such rng satsfes C, D, E Proof Let us frst show that a rng R of the form Ž 71 wth of the form Ž 72, Ž 73 does ndeed satsfy Ž C, Ž D, Ž E It s clear that Ž C s satsfed For Ž E observe that we have ux y Ž 2 for some u, m such that u, form a bass of mm Ths means that we have a complex x Ž u ž/ y 2 0R R RRm0 74 Ž and we have to show that ths complex s exact We flter R wth the m-adc fltraton For ths fltraton t s easy to see that where conssts of the quadratc part of gr R kx, yž, Ž 75

23 GRADED MODULES OF GELFANDIRILLOV 273 Ž The exactness of 74 now follows from the exactness of Ž u 2 x ž/ y 0gr R Ž gr R gr R Rm 0 Ž 76 whch s standard Ths proves Ž E Now let us consder Ž D We assume that R s not commutatve snce otherwse Ž D s trval From Ž 75 t follows that R s a doman, so every element of R s regular Put U y, x We clam that U s normalzng Ths was ndependently observed by Artn and Stafford Assume frst that we are n case Ž 72 One computes Ux qxu x, 1 Uy q yu y,, where represents the non-quadratc terms of Now clearly Ž 77 Ý x, uu Ý y, uu Ž 78 for approprate u,, u, R Substtutng Ž 78 nto Ž 77 and then substtutng the resultng equatons repeatedly nto themselves yelds the formulas Ux Ž qx U Uy Ž q 1 y U Thus U s a normalzng element Ž Case 73 s treated smlarly, startng from Ux xu x, Uy Ž y 2 x Ux, y, Snce R y, x k x, yž s clearly CohenMacaulay of rull dmenson one, we have shown that R satsfes Ž D Now we prove the converse Note that by Ž D, R s automatcally left and rght noetheran STEP 1 proj dm Rm 2 R

24 274 VAN GASTEL AND VAN DEN BERGH Proof We have proj dm RRm 1 proj dm Rm 1proj dm RŽU mum Ž Lemma 53 In partcular proj dm mum s fnte Snce RŽ U R ŽU s commutatve of rull dmenson one ths mples proj dm R ŽU mum 1 Thus proj dm R Rm 2 Assume that the projectve dmenson of Rm s strctly less than 2 It cannot be 0, hence t must be one Ths means that there s a resoluton 0 R n R Rm 0 whch easly yelds that R s the completon of a free k-algebra n n varables If n 1 then R s not noetheran and f n 1 then R s a dscrete valuaton rng and hence Ž D s not satsfed STEP 2 The mnmal resoluton of Rm looks lke Ž Ž x Ž u ž/ y 2 0R R RRm0, 2 where x, y, u, form bases for mm Proof The mnmal resoluton of Rm looks lke g f b a 0R R RRm0 79 Ž Tensorng wth R N and takng ranks yelds that a b 1 We have Ext RŽ Rm, R Ext 1 RŽUŽ Rm, RŽ U 0 for 1 Ž and thus by dualzng Ž 79 we fnd a mnmal resoluton of Ext 2 Ž Rm, R R as rght R-module: f t g a t b 2 0R R R Ext RŽ Rm, R 0 Ž 710 Now Ext 2 Ž Rm, R s annhlated by m and thus dm Ext 2 Ž Rm, R R k R b Hence f b 1 then we see that Ž 710 decomposes as a drect sum of subcomplexes But then so does the dual complex Ž 79, whch s mpossble snce ths s a mnmal projectve resoluton of a smple R-module We conclude that b 1, a 2 We now fnd that the mnmal resoluton of Rm looks lke Ž 79 wth Ž 2 x, y, a bass for mm Snce the dual complex of Ž 79 s a mnmal resoluton of Rm Ž as rght module we fnd that Ž u, s also a bass for 2 mm

25 GRADED MODULES OF GELFANDIRILLOV 275 We can now conclude the proof of Proposton 71 From Step 2 t follows that R s as n Ž 71 wth ux y It s now easy to see that can be put n one of the standard forms Ž 72, Ž 73 Ž PROPOSITION 72 Let R, m, U be as n Proposton 71 Then R U s regular ff U m 2 2 Proof If U mm then by Proposton 54, RŽ U s regular Conversely assume that RŽ U s regular Then 1 dm Ž Ž Ž Ž k m U m U 2 ŽŽ 2 2 m U m, whence U m Remark 73 example, Ths result s false n hgher dmenson Consder, for Rk²² x, y:: Ž x, x, y, x, x, y Ž 2 Ž Then R x, y k x, y s regular, but x, y rad R 8 PROOF OF THEOREM 11 We start by dscussng thngs a bt more generally Let A be a Grothendeck category, G : A A an autoequvalence, and : G d A a natural transformaton such that for all A A Defne Ž GŽ A GŽ Ž A Ž 81 B A A Ž A 04 Then the followng propertes are easly verfed LEMMA 81 Ž 1 B s closed under subquotents, drect sums, and drect products Ž and hence under lmts and colmts Ž 1 2 B s closed under G, G and f A A then ker, coker Ž A B Ž! 3 Let : B A be the ncluson functor The functors, : A B defned by ž / ŽG 1 Ž A! 1 Ž A ker A G Ž A Ž A Ž Ž A coker GŽ A A are respectely the rght and the left adjont to Remark 82 The condton Ž 81 s not automatc A counter example s gven by A ModŽ A wth A k V, where V s a k-vectorspace such

