A cohomological study of local rings of embedding codepth 3

A COHOMO LOGICAL STUD Y OF LOCAL RINGS OF EMBEDDING CODEPTH 3 LUCHEZAR L. A VRAMOV Abstract. The generating series of the Bass num bers µ i R = rank k Ext i R ( k, R ) of lo cal rings R with residue field k ar e computed in closed r ational f orm, in case the embedding dimension e of R and its depth d satisfy e − d ≤ 3. F or eac h s uc h R it is pro ved that there i s a real num b er γ > 1, suc h that µ d + i R ≥ γ µ d + i − 1 R holds for all i ≥ 0, except for i = 2 in tw o explicitly describ ed cases, where µ d +2 R = µ d +1 R = 2. New restrictions are obtained on the mu ltiplicative structures of minimal free resolutions of length 3 o ve r regular l ocal rings. Introduction The pap er concerns coho mological in v ariants of comm utative no etherian lo cal rings. Let R b e such a r ing, m its maximal ideal, and let d deno te the depth of R and e the minimal num b er of g enerator s of m . The num b er e − d is called the emb e dding c o depth of R . It is equal to the length of a minimal free resolution F of b R ov er P , wher e b R is the m -adic completion of R and P a regular lo ca l ring of dimensio n e , for which there is an isomo rphism b R ∼ = P /I ; such an isomorphism alwa ys exists, due to Cohen’s Structure Theorem. F or c ≤ 2 the structure of F , and hence that of b R , is determined by the Hilb ert-Bur ch Theor em. This pa p er is mostly concerned with r ings of co depth 3 , s o we assume c = 3 for the r est of the introduction. There exist then integers l ≥ 2 and n ≥ 1, such that (1) F = 0 − → P n ∂ 3 − − − → P n + l ∂ 2 − − − → P l +1 ∂ 1 − − − → P − → 0 The maps ∂ i are known in a few cas es only . Buchsbaum and Eis enbud describ ed them in [ 14 ] for l = 2, and in [ 1 5 ] when R is Co hen-Macaulay with l = 3 or n = 1 . A. Br own determined ∂ i for cer tain Cohen-Ma caulay rings with n = 2; see [ 13 ]. The pro o fs of thos e theor e ms use the fact that F can b e turned into a g raded- commutativ e DG (that is , differential graded) algebra; see [ 15 ]. Such a structure is not unique in g eneral, but the iso morphism cla ss of the g r aded k -a lg ebra (2) A = F ⊗ P k , where k = R / m , is an inv ariant of R . The p ossible isomorphism classes were determined b y W ey- man [ 34 ] in characteristic zero and by Avramov, Kustin, and Miller [ 11 ] in general. The remark able fact is that for fixed l and n there exist only finitely ma ny p oss ibil- ities for A , de s crib ed ex plicitly by simple m ultiplication tables. These are reviewed in Section 1 , alo ng with other ba ckground materia l. Date : Nov em b er 21, 2018. 2000 Mathematics Subj ect Classific ation. Primary 13D07, 13D40. Key wor ds and phr ases. Lo cal ring, Gorenstein ring, f ree resolution, DG algebra, Bass nu mbers. The author was partly supp orted by NSF grants DMS 080308 2 and DMS 110317 6. 1 2 L. L. A VRAMOV W e ar e interested in classifying non-Gore ns tein rings. A na tural to o l for the task is provided by the Ba ss numb ers µ i R = r ank k Ext i R ( k , R ), which are positive for all i > d when R is not Gorenstein, but v anis h when it is. The Bass series I R R ( t ) = P i > 0 µ i R t i offers a useful format for recording the Bass num b ers of R . As o ur firs t result, Theor em 2.1 , we obtain in clos e d form expressio ns (3) I R R ( t ) = f ( t ) g ( t ) and P R k ( t ) = (1 + t ) e − 1 g ( t ) with f ( t ) , g ( t ) ∈ Z [ t ] , where P R k ( t ) = P i > 0 rank k T or R i ( k , k ) t i is the Poinc ar´ e series of k . That such expressions exist follows fro m [ 11 ], via [ 18 ], and g ( t ) was computed in [ 4 ]. F or the goals of this pap er we