Quantum sections and Gauge algebras

Publicacions Ma emá iques, Vol 36 (1992), 693-714 . A bs ac QUANTUM SECTIONS ANDGAUGE ALGEBRAS L . LE BRUYN * AND F . VAN OYSTAEYEN To he memo y o Pe e Menal Using quan um sec ions o il e ed ings and he associa ed Rees ings one can li he scheme s uc u e en P oj o he associa ed g aded ing e he P oj o he Rees ing . The algeb as o in e es he e a e posi i ely il e ed ings ha ing a non-commu a i e egula quad a ic algeb a o he associa ed g aded ing ; hese a e he so- called gauge algeb as ob aining hei name om special examples appea ing in E . Wi en's gauge heo ies . The pape su eys basic de ini ions and p ope ies bu concen a es en he de elopmen o se e al conc e e examples . 0 . In oduc ion _ Speci ic p oblems in de ining a "scheme" s uc u e on P oj(W), whe e W is he 4-dimensional quan um space o he Rees ing o he Wi en gauge algeb as, may be ackled by i s in oducing such a scheme s uc- u e on P oj(G(W)) _whe e G(W) is he associa ed g aded ing o W and hen ying o li his s uc u e o a scheme s uc u e on P oj(W) . In [LVW] his li ing p oblem is sol ed by using quan um sec ions in- oduced by he second au ho in [VOS], [RVO] and his explains why he de elopmen o he heo y o gauge algeb as he e goes hand in hand wi h ha o quan um sec ions . In ac Noe he ian gauge algeb as a e pa icula Za iski ings in he sense o [LVO, 1, 2, . . .] . Now quan um sec ions a ise in he shea o il a ion deg ee ze o o a mic os uc u e shea o a Za iski ing o e he p ojec i e scheme associa ed o he as- socia ed g aded ing ha is supposed o be commu a i e in [VOS] . The commu a i i y o he associa ed g aded ing is nowhe e essen ial in he s uc u e heo y o he ings o sec ions o hose shea es, excep o cou se *This au ho is suppo ed by an NFWO-g an 694  L . LE BRUYN, F . VAN OYSTAEYEN in he de ini ion o he opological space and he scheme s uc u e o P oj . Howe e , M . A in has ecen ly in oduced in [A ] he quan um p ojec i e space o a quan um n-space gi en by i s g aded quad a ic al- geb a Q as P oj (Q) = Q-g / .F, whe e F is he ull subca ego y o g aded ini e leng h modules, oge he wi h a sui able shi -ope a o . F om his poin o iew i is na u al o y o combine he echniques o Za iski il e ed ings inhe en in he s udy o quan um sec ions wi h he heo y o p ojec i e quan um spaces . This is done by in oducing he class o posi i ely il e ed ings ha ing anon-commu a i e egula in he sense o A in and Schel e [AS] quad a ic algeb a in he sense o Manin, [Man], o he associa ed g aded ing . The es ic ion o posi i e il a ions is no essen ial because he de ini ion o he unde lying p ojec i e scheme may easily be modi ied o deal wi h his ; ne e heless we do es ic o he posi i e case he e . The algeb as oughly de ined abo e a e called gauge algeb as . Since he ing edien e o he heo y lay sp ead ou o e se e al sou ces no all equally a ailable, we ha e concei ed his pape as a su ey pape in oducing necessa y basic de ini ions and p ope ies as well as expanding a ew conc e e examples . A mo e ex ensi e s udy o gauge algeb as is unde aken by he i s au ho in [LB] ; o ecen esul e on o mal quan um sec ions o e o mal schemms we e e o [RVO] . 1 . Fil e ed ings and associa ed g aded ings All ings a e associa i e wi h uni . A il a ion FR on a ing R is gi en by an ascending chain o addi i e subgoups F n R, n E 7L, sa is ying 1 E F O R, F n RF n R C Fn+mR o m, n E 7L . We always assume ha he il a ions conside ed a e exhaus i e, ha is UnE7[FnR = R, and sepa a ed, ha is nnEzFnR = 0 . The ob ious ope a ions induced on he abelian addi i e g oup G(R) = T nE aF n R/F n _ 1 R make G(R) in o a g aded ing wi h G(R) n = FnR/F n _ 1 R, n E 7L . The p incipal sym- bol mapa : R -3 G(R) is de ined by pu ing u(x) = x mod F n _ 1 R whe e n is such ha