26 276 VAN GASTEL AND VAN DEN BERGH 2 that V 0n A For G we take M M for some GLŽ V, whch we extend n the obvous way to A To defne we take V, not -nvarant and we defne : A A as the bmodule map whch sends 1 to Then we put Ž In ths case GŽŽ A ŽGŽ A A, and n partcular B s not G-nvarant Ž Nevertheless 81 holds n the case we are nterested n as the followng lemma shows LEMMA 83 Assume that for all njectes E A we hae that Ž E s surjecte Then Ž 81 holds Proof We have that G s a natural transformaton G 2 G Applyng 1 ths to the map G G : GE E we get a commutatve dagram Žusng Ž 1 E GG Ž E 2 GE Ž GE Ž GE Ž E GE G 1 Ž GE Applyng ths dagram wth E njectve and usng the surjectvty hypothe- 1 Ž Ž Ž ss we fnd that G GE E Now let A A be arbtrary and let 0 A E F GE be an njectve resoluton of A Ths yelds commutatve dagrams 0 GA GE GF GŽŽ A GŽŽ E GŽŽ F E GA GE GF 0 GA GE GF ŽGŽ A ŽGŽ E ŽGŽ F GA GE GF The fact that the rghtmost squares of these dagrams are commutatve yelds the result n general Ž closed under subquotents, extensons, and drect sums and defne D as the full subcategory of A consstng of objects A havng an ascendng fltraton Ž FA such that Now let D B 1 be a G, G -stable localzng subcategory that s, 0,, F A0, F AF A D, A F A Ž 82 0 n1 n n n

27 GRADED MODULES OF GELFANDIRILLOV 277 If there s such a fltraton wth FnAFn1A then we say that A D n Note that D n B n D If A A then there s a maxmal fltraton Ž R A n n on A satsfyng the frst two propertes n Ž 82 wth D B Ths fltraton s gven by n n RnAkerŽ A G A An object A A s n B n f RnAA and t s n B f n RnAA A s n D f n addton Rn1ARnA D We also consder the descendng fltraton on A gven by n n LnAmŽ GA A Ths fltraton satsfes L AL A B If A B then L A n n1 m n R A mn Ž PROPOSITION 84 1 D s a localzng subcategory n A Ž 2 Assume that A s locally noetheran If D s closed under njecte hulls n B then D s closed under njecte hulls n A Proof Ž 1 Only the closedness under extensons s not mmedately clear Let 0 D1 A D20 be an extenson such that D 1, D2 D We consder four cases Ž n a If D 1, D2 D for some n then there s nothng to prove Ž n b Assume D1 D for some n Let F be a fltraton on D2 Ž 1 Ž Ž 1 satsfyng 82 Then A FD Snce by a all Ž FD n n 2 n 2 are n D, we conclude that ths s also true for A Ž n c Assume D2 D for some n Then Ln AD1 and hence Ln A D In the exact sequence n n n 0GRnAG Ž A LnA0 n n n we have that GR A B D D Combnng ths wth Ž b n shows what we want Ž d Assume now that the D are general Usng Ž c 1, 2 we can now use the same reasonng as n Ž b to fnsh the proof Ž 2 Ths asserton can be splt nto two parts Ž a B s closed under njectve hulls n A To prove ths let B A be an essental extenson wth B B and A A We have to show that A B We may clearly assume that AB contans no subobject n B Assume frst that A s noetheran In that case B B n for