need the precise form o f f ( t ) as w ell. In Section 2 the series I R R ( t ) a nd P R k ( t ) a r e computed in parallel. W ork in [ 6 ] and [ 2 ] reduces the pro ble m to finding I A A ( t ) a nd P A k ( t ) fo r the algebra A in (2). T o compute these series we use a battery of change-of-rings results, which are a nalogs of known theo- rems ov e r lo cal rings. T r anslation to the co nt ext of gra de d- commutativ e k -alg ebras requires changes in statements a nd pro o fs; these a re discus sed in Appendix A . It ha s lo ng b een known that that for Gorenstein r ings l is even, see [ 33 ], and that R is Gorenstein if a nd only if A has Poincar´ e duality , see [ 8 ], so n = 1. F urther more, R is complete int ersectio n if and only if A is an exterior alg ebra, see [ 1 ], and then l = c − 1. In Theorem 3.1 we pr ov e that members hip in each o ne of the remaining classes imp os es new restric tions o n the num b er s l and n . The arguments in tro duce ideas that have not b een applied earlier in this con text, suc h as utilizing the DG mo dule structure of Hom P ( F, P ) ov er the DG alg ebra F from (1), a nd analy zing the growth of the Betti n umbers of b R over complete intersection quotient rings of P . In the first three sections the fo cus is on the structur e of rings of co depth 3. The last section is motiv ated by op en problems on the b ehavior of Bass sequences of lo cal r ings in genera l. In the intro duction of [ 17 ], Chr istensen, Striuli, and V e liche collect prec ise questio ns and give a comprehensive s ur vey of earlier results. Theorem 4 .1 gives complete answers in co depth 3 : When R is not Gorens tein (4) µ d + i R ≥ γ µ d + i − 1 R holds for some r eal num be r γ > 1 and every integer i ≥ 1, with a single exception: (5) µ d +2 R = µ d +1 R = 2 when b R ∼ = P / ( wx, w y , z ) and w is P -regular, x, y is a P -regular s e q uence, and z is P / ( w x, wy )-regular . In particular, we recover the asymptotic information known from earlier w or k: The Bass sequence of R even tually is either co nstant or gr ows exp o nentially , see [ 4 ]; it is unbounded when R is Cohen-Ma caulay , but not Gor enstein, see Jor gensen and Leuschk e [ 2 4 ]; if it is unbounded, then (4 ) holds for i ≫ 0, see Sun [ 31 ]. Neither the inequalities in (4), nor the description of the exce ptio ns in (5), are formal consequences of the ra tional expressions in (3). In fac t, extracting infor- mation on the T aylor co efficients of a ratio nal function fro m expr essions for its nu merator and denominator is cla s sically known to be a very har d problem. Our approach is to prove first that µ d + i R > µ d + i − 1 R holds, with the exceptions in (5), by drawing on three distinct sour ces—the express io ns of the co efficients o f f ( t ) and g ( t ) from Theorem 2.1 , the r e lations betw ee n thos e coefficients implied by Theorem 3.1 , and certain growth prop er ties of the Betti num b ers of k that are satisfied whenever R is no t complete intersection. Once the growth of the Bass sequence is established, Theo rem 4.1 ea sily follows from results in [ 4 ] and [ 31 ]. LOCAL RINGS OF EMBEDDING CODEPTH 3 3 Since no additional effort is involv e d, all the results in the pap er are stated and prov ed for lo ca l rings o f embedding co depth at most 3 . 1. Back ground In this pap er we say that ( R, m , k ) is a lo ca l ring if R is a commutativ e no etheria n ring, m its unique maximal ideal, a nd k = R/ m . Recall the inv ariants edim R = r ank k ( m / m 2 ) and depth R = inf { i ∈ Z | µ i R 6 = 0 } . 