x E F n R - F,-,R . The Rees ing R = ®n,EZFnR may be iden i ied wi h he sub ing EnEa F n ,RXn o he polynomial ing R[X, X -1 ] . The no ion o he Rees ing o a Z- il a ion ex ends in a na u al way he no ion o he blow-up ing o an I-adic il a ion used in singula i y heo y and commu a i e Za iski ings . A il a ion FR is comple e when Cauchy-sequences con e ge in R, o equi alen ly R = limR/FnR ; when F,,,R = 0 o n< 0 hen we say ha FR is posi- n i e o R is posi i ely il e ed and i is clea ha posi i e il a ions a e comple e . Comple e il e ed ings ha ing a Noe he ian associa ed g aded ing a e an impo an clase o Za iski ings in he sense o [LVO1], in pa icula such a ing R has a Noe he ian Rees ing R . QUANTUM SECTIONS AND GAUGE ALGEBRAS  69 5 Example 1 . The n h Weyl algeb a A n (O) is he algeb a gene a ed by 2n inde e mina es x, .... , x n and yi, . . . , yn sa is ying he commu a- o ela ions : [xi, x j ] = [yi, y j ] = 0, [xi, y j ] = bij . His o ically, An(O) has been in oduced as he ope a o algeb a gene a ed by he compo- nen s xi o he posi ion ec o and he componen s p j = ihy j o he momen um ec o o a quan um pa icle in n-dimensional space . The non- anishing o he commu a o [x j , p j ] = ice exp esses ha one canno ha e simul aneous knowledge o posi ion and momen um o a pa icle in he quan um case . The associa ed g aded ing G(AnP) is he poly- nomial ing C[xi, . . ., xn, Y, . . . , yn] ha is he ope a o algeb a o he classical (Le . non-quan um) si ua ion . So in a sense he il e ed da a may be iewed as quan iza ions o he associa ed g aded da a . The Rees ing An(O) - is he posi i ely g aded algeb a gene a ed by he deg ee one elemen X and xiX, yiX, 1 < i <_ n sa is ying he commu a i e ela ions : [xi, xj ]Xz = [yi, yj ]Xz = 0 and [xi, yj ]Xz = Ói7X2 . Since X is cen al in AnP_ we may subs i u e new inde e mina es X = X,Xá = xiX, Yi = yiX sa is ying homogeneous ela ions A1 X i ] _ [Y¡, Y j ] = 0 and [X,, YI ] = Sij X 2 . The e o e we may iew A n (C) i as a quad a ic ex ension o he en eloping algeb a o he .Heisenbe g algeb a (see below) . No e ha we may specialize X o 1 and we ob ain An(C) asa specializa ion o i s Rees ing ; mo eo e specializing X o 0 yields G(An,(O)) as a specializa ion . 1 .1 . Lemma . we le X s and o he canonical homogeneous cen- al egula elemen o deg ee one in R hen a . R _ /(1 - X)R =R b . R/XR = G(R) . The obse a ions in h_e lemma exp ess ha R is a "de o ma ion" o G(R) ia he Rees ing R . No e ha o any g aded ing wi h a cen al egula homogeneous elemen he cons uc ion in a in he lemma yields he dehomogenized il a ion co esponding o a g ada ion, c . [LV02] . A special case o his dehomogeniza ion p inciple is well-known in p o- jec i e algeb aic geome y (a ine models) and i is also e iden in he ela ion be ween de e minen al ings and Schube cycles . Example 2 . Le g be a ini e dimensional Lie algeb a, say g = Cxl + . . +Cx n wi h de ining ela ions [xi, xj] = 1 : a ~xk sa is ying he Jacobi iden i y . By de ini ion all commu a o s d op in il a ion deg ee o he usual il a ion de ined on he uni e sal en eloping algeb a U(g) . One easily checks ha G(U(g)) = C[xl, . . . , xn] (Poinca é, Bi kho , Wi ) . The Rees ing U(g)- is gene a ed by X, X, = x1X, . . . , X n = x,,,X, 696  L . LE BRUYN, F . VAN OYSTAEYEN sa is ying = [Xi, Xj] = [xi, xj]X 2 =Ek a 1 3 - XkX .  