28 278 VAN GASTEL AND VAN DEN BERGH some n From the exact sequence n n 0 B A G Ž LnA 0 n we deduce that AB G Ž L A n Hence LnA contans no subobject n B Thus LnAB0 and hence LnA0 Ths yelds A B and we are through Now assume that A s general By hypothess A I A where the A are noetheran By lookng at the pars Ž B A, A we fnd that A B Hence A B Ž b D s closed under njectve hulls n B To prove ths assume that D B s an essental extenson wth D D and B B Snce B R B, by consderng all the pars Ž R BD, R B n n n n, we may reduce to the case B B n We then use nducton on n If n1 then B B, D D and the result follows from the hypotheses on D Assume now n 1 We have the standard exact sequence 1 0 RBB 1 G Ž L1B 0 Ž 83 n1 Clearly RB B,LB B Lookng at the pars Ž RBD,RB and Ž LBD,LB 1 1 and nducton reveals that RB,LB 1 1 D Hence from Ž 83 we deduce that B D From now on we assume that A s locally noetheran Let Ž T Jbe the smple objects n B It s easy to see that these are also the smple objects 1 n B Defne t : J J by G Ž T T t Clearly t s a permutaton of J We let D, C be the mnmal localzng subcategores of B and B Ž contanng T J Clearly C D For, j J we wrte D j, C j f T,T j are respectvely n the same connected component of D and C In other words, s the transtve closure of the relaton Ext 1 Ž, 0on smple objects Wth a reasonng smlar to Lemma 62 one shows that C j p : D t p j Ž 84 Let J be a unon of equvalence classes for D, stable under t, t 1 By Ž 84, s then also a unon of equvalence classes for C We denote by D, C the mnmal localzng subcategores of D and C contanng Ž T It s standard that CŽJ C C Ž 85 D D ŽJ D Let E be the njectve hull of T n C Put E E Then E s an njectve cogenerator of C The njectve hull of T n D s gven by F RE 1 We also put F F

29 GRADED MODULES OF GELFANDIRILLOV 279 PROPOSITION 85 Assume that Ž E s surjecte Let C End Ž E, D End Ž F wth the natural topology Ž as n Theorem 31 C D Then there s a regular normalzng element N radž C wth the followng propertes Ž 1 D CN Ž as pseudocompact rngs Ž 1 2 Put N N Let e C be the dempotent correspondng to the projecton E E Then Ž e e t Ž 3 Let U C There s an somorphsm as C -modules p: HomŽ U, E HomŽ GU, E whch s functoral n U Ž 4 There s a commutate dagram N Hom U, E RHomŽ U, E Ž HomŽ Ž U, E HomŽ U, E HomŽ GU, E p Ž 86 Ž 5 s a homeomorphsm Proof By constructon we have an exact sequence 1 0 F E G Ž E 0 Applyng Hom Ž, E and usng the fact that Hom Ž F, E C C Hom Ž F, F we obtan an exact sequence D Ž 1 r s C 0Hom G E, E C D 0 Ž 87 Here rž f f, sž g ff IfUs a fnte length object n D then 1 one checks that s Ž Ž U Ž U D C and hence s s contnuous Now choose somorphsms : G 1 Ž E Et and let The map whch sends h to h defnes an somorphsm C Ž 1 Hom G E, E Put N, as an element of C Then Ž 87 C becomes an exact sequence N 0 C C D 0 for whch we deduce that N s regular and normalzng The smple pseudocompact C modules are of the form Hom Ž T, E C and f f : T E s a map n C then f has ts mage n F and thus s annhlated by Hence Nf f 0 and thus N radž C

30 280 VAN GASTEL AND VAN DEN BERGH We now show that N satsfes Ž 2 e s the composton of the projecton p : E E and the njecton q : E E The fact that NeN 1 et now follows from the followng commutatve dagram p q E E E Ž 1 Ž 1 Ž 1 G E G E G E 1 1 G Ž p G Ž q G Ž E G Ž E G Ž E pt qt t E E E Ž Now we prove 3 Defne the map Ž Ž Ž p : Hom U, E Hom GU, E : f G 1 f We nvestgate the behavour of p wth respect to left multplcaton by Ž Ž 1 Ž 1 1 an element g of C We fnd p gf G gf G g f Ž 1 G g pž f Now we look at the followng commutatve dagram E Ž 1 G g E 1 g 1 1 G E G E E Ž Ž g E Ž 1 1 From ths commutate dagram we deduce that G g N gn 1 Ž gso we conclude that to make p a map of C-modules, t suffces to twst HomŽ U, E by Now we prove Ž 4 The commutatvty of Ž 86 amounts to the dentty Ž 1 Ž Ž Ž 1 G Nf U f for f n Hom U, E Snce G Nf Ž Ž 1 G G E f Ž E GŽ f ths follows from the fact that s a natural transformaton Fnally we note that Ž 5 follows from Lemma 319 and Ž 3 Now we specalze to the stuaton of Theorem 11 Thus B QchŽ Y for a CohenMacaulay curve Y and J Y Furthermore Tx P x, t It s also clear that x y x y and thus the equvalence classes for D D