1.1. The following notation is fixed for the rest of the pap er: e = edim R , d = depth R , c = e − d , and h = dim R − d . W e write K for the K oszul co mplex on a minimal s et of g enerators of m . It is a DG alg ebra ov er R , so its homolo gy is a graded algebra with H 0 ( K ) = k . W e set A = H( K ) and fix notation for the ranks of some k -vector spaces asso ciated with A : l = rank k A 1 − 1 p = rank k ( A 2 1 ) m = ra nk k A 2 q = rank k ( A 1 · A 2 ) n = ra nk k A 3 r = r ank k ( δ 2 ) where δ 2 : A 2 → Hom k ( A 1 , A 3 ) is defined by δ 2 ( x )( y ) = xy fo r x ∈ A 2 and y ∈ A 1 . 1.2. Let b R deno te the m -adic co mpletion of R . Cohen’s Structure Theor em yields b R ∼ = P /I for some r egular lo c a l ring ( P, p , k ) with dim P = e ; that is, I ⊆ p 2 . When I can b e gener ated b y a r egular sequence R is sa id to b e c omplete inter- se ction ; this prop erty is indep e nden t of the choice o f presentation, see [ 16 , 2.3.4(a)]. Let F b e a minimal free resolution o f b R over P and L b e the K oszul complex on a minimal genera ting set of the ma ximal ideal of P . There ar e natural maps (1.2.1) K = R ⊗ R K ≃ − → b R ⊗ R K ∼ = − → b R ⊗ P L ≃ ← − F ⊗ P L ≃ − → F ⊗ P k ; the sy m b ol ≃ denotes a quasi-iso morphism. In particular, a n eq ua lity (1.2.2) rank k A i = r ank P F i holds for every integer i . The Auslander-Buchsbaum Equality now yields (1.2.3) max { i | A i 6 = 0 } = c = pd P b R . Krull’s Principal Idea l Theo rem g ives the inequa lities b elow; the fir st eq ua lity is ( 1.2.2 ) fo r i = 1; the third one comes from the catenarity of the regular ring P : (1.2.4) l + 1 = rank k ( I / p I ) ≥ height P ( I ) = e − dim R = c − h ≥ 0 . By definition, the seco nd inequality b ecomes an equa lity if a nd only if R is r e gular . Since P is Cohen-Macaulay , the first inequality b ecomes a n equalit y precisely when I is genera ted b y a reg ular sequence; that is, when R is complete intersection. When c ≤ 3, the equality P i > 0 ( − 1) i rank P F i = 0 , ( 1.2.2 ), a nd ( 1.2.3 ) give (1.2.5) m = l + n . The following classification is the starting p oint for o ur work and is used through- out the pap er. As alwa ys, V k denotes the exterior algebra functor . The functor s Σ and Hom k ( − , Σ 3 k ) and the construction ⋉ are defined b elow, in 1.5 . 4 L. L. A VRAMOV 1.3. If c ≤ 3, then up to isomo rphism A is describ ed b y the following table, wher e B , C , and D ar e graded k -algebr as, and W a gr aded B -mo dule w ith ( B + ) W = 0 : Class [range] c A B C D C ( c ) [ c ≥ 0] c B V k Σ k c S 2 B ⋉ W k T 3 B ⋉ W C ⋉ Σ ( C / C > 2 ) V k Σ k 2 B 3 B ⋉ W C ⋉ Σ C + V k Σ k 2 G ( r ) [ r ≥ 2] 3 B ⋉ W C ⋉ Hom k ( C, Σ 3 k ) k ⋉ Σ k r H ( p, q ) [ p, q ≥ 0] 3 B ⋉ W C ⊗ k D k ⋉ ( Σ k p ⊕ Σ 2 k q ) k ⋉ Σ k No tw o alg ebras A in the table a re isomo rphic, and neither ar e any tw o algebras B . The table is compiled as fo llows. If c ≤ 1, then A i = 0 for i > c and A 1 ∼ = k c , b y ( 1.2.3 ) and ( 1.2.2 ), whence A ∼ = V k Σ k c . If c = 2, then F is given by the Hilb ert- Burch T he o rem; an explicit multiplication o n F , see [ 5 , 2.1.2 ], yields A ∼ = V k Σ k 2 or A ∼ = k ⋉ W . When c = 3 the po ssible isomo rphism clas ses of A a re determined in [ 34 , P ro of o f 4.1] when k has character is tic 0, and in [ 11 , 2.1] in gener al. In so me case s , the class of a ring and its structure determine each o ther: 1.4. Let R b e a lo ca l ring with edim R − depth R = c ≤ 3. 1.4.1. The r ing R is complete in tersection of co dimension c if and only if it is in C ( c ), as prov ed by Ass m us [ 1 , 2.7], see also [ 16 , 2.3.11 ]; for such rings l = c − 1 . 