Le us p o ide a mo e conc e e example and conside he 2-dimensional non-Abelian Lie algeb a g =Cx + Cy sa is ying [x, y] = x . Then U(g) - is he egula algeb a in he sense o A in-Schel e [AS] o [ATV1-2] gene a ed by X = xZ, Y = yZ and Z (now Z plays he ole o he cen al elemen X be o e) sa is ying he quad a ic ela ions : X Z - ZX = 0, YZ - ZY= 0, XY - YX - X Z= 0 . We poin ou ha in he classi ica ion o [ATV1] his algeb a is o ype SI, in pa icula i is no o gene ic, i .e . ellip ic, ype . The la e p ope y is one ha U(g)- will sha e wi h all o he gauge algeb as de ined la e in his pape . Simila ly, when g = s12, s1 2 = Cx + Cy + Cz wi h [x, y] = 2y, [x, z] = -2z, [y, z] = x hen he Rees algeb as is he 4-dimensional quad a ic algeb a (o quan um space) gene a ed by X = xT, Y = yT, Z = 2T and T (now playing he ole o he cen al elemen o deg ee one), sa is ying : XT - TX = 0, YT-TY = 0, ZT-TZ = 0, XY-YX-2YT = 0, XZ-ZX+2ZT = 0, YZ-ZY- XT = 0 . Bu again u(s12)_ is no a Sklyanin algeb a in he sense o [SS] . Mos o he algeb as we shall conside in his pape will be posi i ely il e ed howe e some localiza ions o hese will be o in e es oo and so non-posi i e il a ions will appea na u ally . Ex eme amongs he non- posi i e il a ions a e he so-called s ongly il e ed ings . The il a ion FR is said o be s ong i F n RF n ,R = F n+  ,,R holds o e e y n, m E 7L . I is easy o check ha FR is s ong i and only i G(R) is a s ongly g aded ing, i .e . G(R)nG(R),n = G(R) n+n ,, o n, m E 7L ; o i and only i R is a s ongly g aded ing . We say ha G(R) is d-s ongly g aded i G(R) n dG(R) m d = G(R)(n+m)d o n, m E 7L, and d and is minimal as such . Fo a commu a i e posi i ely g aded ing A, P oj(A) is locally s ongly g aded in he sense ha o any Za iski open UC P oj(A) he g aded ing o sec ions is d-s ongly g aded o some d (depending on U) ; when A is gene a ed o e Ao by A l as a ing hen i is e en locally s ongly g aded because e e y g aded ing o sec ions will con ain a uni o deg ee one . Kashiwa a's ing o ge ms o mic o-di e en ial ope a o s on holonomic unc ions p o ides an in e es ing example o a s ongly il e ed ing . Example 3 . Le z =(z,, ... , z n ) be coo dina es in Cn and 1 = . . . , in) he coo dina es o co angen ec o s . Pu Tó (Cn) = {(z, ~), 0}, his is an open subse o C2n and zl, . . . , zn, ~i, . . . , ~n a e holo mo phic unc ions on Tó (en) . Take p = (z*, ~*) ETó(e n ) . Le O p be he local ing o ge ms o holomo phic unc ions (isomo phic o he local ing o con e gen powe se ies in 2n a iables . Following J .E . Bjd k (p . 136, [B]) we le O p (m) be he ~-homogeneous elemen s o o de m . QUANTUM SECTIONS AND GAUGE ALGEBRAS  69 7 I U is an open se in Tó (en) hen O(m)(U) is he se o holomo phic unc ions in U which a e ~-homogeneous o o de m . The ing E p is consis ing o he E , (z, ~) such ha E 0( )(U) o some open neighbou hood U o p and sa is ying he condi ions . i . = 0 o all > w, o a ce ain in ege w . ii . The e exis cons an s A and , such ha lu =sup{i (z,~),(z,~) E U} < A(1 i!)Kwl o all . I F = E , G= Eg N , E E p hen FG = E(a!) á aa in mul i-index no a ion . Now, i F= E , E Ep, hen he unique la ges w E Z such ha  , :7~ 0 is called he o de o F ; his de ines he il a ion o E p and u(F) = w is hen he p incipal symbol . Wi h espec o his il a ion G(E p ) = OZ,,_1[T,T-1] is a s ongly g aded ing which is mo eo e a egula Noe he ian ing o pu e dimension 2n . The ac ha E p is a Za iski ing (see Sec ion 4) en ails all he desi ed p ope ies o E p . Al hough we a e mainly conce ned wi h ings he e i is use ul o es- ablish he co esponding module heo y as well . An R-module M is il e ed i he e is an ascending chain o addi i e subg oups F n M,n E 7L, sa is ying F n RF n M C Fn+n,M o n,m E 7L . The ca ego y R- il is ob ained by aking he il e ed R-modules and he R-linea maps p e- se ing il a ion deg ee o he objec s and mo phisms . We w i e FM o he il a ion o M and G(M) = ®,EZFnM/Fn_1M o he associ- a ed g aded G(R)-module . The G o hendieck ca ego y o g aded G(R)- modules _ will be deno ed _ by G(R)-g . Simila ly, we may de ine he Rees module M o FM by M = ®nEaFnM and iden i y i wi h a submod- ule o M[X, X -1 ] . Again we always assume ha FM is exhaus i e, M = U n M_and_sepa a ed Le . n n F n M = 0 . We m_ay ex end Lemma 1 .1 . o : M/XM = G(M), M/(1- X)M -- M, ú(x) = M[X,X-1] whe e (-)(X)  _ s ands o he objec localized a he cen al mul iplica- i e se o homogeneous elemen s {1 ,X, X2, . . . } . In R-g we ha e a ull subca ego y FX consis ing o he X- o sion ee g aded R-modules . 