31 GRADED MODULES OF GELFANDIRILLOV 281 Ž are sngletons From 84 t follows that the equvalence classes for C are gven by the -orbts Fnally we have for Y D O ˆ Ž 88 Ł x Wth these data, the proof of Theorem 11 s now a smple matter of translaton, usng the results n Sects 6, 7, Proof of Theorem 11 Ž 1 The fact that exsts and s a bjecton follows easly from Lemmas 81 and 83 Ž 2 Ths follows from Ž 85 ŽŽ 3, 4 By Theorem 31 and functor F gven by M HomŽ M, E defnes an equvalence between the dual of Cf, z and the category of left pseudocompact modules of fnte length over the rng Cz C where s the -orbt of z By Proposton 85, D C Ž N for N radž C such 1 that N N s a homeomorphsm and such that Ž e y e y Thus C satsfes the hypotheses of Proposton 61 From that proposton t follows that we can put C and N n the requred matrx forms and that we have R Oˆ f and RŽ U Oˆ f Y, z Y, z To fnd out the exact form of R, we frst note that by Ž 88, D s locally noetheran and hence so s C by Proposton 325 Furthermore by Proposton 84 every object n C has fnte njectve dmenson Thus C has fnte global dmenson, and by Propostons 65, 52 ths means that R has fnte global dmenson Hence the hypotheses for Proposton 71 are satsfed and thus R does ndeed have the form Ž 11 or Ž 12 Now note that f 2 I then Proposton 65 actually tells us that gl dm Oˆ Y, z Thus z s regular on Y Also by Proposton 72 ths mples that U rad 2 Ž R The essental mage of F s gven by the pseudocompact left Cz-mod- ules of fnte length From Proposton 43 t follows that such modules correspond precsely to the fnte dmensonal left modules over Cz satsfy- ng V ev Under the dualty V V Y, x such modules correspond to the fnte dmensonal rght C-modules W satsfyng W We We now clam that n fact every fnte dmensonal C z representaton s pseudocompact Ths s clear f O Ž z, so assume O Ž z In that case the statement depends upon the fact that cardž k Clearly we may reduce to the case that W s smple Then W s annhlated by the Jacobson radcal of A, whch accordng to 8 s precsely gven by the common annhlator of the pseudocompact smple modules In other words radž C z s gven by the lower trangular matrces, havng only non-unts on the dagonal Thus W s a Ł k-module A fnte dmensonal smple module over a commutatve k-algebra s clearly one dmensonal Hence

32 282 VAN GASTEL AND VAN DEN BERGH dm W 1 and we have a correspondng character : Ł k k Choose až a Łk n such a way that a aj f j Then there exst b k such that Ž a b 0 But a b 0 for at most one, and hence the deal generated by a b s ether mproper or the kernel of the projecton map pr : Ł k k The frst case s clearly mpossble and the second case mples that s gven by projecton on the th factor Hence W s pseudocompact Puttng FŽ M HomŽ M, E fnshes the proof of ŽŽ 2, 3 Ž 5 Snce F Ž P x s by constructon the th smple module of C z, t s gven by CeradŽ Ce Hence FŽ P Ž Ž z z x Cerad z Ce z ec z radž ec z Ž 6 Ths amounts to the constructon of a natural somorphsm between Ž F M and FGŽ M for M C Snce Ž F M ŽHom Ž f, z C M, E Hom Ž M, E and FGŽ M HomŽ G M, E C, we can use p wth p as n Proposton 853 Ž 7 Ths dagram can be obtaned by dualzng Ž 86 Ž 8 Let M QchŽ Y Then F M Hom Ž M, E A Hom Ž M, F Ł Hom Ž M, F QchŽY QchŽY where as before F s the njectve hull of kž z n QchŽ Y It follows from Matls dualty that Hom Ž M, F s the completon of M at z QchŽY REFERENCES 1 Ajtabh, Modules over ellptc algebras and quantum planes, Proc London Math Soc Ž 3 72 Ž 1996, M Artn and J J Zhang, Noncommutatve projectve schemes, Ad Math 109 No 2 Ž 1994, M Artn and W Schelter, Graded algebras of global dmenson 3, Ad Math 66 Ž 1987, M Artn, J Tate, and M Van den Bergh, Some algebras assocated to automorphsms of ellptc curves, n The Grothendeck Festschrft, Vol 1, pp 3385, Brkhauser, Basel, M Artn, J Tate, and M Van de Bergh, Modules over regular algebras of dmenson 3, Inent Math 106 Ž 1991, M Artn and M Van den Bergh, Twsted homogeneous coordnate rngs, J Algebra 188 Ž 1990, M Artn, A conjecture about graded algebras of dmenson 3, n Proceedngs, Colloqum n Honor of Maurce Auslander, 1996, n press 8 P Gabrel, Des categores abelennes, Bull Soc Math France 90 Ž 1962, A Grothendeck, Sur quelques ponts d algebre ` homologques, Tohoku ˆ Math J Ž 2 9 Ž 1957,