1.4.2. The ring R is Gor enstein, but no t co mplete int ersectio n, if and o nly if it is in G ( r ) with l = r − 1 and n = 1; for such ring s l is even a nd l ≥ 4. Indeed, R is Gorenstein if and only if A has Poincar´ e duality , by [ 8 ], and then l is even, by J. W atanab e [ 33 , Thm.]; a lternatively , see [ 15 , 2 .1] or [ 16 , 3.4 .1]. 1.4.3. The r ing R is Golo d if and only if it is in S or in H (0 , 0). By definition, R is Golo d if and o nly if a ll Massey pro ducts of elements of A + are trivial. The binary ones ar e just ordinar y pro ducts. Mas sey pro ducts of three or more elements have deg ree at least 4, and A i = 0 for i ≥ 4. Thus, R is Golo d if and only if A 2 + = 0 . B y 1.3 , this o ccurs precisely for the r ings in S or H (0 , 0). W e recall a mo dicum of no tation and facts c o ncerning DG mo dules. 1.5. Let E b e a DG algebra over a comm utative ring S . W e assume E i = 0 fo r i < 0 and that E is gr ade d-c ommu tative , meaning that xy = ( − 1) ij y x holds for all x ∈ E i and y ∈ E j and x 2 = 0 when i is o dd. The DG algebra E acts on its mo dule M from the left. All differen tials hav e degre e − 1 . A morphism of DG mo dules is a degree ze ro E -linear map that comm utes with the differentials; if it induces isomorphisms in homolog y in all degrees , it is called a quasi-isomorphism . F or every s ∈ Z , set ( Σ s M ) j = M j − s for j ∈ Z . The identit y maps o n M j define a bijectiv e map ς s : M → Σ s M of degree s . Setting ∂ Σ s M ( ς ( m )) = ( − 1) s ς s ( ∂ M ( m )) and xς s ( m ) = ( − 1 ) is ς ( xm ) for every x ∈ E i turns Σ s M into a DG E -mo dule. Recall that Hom S ( M , Σ s S ) denotes the DG E -mo dule with Hom S ( M , Σ s S ) j = Hom S ( M s − j , S ), differential ∂ ( µ )( m ) = ( − 1) j +1 µ∂ ( m ) for µ ∈ Hom S ( M , Σ s S ) j , and E acting by ( xµ )( m ) = ( − 1) ij µ ( xm ) fo r x ∈ E i . Set M ∗ = Hom S ( M , S ). The trivial ext ension E ⋉ M is the DG algebra with underlying complex E ⊕ M and pro duct ( x, m )( x ′ , m ′ ) = ( xx ′ , xm ′ + ( − 1) j i ′ x ′ m ) for x ′ ∈ E i ′ and m ∈ M j . LOCAL RINGS OF EMBEDDING CODEPTH 3 5 1.6. Let E be a DG algebr a over S , a nd let M and N b e DG E - mo dules Mo dules T or E i ( M , N ) and Ext i E ( M , N ) over the ring S are defined for every int eger i , see [ 9 , § 1]. If E is a ring, considered a s a DG algebra concentrated in degree 0, a nd M and N are E -mo dules, tre a ted as DG mo dules in a simila r way , then these derived functor s coincide with the classical ones. When k is a field E → k is a homomorphism of DG algebras, and the k -vector s paces T or E i ( M , k ) and Ext i E ( k , N ) have finite rank for ea ch i and v a nis h for i ≪ 0, we set P E N ( t ) = X i ∈ Z rank k T or E i ( M , k ) t i ∈ Z [ [ t ] ][ t − 1 ] . (1.6.1) I N E = X i ∈ Z rank k Ext i E ( k , N ) t i ∈ Z [ [ t ] ][ t − 1 ] . (1.6.2) Every mo rphism of DG algebra s ε : E ′ → E induce s natur a l homomo r phisms of S -mo dules T o r E ′ i ( M , N ) → T or E i ( M , N ) and Ext i E ( M , N ) → Ext i E ′ ( M , N ) for each i ∈ Z . These maps a re bijective when ε is a qua s i-isomor phis m. A gr ade d algebr a ov er S is a DG algebra with zero differential; a gr ade d mo dule ov er a gra ded algebra is a DG mo dule with zero differential. 