1 .2 . Lemma . The unc o - : R- il ----> R-g de ines an equi alen e o ca ego ies be ween R- il and .FX . The il e ed mo phism in R- il co esponding o he mo phisms in .FX a e he s ic mo phisms ( ecall ha a il e ed mo phism : M --> N is s ic i FNN nIm = (F n M)) . The unc o G : R- il , G(R)-g is no eally exac bu á is exac o_n s ic mo phims an_d sequences_o s ic mo phisms ; he unc o D : R- g -+ R- il M H M/(X - 1)M, is exac . Wemay de ine a p incipal 69 8  L . LE BRUYN, F . VAN OYSTAEYEN symbol map o m : M --> G(M) by pu ing o m (m) = m mod F,, M when m E F n M - F n _1M . Fo E R such ha QR(a)UM(m) qÉ 0 we ha e QR( )om(m) = om( m) . In pa icula , when G(R) is a domain hen a = R is mul iplica i e . A il a ion FM is said o be a good il a ion i he e exis m,, ... M and d, ... d EZ such ha o all n E 7L, F n M = ~i-1 Fñ_d ¡ Rm i . The u ili y o he Rees objec s is ha p ope ies con- ce ning he il a ion FM_ a e ansla ed o p ope ies in .FX conce ning he X-adic il a ion on M . 1 .3 . P oposi ion . Wi h no a ion as be o e a . FM is sepa a ed i and onl_y i M is X-adically sepa a ed . b . FM is good i and only i M is ini ely gene a ed . c . F_1R C J(FoR) i and only i X E J 9(R), whe e J 9 (-) s ands o he g aded Jacobson adical c . [NVO] . d . FM is comple e i and only i M_is X-adically comple e e . FM is p ojec i e i and only i M is p ojec i e (simila o la - ness) . . Amap : M  + N is s ic i and only i Coke E .F X . 2 . Quan um sec ions as de o ma ions o localiza ions In his sec ion we es ic a en ion o posi i ely il e ed ings R ha ing a commu a i e Noe he ian domain o he associa ed g aded ing . The essen ial pa o his sec ion can and will be conside ably gene alized in a u he sec ion ( o Za iskian il a ions) . Conside a mul iplica i ely closed se S in R, 1 E S, 0 1 S, such ha u(S) is mul iplica i ely closed in G(R) ( his holds au oma ically when G(R) is a domain because u is mul iplica i e in ha case) . Since Q(S) consis s o homogeneous elemen s, (S) -1 G(R) is a g aded ing . In gen- e al S is no an O e se and so one canno necessa ily o m S`R . This d aw-back may be o e come by in oducing he algeb aic mic olocaliza- ion o R a o,(S), we ollow he ideas o [AVV] . To he se S_we associa e he mul iplica i ely closed se S in R, S __ {s, s _ = sX' E R n o s E S such ha s E F,, R-F_ n _1R} . Clea ly 1 E S, 0 1 S and S consis s o homogeneous elemen s_o R ._ Fo n_ E N_ he e is a canonical _ epimo phism o _ g ade _ d ings : in, R/XnR -> R/XR -- G(R) . Le S(n) be he image o S in R/XnR . Now ke ~i n i _ s nilpo en o index n and ~!n(S(n)) = o,(S) is an O e se in G(R), hence S(n) is an O e se in QUANTUM SECTIONS AND GAUGE ALGEBRAS  69 9 R/X'R . The e o e we can de ine : B = Q' (R) = 1lim 95(n)-1(R/XnR), n whe e lim s ands o he g aded in e se limi ( he di ec sum o he 9 in e se limi s o he sys ems ob ained in each deg ee o he g ada ion) . 2 .1 .  Lemma . The i  map js = - ,ú ---_> B is a g aded ing mo phism . We ha e B E .FX and also B/js(R) E .FX, mo eo e B/XB = o,(S)-1G(R), c . [AVV] . A e his lemma we may ake he dehomogeniza ion o B, Q S '(R) _ B/(1 - X)B and we ob ain a il e ed ing Qs(R) such ha js : R Ql(R) is a s ic il e ed mo phism, ha is he il a ion o R is induced by he il a ion o B= Q" (R), and G (B) -- a(S) -1 G(R) . Tha Q 5 ` (R) is comple e (bu no posi i ely il e ed) is easily e y ied . In ac Q S '(R) is no hing bu he mic o-localiza ion a u(S) as de ined by T . Sp inge in [Sp ] . This ollows om he ac ha B has he uni e sal p ope y men ioned in he ollowing . 