1.7. L e t K b e the Kosz ul complex K describ ed in 1.1 . The natural map K → k turns k into a DG K - mo dule. F rom [ 2 , 3.2 ] and [ 6 , 4.1], resp ectively , we get P R k ( t ) = (1 + t ) e · P K k ( t ) and I R R ( t ) = t e · I K K ( t ) . If the resolution F in 1.2 has a structure o f DG algebra ov er P , then the natural surjection F 0 = P → k turns k into a DG F -mo dule. The maps in ( 1 .2.1 ) then ar e morphisms of DG alg ebras, so we g et isomor phisms of DG alg e br as (1.7.1) F ⊗ P k = H( F ⊗ P k ) ∼ = A . The inv ariance under quasi-iso morphisms of the DG derived functor s in 1.6 g ives P K k ( t ) = P A k ( t ) and I K K ( t ) = I A A ( t ) . When c ≤ 3 we hav e F i = 0 for i > 3, see ( 1.2.3 ), s o F s upp o r ts a structur e of DG alg ebra ov er P b y [ 15 , 1.3]; s ee also [ 5 , 2 .1.4]. Thus, in this ca se we hav e P R k ( t ) = (1 + t ) e · P A k ( t ) . (1.7.2) I R R ( t ) = t e · I A A ( t ) . (1.7.3) T echniques fo r computing Poincar´ e ser ies a nd Bass series over g raded alg ebras are presented in Appendix A , along with a num b er of examples. 2. Bass series Our g oal in this is section is to prov e the fo llowing res ult. Theorem 2.1. L et ( R, m , k ) b e a lo c al ring, set d = depth R and e = edim R , and let l , n , p , q , and r b e the numb ers defin e d in 1.1 . When e − d = c ≤ 3 ther e ar e e qualities P R k ( t ) = (1 + t ) e − 1 g ( t ) and I R R ( t ) = t d · f ( t ) g ( t ) wher e f ( t ) and g ( t ) ar e p olynomials in Z [ t ] , liste d in t he fol lowing table: 6 L. L. A VRAMOV Class g ( t ) f ( t ) C ( c ) (1 − t ) c (1 + t ) c − 1 (1 − t ) c (1 + t ) c − 1 S 1 − t − l t 2 l + t − t 2 T 1 − t − l t 2 − ( n − 3) t 3 − t 5 n + l t − 2 t 2 − t 3 + t 4 B 1 − t − l t 2 − ( n − 1) t 3 + t 4 n + ( l − 2) t − t 2 + t 4 G ( r ) 1 − t − l t 2 − nt 3 + t 4 n + ( l − r ) t − ( r − 1) t 2 − t 3 + t 4 H (0 , 0) 1 − t − l t 2 − nt 3 n + l t + t 2 − t 3 H ( p, q ) 1 − t − l t 2 − ( n − p ) t 3 + q t 4 n + ( l − q ) t − pt 2 − t 3 + t 4 p + q ≥ 1 All of the Poincar´ e series and a smattering o f the Bass ser ies ab ove are known: R emark 2.2 . Since r ings in C ( c ) are co mplete intersection, see 1.4.1 , the formula for P R k ( t ) is due to T ate [ 32 , Thm. 6]; we have I R R ( t ) = t d bec ause R is Gorens tein. When R is in G ( r ) with r = l + 1 and n = 1, it is Go renstein by 1.4.2 . The formula for P R k ( t ) then is due to Wiebe [ 36 , Sa tz 9 ]; the other for mula gives I R R ( t ) = t d . When R is Golo d, P R k ( t ) is g iven by Golo d [ 19 ] and I R R ( t ) by Avramov and Lescot [ 12 ]. In view of 1.4.3 , this cov ers the rings R in S and H (0 , 0); for R in S , Schej a [ 2 9 , Satz 9] computed P R k ( t ) and Wiebe [ 3 6 , Satz 8] calcula ted I R R ( t ). The for mulas for P R k ( t ) in the r emaining cases were obtained in [ 4 , 3.5 ]. In the pro of that follows the series P R k ( t ) and I R R ( t ) are computed s im ultaneously and in a uniform manner. A s e parate calcula tion is needed for each class. Pr o of of The or em 2.1 . By ( 1.7.2 ) a nd ( 1.7.3 ), it suffices to establish the equalities 1 P A k ( t ) = (1 + t ) · g ( t ) and I A A ( t ) P A k ( t ) = t − c · (1 + t ) · f ( t ) . (Class C ( c )) . The formulas come from ( A.3.1 ) and ( A.4.1 ), r esp ectively . (Class S ) . The for mulas come from ( A.2.1 ) and ( A.2.2 ), resp ectively . (Class T ) . The ex act se q uence 0 → Σ 2 k → C → C /C > 2 → 0 and formulas ( A.1.3 ) and ( A.1.1 ) give P C Σ ( C /C > 2 ) ( t ) = t (1 + t 3 P C k ( t )). Now ( A.5.1 ) and ( A.3.1 ) yield 1 P B k ( t ) = (1 − t 2 ) 2  1 − t 2  1 + t 3 1 (1 − t 2 ) 2  = 1 − 3 t 2 + 3 t 4 − t 5 − t 6 . The isomor phism of k -algebr as B ∼ = E /E > 3 with E = V k Σ k 3 and ( A.9.1 ) give I B B ( t ) P B k ( t ) = t − 4  1 − (1 − 3 t 2 + 3 t 4 − t 5 − t 6 )  − t = 3 t − 2 − 3 + t 2 . LOCAL RINGS OF EMBEDDING CODEPTH 3 7 Using formulas ( A.5.1 ) and ( A.8.1 ) we now obtain: 1 P A k ( t ) = (1 − 3 t 2 + 3 t 4 − t 5 − t 6 ) − t  ( l − 2) t + ( l + n − 3) t 2 + nt 3  = (1 + t )  1 − t − l t 2 − ( n − 3) t 3 − t 5  . I A A ( t ) P A k ( t ) = (3 t − 2 − 3 + t 2 ) +  ( l − 2) t − 1 + ( l + n − 3) t − 2 + nt − 3  = t − 3 (1 + t )( n + l