2 .2 . Lemma . Fo s E S, js(s) is in e ible in B and i B' is ano he il e ed ing such ha FB' is comple e, js : R --- ~ B' is a s ic inclusion and o e e y s E S wi h o,(s) E G(R) n we ha e s -1 E B' wi h u(s -1 ) E G(B')_ n , hen he e exis s a s ic ac o iza ion h : B -> B' such ha hjs = js , (c . [AVV]) . Since Q S '(R) is comple e and G(Q'(R)) = u(S) -1 G(R) is a commu a- i e Noe he ian domain, his ing will be in he class o ings we conside ( hough no posi i ely il e ed) . We may iew Q S '(R) as a "de o ma ion" o o,(S)-1G(R) ia he co esponding Rees ing . When conside ing he s uc u e o P oj(G(R)) we ha e o es ic a - en ion o he pa o deg ee ze o o o,(S)-1G(R), say G(R)(a(s)) = (Q (S)-1G(R))o . In pa icula we ha e : FOQs(R)/F_1QS(R) = G(R)Q(s)) . We de ine he quan um sec ions o R a S o he ing FoQ1(R) = Qs(R)0 equipad wi h he induced il a ion . We deno e his ing by R(s ) . The F-sa u a ion o S is S = { E R, u( ) E u(S)} . I is clea ha S is mul iplica i ely closed, 1 E S, 01 S, and a(S) = o,(S) . 2 .3 . P oposi ion . Wi h no a ion and con en ions as be o e we ha e Q" (R) = Q" (R) . Mo eo e ; S is an O e se in R and o he localizad il a ion on (S) -1 R (being he one induced on i om Q' (R)) we ha e ha Q S '(R) = ((S) -1 R) ^ , i .e . he mic olocaliza ion can always be ob- ained as a classical localiza ion ollowed by a comple ion . 700  L . . LE BRUYN, F . VAN OYSTAEYEN Combining P oposi ion 2 .3 . wi h o egoing p ope ies and P o_pos_i ion 1 .3 .d . we know o a sa u a ed S, i .e . S = S_  _  Qs(R) - : Qs(R) is he g aded X-adic comple ion o  The de ini ion o he g aded comple ion yields ha Q'(R)o = 1S)-1RIXn .(S)-1R)o) n =1~((S)-1R)o/(Xn(S)-1R)o) n =lS)-1R)o/Xn((S)-1R)-n) n llimF o S -1 R/F- n S -1 R = (FoS_1R)n n whe e A s ands o he comple ion wi h espec o he il a ion induced F5 -1 R (o by FQ"(R)) in FOS -1 R . This p o ides a way o calcula e quan um sec ions e ec i ely by i s calcula ing (S -1 R)o and hen a1- lowing he sui able comple ion . Example 4 . [RVO] Conside he i s Wey] . algeb a R = A, (C) . F om he o egoing sec ion we know ha G(A 1 (C)) = C [X, y] and A1 (C) ^' C [X, Y, Z] / (XY - YX - X2, YZ - ZY, X Z- ZX ), whe e we lla e pu X = xZ, Y= yZ, Z being he egula cen al homogenous elem _ en o deg ee one . Conside S= {1, x, x2 . . . } in C [x, y] . The sa u a ion S o S consis s o all elemen s such ha a( ) =Xn o some n and his is an O e se . No e ha all elemen s o _ he o m A + x, A E C, a e con ained in S . The homogeneous O e se is S= { Zn, = xn+Ek+ <n CUMxkya} and we may w i e Zn as a homogeneous o m o deg ee n in he new a iable : X = xZ, Y =yZ and Z, e .g . x2 +y+1 is li ed o X 2 +YZ+Z 2 . In (S --1 R)o we ind he elemen s YX -1 , X -1 Y, ZX -1 , X -1 Z and hese sa is y he ela ions : YX -1 - X -1 Y = (ZX -1 ) 2 , ZX -1 - X -1 Z = 0 . So we ha e o comple e he commu ing ela ions o he gene a o s p = YX -1 and q = ZX -1 . Now qp = ZX -1 YX -1 = Z(YX -1 - Z 2 X -2 )X -1 = YZX -2 -Z 3 X -3 = XY -1 ZX -1 -Z 3 X -3 = pq - q3 . Vence we a i e a [p, q] = q 3 . No e ha Z E J 9 (R) hence Z C- J 9 (9 - 1 ,ú) and he e o e e e y elemen o he o m Xn +ZF n - 1 (X, Y, Z), whe e F n -1 (X, Y, Z) is homogeneous o deg ee n - 1, has o yield an in e ible 1 + ZX -1 (X 1- nF n _ 1 (X,Y_ _ Z)) in he comple ion o _(S -1 R)o because ZX -1 E J 9 (S - 1R)o, X 1-n Fn_1(X, Y, Z) E (S -1 R)o . The e o e we may Conside he algeb a C (p, q)/(pq - qp = q3 ) as de e mining he quan um sec ions up o comple ion . QUANTUM SECTIONS AND GAUGE ALGEBRAS  70 1 Example 5 . Conside he wo-dimensional Lie algeb a g = Cx+Cy wi h [x, y] = x . Then U(g) - is he quad a ic algeb a gene a ed by X, Y, Z sa is ying : XZ-ZX = 0, YZ-ZY = 0, XY-YX = XZ . Le u(S) be {1, y, y2, . . . } in C [x, y] = G(U(g)) .  The sa u a ed O e se in U(g) is S= { = yn + Ek+ < . . aklxky'} and his li s o a homogeneous O e se in U(g) - ,_S = { Z' 2 , E S ha ing Q( ) = yn} . Some ela ions in deg ee ze o o S -1 U (g) - de i e om he de ining ela ions abo e ZY -1 = Y - 'Z, Y -1 X - XY -1 =Y -1 XZY -1 . Conside he canoncial gene a o s p = XY -1 and q =ZY -1 , hen we calcula e qp = ZY-1XY-1 = Z(XY -1 +Y -1 XZY -1 )Y -1 = XZY -2 + ZY-1XY-1ZY-1 = pq + qpq . Yielding he a he odd ela ion [p, q] = -qpq . Howe e , since Y-Z E S we mus in e 1 - q in he quan um sec ions so we may ew i e he commu a ion ela ion as qp = p 11 4 and so we may look a he skew polynomial ing C [[q]] [p, -y] whe e ,y is he au omo phism de ined by p H 1 q q and see ha C [[q]] [p, -y] de e mines he quan um sec ions o U(g) (no e ha as in Example 4, i su iced o in e one elemen , he e Y, in o de o ind up o comple ion he quan um-sec ions) . One should no conclude om he examples 4 . and 5 . ha quan um-sec ions o some Q(S) o he ype {1, a, a 2 . . . . } a e always ha easy o ob ain . Example 6 . Le g be he Lie algeb a 812 and change he 812 - basis such ha [Y, Z] = X, [Z, X] =Y and [X, Y] =Z . Conside he mul iplica i e se Q (S) = {1, X, X2 ,.. . } . The eade may check ha he commu a ion o mulas de e mining he quan um sec ions o U(sl2) a Q(S) may be gi en as [A, B] = (A2 + B2 + 1)C C 2 C2 [A,C]=AC 1+CZ +B 1 +C 2 C z C Z [B,C]=BC 1+C 2-A 1+C 2 whe e A= YX -1 , B= ZX -1 and C= TX-1 . Again i is use ul o in oduce he quan um sec ion o il e ed mod- ules . Fi s , in a way o mally simila o he way Q'(R) had been con- s uc ed we may de ine Q'(M) o any sepa a ed il e ed R-module M 70 8  L . LEBRUYN, F . VAN OYSTAEYEN good il a ion . ii . E e y good il a ion is sepa a ed . I u ns ou h_a i and ii a e equi alen o he ollowing : R is Noe he ian and X E J 9 (R) ( o se e al equi alen s a emen s we e e o [LVO1], and in his case he il a ion FR is said o be Za iskian . E e y comple e il e ed ing R such ha G(R) is Noe he ian is a Za iskian ing and in pa icula he posi i e case wi h Noe he ian associa ed g aded ing is also a pa icula case . The esul s o Sec ion 1 and Sec ion 2 emain alid o Za iski ings in gene al ; e en o Sec ion 3 one can do a lo bu one has o de ine a sui able opological space i s . No e ha we will assume he condi ions i and ii o bo h le and igh modules, so he Za iski ings men ioned he e a e le and igh Za iski ings, as in [LVO, 1,2] . We men ion some undamen al esul s s emming om [LVO, 1, 2, . . .] . 4 .1 .  Theo em . Le R be a Za iski ing . I G(R) has ini e global dimension hen i . gldim R = 1 + gldim G(R) ii . gldim R <_ g gldim G (R) = g gldim R - 1, whe e g gldim s ands o he_gldim in he g aded ca ego y . iii . gldim R= gldim R 4 .2 . Theo em . Le R be a Za iski ing . I _ G(R) i .s a egula Noe he- ian (in he sense o Auslande ) hen R and R a e egula Noe he ian . Recall ha a non-commu a i e ing is egula in he sense o Auslan- de i i has ini e global dimension and e e y ini ely gene a ed le o igh module sa is ies he Auslande condi ion ; ecall ha a ini ely gene a ed R-module M sa is ies he Auslande condi ion i o e e y 0 <_ k < lc = gldim R and any nonze o R-submodule N o Ex k (M, R) we ha e jR(N) > k, whe e jR(-) s ands o he g ade numbe , i .e . he smalles na u al numbe j such ha Ex j(-, R) =,A 0 . 4 .3 .  