t − 2 t 2 − t 3 + t 4 ) . (Class B ) . Using ( A.1.1 ), ( A.1.3 ), a nd ( A.3.1 ) we obta in P C Σ C + ( t ) = t · P C C + ( t ) = t · t − 1 · ( P C k ( t ) − 1) = 1 (1 − t 2 ) 2 − 1 = 2 t 2 − t 4 (1 − t 2 ) 2 . F rom formulas ( A.5.1 ) and ( A.3.1 ) we now get 1 P B k ( t ) = (1 − t 2 ) 2 − t (2 t 2 − t 4 ) = 1 − 2 t 2 − 2 t 3 + t 4 + t 5 . Assume, fo r the moment, that there is a n exac t sequence o f graded B -mo dules (2.2.1) 0 − → Σ − 2 B + ⊕ Σ − 1 k − → Σ − 2 B ⊕ Σ − 3 B − → B ∗ − → 0 where B ∗ = Hom k ( B , k ). W e then have a string of equalities, where the fir st one comes from ( A.1.2 ), and the second o ne from ( A.1.3 ) and ( A.1.1 ) applied to ( 2.2.1 ): I B B ( t ) P B k ( t ) = P B B ∗ ( t ) P B k ( t ) = t ( t − 2 · t − 1 · ( P B k ( t ) − 1) + t − 1 · P B k ( t )) + t − 3 + t − 2 P B k ( t ) = 1 + t − 2 + t − 3 · 1 P B k ( t ) = t − 3 + t − 2 − 2 t − 1 − 1 + t + t 2 . F ormulas ( A.5.1 ) and ( A.8.1 ) now y ield: 1 P A k ( t ) = (1 − 2 t 2 − 2 t 3 + t 4 + t 5 ) − t  ( l − 1) t + ( l + n − 3) t 2 + ( n − 1) t 3  = (1 + t )  1 − t − l t 2 − ( n − 1) t 3 + t 4  . I A A ( t ) P A k ( t ) = ( t − 3 + t − 2 − 2 t − 1 − 1 + t + t 2 ) +  ( l − 1) t − 1 + ( l + n − 3) t − 2 + ( n − 1) t − 3  = t − 3 (1 + t )  n + ( l − 2) t − t 2 + t 4  . It rema ins to construct the sequence ( 2.2.1 ). T o do this we use the mo dule structures on s us pe ns ions and dual mo dules , describ ed in 1.5 . Recall fro m 1.3 that B = C ⋉ Σ C + , with C = V k Σ k 2 . Choo se a basis { a 1 , a 2 } for C 2 . With b i = ( − 1) i ς ( a i ) for i = 1 , 2 , a 3 = a 1 a 2 and b 3 = a 1 b 2 the set a = { 1 , a i , b i } 1 6 i 6 3 is a k -basis for B . T he non-ze ro pro ducts of elements of a ar e listed b elow: (2.2.2) a 1 a 2 = − a 2 a 1 = a 3 and a 1 b 2 = a 2 b 1 = b 1 a 2 = b 2 a 1 = b 3 . 8 L. L. A VRAMOV Let α ∈ ( B ∗ ) − 2 , resp ectively , β ∈ ( B ∗ ) − 3 be the k -linea r map that sends a 3 , resp ectively , b 3 to 1, and the remaining elements o f a to 0. The map defined by π  ς − 2 ( x ) , ς − 3 ( y )  = xα − ( − 1) i y β for x ∈ B i and y ∈ B i +1 is a morphism π : Σ − 2 B ⊕ Σ − 3 B → B ∗ of gra ded B - mo dules. Its image contains the basis o f B ∗ dual to a , s o π is surjective. Set U = K e r( π ). The surjectivity of π implies rank k U = 7. Set u j =  ς − 2 ( a j ) , ( − 1) j ς − 3 ( b j )  , v j =  ς − 2 ( b j ) , 0  , w =  0 , ς − 3 ( a 3 )  , and u = { u j , v j , w } j =1 , 2 , 3 . It is ea sy to see that u is in U and is linearly indep endent ov er k . Thus, u is a k -basis of U , so there is an isomorphism of vector s pa ces υ : Σ − 2 B + ⊕ Σ − 1 k ∼ = − → U satisfying υ ( a j ) = u j and υ ( b j ) = v j for j = 1 , 2 , 3, a nd υ (1) = w . Simple calcula - tions, using ( 2.2.2 ), yield υ ( bu ) = bυ ( u ) for all b ∈ a and u ∈ u . This means that υ is B -linear, and so v alidates the exac t sequence ( 2.2.1 ). (Class G ( r )) . F or mulas ( A.6.1 ) and ( A.6.2 ) give 1 P B k ( t ) = 1 − r t 2 − rt 3 + t 5 . I B B ( t ) P B k ( t ) = t − 3 − rt − 1 − r + t 2 . F rom formulas ( A.5.1 ) and ( A.8.1 ) we now obtain: 1 P A k ( t ) = (1 − r t 2 − rt 3 + t 5 ) − t  ( l + 1 − r ) t + ( l + n − r ) t 2 + ( n − 1) t 3  = (1 + t )(1 − t − l t 2 − nt 3 + t 4 ) . I A A ( t ) P A k ( t ) = ( t − 3 − rt − 1 − r + t 2 ) +  ( l + 1 − r ) t − 1 + ( l + n − r ) t − 2 + ( n − 1) t − 3  = t − 3 (1 + t )  n + ( l − r ) t − ( r − 1) t 2 − t 3 + t 4  . (Class H ( p, q )) . When p = 0 = q we have A = k ⋉ W , so ( A.5.1 ) and ( A.7.1 ) give 1 P A k ( t ) = 1 − t  ( l + 1) t + ( l + n ) t 2 + nt 3  = (1 + t )(1 − t − l t 2 − nt 3 ) . I A A ( t ) P A k ( t ) = ( l + 1) t − 1 + ( l + n ) t − 2 + nt − 3 − t = t − 3 (1 + t )( n + l