Co olla y . I 9 is a Noe he ian gauge algeb a hen 9 and 1 a e Noe he ian egula algeb as . I _ G(9) is an n-dimensional quan um space hen : gldim  = 1 + n, gldim 9 <n . Mo eo e , G is an n + 1-dim quan umspace . The "scheme"- heo e ic ea men o gauge algeb as has i s oo s in ying o unde s and he geome y o so-called innocen quan um spaces, i .e . quan um spaces ha a e Noe he ian and posessing a cen al elemen o deg ee one . The innocen quan um space co esponding o a gauge QUANTUM SECTIONS AND GAUGE ALGEBRAS  70 9 algeb a g is i s Rees ing 9 . In [A ] he quan um p ojec i e space o a quan um n-space Q is de ined o be P oj(Q) = Q-g .F, whe e *F is he ull subca ego y o ini e leng h modules, oge he wi h a shi ope a ion [A , De ini ion 1 .2 .] . Fo a commu a i e ing, o one ha is a ini e module o e i s cen e , one may eco e he unde lying scheme s uc- u e om Se e's heo em . In gene al howe e he "scheme" s uc u e o P oj(Q) as de ined abo e is a om being unde s ood . A i s eeling o he unde lying p oblems can be ob ained by conside ing pa icula modules, i.e . he poin - and line-modules, c . [ATV2], and he a poin modules in oduced in [A ] . I we es ic a en ion o he geome y o innocen quan um spaces we can use he gauge algeb a and i s quan- um sec ions o pu a "scheme" s uc u e on P oj(~) which educes he s udy o ( a ) poin -modules o ha o ini e dimensional ep esen a- ions o algebas, c . [LB] . In ac , we may iew P oj(U) as an a ine piece co esponding o he gauge algeb a g and a piece a in ini y iden i- ied o P oj(~/ ~) = P oj(G(Cg)) . Tha is a lowe dimensional p ojec i e quan um space . Assume by induc ion ha we ha e been able o pu a scheme s uc u e on P oj(G(g)) wi h a ine open se s co esponding o some g aded O e se s S,, . . . , Sk and i is no es ic i e o assume ha each o hese O e se s may be gene a ed by a single elemen . Thenwe may co e P oj(~) by open se s co esponding o he quan um sec ions wi h espec o he Si,¡ = l, . . . , k plus he app op ia e glueing mo - phisms . No e ha hese quan um sec ions and hei "glues" wi h Gmay be iewed as a scheme s uc u e on P oj(1) . Le us p o ide some easy examples he e . Example 11 . Reconside he i s Weyl algeb a A l (O) . We use he calcula ions made abo e o ex end he schema ic pic u e in Example 7 by glueing o he open se co esponding o Y(Z) i.e . he a ine piece co esponding o (Aj(C) . We ob ain he ollowing diag am o glueing da a C{XZ -1 ,YZ -1 ,ZX -1 } ~- Aj(C)=C{XZ-1,YZ-1} ~ C{XZ -1 ,YZ -1 , ZY -1 } [XZ -1 ,YZ -1 ] = 1 C{YX -1 , ZX -1 }  C{XY -1 , ZY -1 } [YX -1 , ZX -1 ] = ( ZX -1 ) 3  [XY -1 , ZY -1 ] = (ZY-1)3 1 e{ZX -1 ,YX -1 , XY -1 } To ind i s poin modules we ha e o s udy he one-dimensional ep e- 710  L . LEBRUYN, F . VAN OYSTAEYEN sen a ions and hei glueing da a . V((ZX_ 1) 3)  V((ZY_ 1 )3) V((ZX -1 ) 3 ) n V((ZY -1 ) 3 ) co esponding o he ac ha he associa ed deg ee 3 di iso o he quan um 3-space A1(C)- is Z 3 . This may be pic u ed in he usual p2 . y(y)-poin s z=0 whe e he ci cle means ha he in e sec ion poin is missing and we ha e d awn a double copy o he same z = 0 locus . Example 12 . Reconside he si ua ion o Example 8 . Using he compu a ions and no a ions o ha Example 8, we ob ain he ollowing diag am o glueing da a whe e QUANTUM SECTIONS AND GAUGE ALGEBRAS  71 1 U(g) C{XZ -1 , ZX -1 ,YZ -1 } -- C{XZ -1 ,YZ -1 } --, C[[ZY-1,YZ-1]][XY-1,7] [XZ -1 , YZ -1 ] = XZ-1 C{YX -1 , ZX-1}  CpY-1]][XY-1,7] [YX -1 , ZX -1 ] = (ZX-1)z Only he op le co ne (glueing he en eloping algeb a o he excep- ional quan um space) esembles he commu a i e case . The o he wo co ne s ha e sh unk in dimension . I may be help ul in unde s an_ding hese phenomena o look a he pic u e o poin -modules in P oj(U(g)) pic u ed in he usual p 2 C[[ZY-1]][XY-1,YX-1,7] he y(x)-poin s a e z = 0 wi hou he poin a, he y(z)-poin s a e x = 0 wi hou he poin a, he y(y)-poin s a e a . z=0 Example 13 .  