t + t 2 − t 3 ) . When ( p, q ) 6 = (0 , 0), using ( A.1.4 ), ( A.2.1 ), and ( A.2.2 ) w e get 1 P B k ( t ) = (1 − pt 2 − q t 3 ) · (1 − t 2 ) . I B B ( t ) = q t − 2 + pt − 1 − t 1 − pt 2 − q t 3 · t − 1 = q t − 3 + pt − 2 − 1 1 − pt 2 − q t 3 . LOCAL RINGS OF EMBEDDING CODEPTH 3 9 F rom ( A.5.1 ) and ( A.8.1 ) we now obtain 1 P A k ( t ) = (1 − pt 2 − q t 3 )(1 − t 2 ) − t  ( l − p ) t + ( l + n − p − q ) t 2 + ( n − q ) t 3  = (1 + t )  1 − t − l t 2 − ( n − p ) t 3 + q t 4  . I A A ( t ) P A k ( t ) = ( q t − 3 + pt − 2 − 1  (1 − t 2 ) +  ( l − p ) t − 1 + ( l + n − p − q ) t − 2 + ( n − q ) t − 3  = t − 3 (1 + t )  n + ( l − q ) t − p t 2 − t 3 + t 4  . These formulas gives the des ired expr essions for P A k ( t ) and I A A ( t ).  3. Classifica tion W e s ig nificantly tigh ten the class ification of rings R of embedding co de pth 3, recalled in 1.3 , by proving that membership in each one of the classes describ ed there impo ses non-tr iv ial restr ictions on the numerical inv a riants of R . Comparis o n with existing exa mples ra ises intriguing questions, discussed at the end of the sec tio n. Theorem 3.1. L et ( R, m , k ) b e a lo c al ring with edim R − depth R = c ≤ 3 . When R is not Gor enst ein the invariants fr om 1.1 satisfy the fol lowing r elations. Class c h ≤ l ≥ n ≥ p q r S 2 1 2 − h 0 = n 0 0 0 T 3 1 3 − h 2 3 0 0 B 3 1 4 − h 2 − h 1 1 2 G ( r ) 3 1 max { 4 − h, r + 1 } 2 − h 0 1 r H ( p, q ) 3 2 max { 3 − h, p, q + 1 , 2 } max { 2 − h, p − 1 , q , 1 } p q q The no tation used in the theorem remains in force for the rest o f the section. R emark 3 .2 . The entries in the co lumns for c , p , q , and r are read off directly from the desc r iption of the gr aded alg ebra A in 1.3 . Some n umerical equalities determine the str uc tur e of the r ing R . Corollary 3.3. Assume c = 3 and R is not c omplete interse ction. The fol lowing c onditions t hen ar e e quivalent. (i) l = q + 1 . (ii) l = p and n = q . (iii) R is in H ( p, q ) with n = p − 1 . (iv) b R ∼ = P / ( J + z R ) , wher e ( P , p , k ) is a r e gu lar lo c al r ing, J an ide al of P with J ⊆ p 2 and rank k ( J / p J ) = l ≥ 2 , and z a P / J -re gu lar element in p 2 . When l o r n is s mall the theorem is complemented by more pr ecise results. 3.4. Let R b e a lo ca l ring with c = 3. 3.4.1. If l = 2 , then b y [ 2 , Pr o of of 7.2] one of the following cases oc c urs: (a) h = 0 and R is in C (3). (b) h = 1 and R is in H (2 , 1) with n = 1, or in H (0 , 0) or H (1 , 0) with n ≥ 1 , o r in H (2 , 0) or T with n ≥ 2. (c) h = 2 a nd R is in H (0 , 0) with n ≥ 1. 10 L. L. A VRAMOV 3.4.2. If l = 3 and h = 0, then b y [ 3 , Pr o of of Thm. 2] R is in one of the classes : (a) H (3 , 2) with n = 2. (b) T with o dd n ≥ 3. (c) H (3 , 0) with even n ≥ 4. 3.4.3. If l ≥ 4 , h = 0, n = 2, and p > 0 , then by [ 13 , 4.5 ] R is in one of the cla sses: (a) B with even l . (b) H (1 , 2) with o dd l . W e start the pro of of the theorem with so me general co nsiderations. Lemma 3.5. Write b R in the form P /I , with P r e gular and dim P = e ; se e 1.2 . When R is n ot c omplete interse ction the fol lowing a ssertions hold. (1) l ≥ c − h = e − dim R ≥ 1 hold. (2) l = 1 implies c = 2 and h = 1 ; furthermor e, I = ( wx, wy ) wher e w is an b R -r e gular element and x, y is an b R -r e gular se quenc e. (3) h ≤ 2 holds; furt hermor e, h = 2 implies that R is in H (0 , 0) . (4) If R is not Gor enstein and c = 3 , then n ≥ 2 − h . Pr o of. (1) F o r mula ( 1.2.4 ) g ives l + 1 > c − h = e − dim R ≥ 1. (2) When l = 1 the ideal I is minimally ge nerated by tw o elements, see ( 1.2.4 ); say , I = ( u, v ). As the regular lo cal r ing P is