The quan ized Weyl algeb a A l (C, q) as a gauge algeb a . We use no a ion and calcula ions as in Example 9 . Pu R = Al (C, q) . Bo h X and Y a e no malizing in he quan um plane C e [X,Y], so we may de ine a "scheme" s uc u e on P oj(R) by gi ing 712  L . LE BRUYN, F . VAN OYSTAEYEN he ollowing glueing da a . C{XT -1 ,TX -1 ,YT -1 } -- {XT -1 ,YT -1 } -- . C{XT-1,YT-1,TY-1} XT -1 , YT -1 - qYT -1 XT -1 = 1 C{YX -1 ,TX -1 } C{XY -1 ,TY -1 } YX -1 TX -1 -qTX -1 YX -1 =(TX -1 ~ XY -1 TY -1 - 1- ' TY -1 XY -1 =-Q(TY -1 ~ C{YX -1 , XY -1 ,TX -1 } and again one can isualize ' he poin modules as poin s in p 2 . The pic u e co esponds o he ac ha he associa ed deg ee 3 di iso is T(T Z + (q - 1)XY) whe e C is he conic de ined by TZ + (q - 1)XY . Y(T) poin s a e C - {a, b} Y(X) poin s a e C - {b} U (T = 0) - {b} Y(Y) poin s a e C - {a} U (T = 0) - {b} =0 The scheme s uc u e o A l (C, q) - desc i _ bed abo e is c i ical in de in- ing he scheme s uc u e on P oj (W) whe e W is he 4-dimensional quan- um space o he Rees ing o he Wi en gauge algeb as . Fo , i is pos- sible o change he pola iza ion on P oj(G(W)) as in [A ] o ob ain P oj (Al (C, q) -) and use he o egoing in o de o de ine a scheme s uc- u e on P oj(G/W)) ha is hen li ed ia quan um sec ions o P oj(W) as in [LVW] . In a simila way one can s udy he a -poin modules o mul iplici y n (as in [A ]) in an innocen p ojec i e quan um space by glueing oge he he n-dimensional ep esen a ion o he scheme compo- nen s . QUANTUM SECTIONS AND GAUGE ALGEBRAS  71 3 Re e en es [A ] M . ARTIN, Geome y o Quan um Planes, P ep in MIT (1991) . [AS] M . ARTIN AND W . SCHELTER, G aded Algeb as o Global Di- mension Th ee, Ad . Ma h . 66 (1987), 171-216 . [ATV1] M . ARTIN, J . TATE AND M . VAN DEN BERGH, Some Al- geb as Associa ed o Au omo phisms o Ellip ic Cu es, in "The G o hendieck Fes sch i , Vol . I," Bi khause (1990), pp . 33-85 . [ATV2] M . ARTIN, J . TATE AND M . VAN DEN BERGH, Modules o e Regula Algeb as o Dimension Th ee, MIT . [AVV] M .-J . ASENSIO, M . VAN DEN BERGH AND F . VAN OYS- TAEYEN, A New Algeb aic App oach o Mic olocaliza ion o Fil e ed Rings, ' ans . Ame . Ma h . Soc . [B] J . E . BJóRK, "Rings o Di e en ial Ope a o s," Ma h . Lib a y 21, No h . Holland, Ams e dam, 1979 . [LB] L . LE BRUYN, Gauge Algeb as, In p epa a ion . [LVO1] LI HUISHI ANDF . VAN OYSTAEYEN, Za iskian Fil a ions, Comm . i n Algeb a 17(12) (1989), 2945-2470 . [LVO2] LI HUISHI ANDF . VAN OYSTAEYEN, Global Dimension and Auslande Regula i y o Rees Rings, Bull . Soc . Ma h . Belg . [LVW] L . LE BRUYN, F . VAN OYSTAEYEN AND L . WILLAERT, Quan um Sec ions o Schema ic Algeb as, UIA p ep in , No embe (1992) . [Man] YU 1 MANIN,"Quan um G oups and non Commu a i e Geom- e y," Publ . Cen e P ech . Ma h . Mon éal, 1988 . [RVO1] A . RADWAN AND F . VAN OYSTAEYEN, Cohe en Shea es o e Mic os uc u e Shea es, UIA p ep in (1991), P oceedings o he Colma mee ing . [RVO2] A . RADWAN AND F . VAN OYSTAEYEN, Mic o-s uc u e Shea es and Quan um Sec ions o e Fo mal Schemes, UIA p ep in (1991), o appea in Bull . Soc . Ma h . Belg . (1993) . [SVO] R . SALLAM AND F . VAN OYSTAEYEN, A Mic o-s uc u e shea and Quan um Sec ion o e a P ojec i e Scheme, UIA p ep in (1990), J . o Algeb a (1992) . [Sp ] T . SPRINGER, Algeb aic Mic olocaliza ion, in "Sém . M . P . Malli- a in," Lec u e No es in Ma h ., Sp inge Ve lag, Be lin, 1984 . [Schap] P . SCHAPIRA, "Mic odi e en ial Sys ems in he Complex Plane," G undleh en de Ma h . Wiss . 269, Sp inge Ve lag, Be lin, 1985 . 714  L . LE BRUYN, F . VAN OYSTAEYEN [SS] S . P . SMITH AND J . T . STAFFORD, Regula i y o he Fou Di- mensional Sklyanin Algeb a, P ep in Ann . A bo (1990) . [Gin] V . GINSBURG, Cha ac e is ic Va ie ies and Vanishing Cycles, In en . Ma h . 84 (1986), 327-402 . [WIT] E . WITTEN, "Gauge Theo ies, Ve ex Models and Quan um G oups ." [WOR] S . L . WORONOWICZ, Twis ed SU(2)-G oups, an Example o a non-commu a i e Di e en ial Calculus, Publ . R .LM .S, Kyo o Uni- e si y 23 (1987), 117-181 . Uni e si y o An we p UIA Depa men o Ma hema ics 2610 W¡I ijk BELGIUM Rebu el 14 d'Oc ub e de 1991