factor ial, we ha ve u = wx and v = wy with relatively prime x, y . The sequence x, y is r e gular, so p d P b R = 2. As b R is not complete intersection, the element w is non-zero and not inv ertible, so h = 1. (3) F ro m (1) we s ee that h ≤ c − 1 ≤ 2 holds, a nd e quality implies e − dim R = 1. The ring R then is Golo d by [ 5 , 5.2 .5], and so it is in H (0 , 0) by 1.4.3 . (4) F or a ny maxima l R -regula r sequence x standard results, see [ 16 , 1.6.16 ], give A 3 ∼ = (( x ) : m ) / ( x ) ∼ = Hom R/ ( x ) ( k , R/ ( x )) ∼ = Ext d R ( k , R ) 6 = 0 , so n = µ d R ≥ 1. No w r ecall that R is Go renstein if and only if h = 0 and µ d R = 1 .  It is pr ov ed in [ 11 ] that for several clas ses of lo cal rings R , including those of embedding co depth at most 3, there is a complete intersection ring Q and a Golo d homomorphism Q → R . This w as used to sho w that for ev ery finite module M ov er such a ring, P R M ( t ) repr esents a rational function with fixed denominator. In the pr o of of the next lemm a w e turn the tables: By apply ing the form ulas for P Q k ( t ) and P R k ( t ) from Theore m 2.1 we e xpress the Betti num b ers β Q i ( R ) in terms of the numerical inv ar iants of R , defined in 1.1 , then us e informatio n on the asymptotic b ehavior of B etti n umbers ov er complete intersections, obtained in [ 7 ]. Lemma 3.6. Set τ R = 1 for R in T and τ R = 0 otherwise. If c = 3 and R is not c omplete int erse ction, then t he fol lowing dichotomy holds: (a) l ≥ q + 2 and n ≥ p − τ R , or (b) l = q + 1 and n = p − 1 − τ R . Pr o of. Parts (1) and (2) of Lemma 3.5 imply l ≥ 2 and h ≤ 2. If h = 2, then Lemma 3 .5 (3) s hows tha t R is in H (0 , 0). When R is in H (0 , 0) the fir st pair holds and the second fails . Until the end o f the pro of we assume h ≤ 1 and p + q ≥ 1. W e choose an isomo rphism b R ∼ = P /I , as in 1.2 . By taking a close lo ok a t some arguments in [ 11 ], we s et out to show next that I contains a re g ular sequence x, y , such that for Q = P / ( x, y ) the induced map Q → b R is a Golo d homomorphism. LOCAL RINGS OF EMBEDDING CODEPTH 3 11 F or R in T suc h a sequence is found in the pro of of [ 11 , 6 .1]. It is also shown there that if R is in G ( r ), B , o r H ( p, q ) with p + q ≥ 1, then for some x ∈ I and P = P / xP the map P → R is Golo d. As the ideal I = I /xP of P has p ositive height, we can choos e a minimal generator y of I so that its image in R is regular . The natural ma p from Q = P /y P to R is Go lo d b y [ 11 , 5.13]. B y the definition of Golo d homo mo rphisms, see Levin [ 2 6 ], the following eq ua lity then holds: (3.6.1) P Q b R ( t ) = 1 t ·  1 + t − P Q k ( t ) · 1 P R k ( t )  . Insp e cting the tabulated v alues of p and q , see Remark 3.2 , we note that the v arious for ms of P R k ( t ) lis ted in Theorem 2.1 admit an uniform e x pression, na mely , (3.6.2) P R k ( t ) = (1 + t ) e − 1 1 − t − l t 2 − ( n − p ) t 3 + q t 4 − τ R t 5 . Now Q is in C (2), so P Q k ( t ) is given by Theorem 2.1 , hence ( 3 .6 .2 ) and ( 3.6.1 ) yield (1 − t ) · P Q b R ( t ) = 1 − t t  1 + t − (1 + t ) e − 2 (1 − t ) 2 · 1 − t − l t 2 − ( n − p ) t 3 + q t 4 − τ R t 5 (1 + t ) e − 1  = 1 − t t (1 − t ) 2 (1 + t )  (1 − t 2 ) 2 − (1 − t − l t 2 − ( n − p ) t 3 + q t 4 − τ R t 5 )  = 1 t (1 − t 2 )  t + ( l − 2 ) t 2 + ( n − p ) t 3 − ( q − 1) t 4 + τ R t 5 )  = 1 + ( n − p ) t 2 + τ R t 4 1 − t 2 + ( l − 2) t − ( q − 1) t 3 1 − t 2 = 1 + ( l − 2) t + ( n + 1 − p ) t 2 + ∞ X i =1 ( l − 1 − q ) t 2 i +1 + ∞ X i =1 ( n + 1 − p + τ R ) t 2 i +2 . The co mp o site equa lity of formal p ow er pro duces numerical equalities (3.6.3) l − 1 − q = β Q 2 i +1 ( b R ) − β Q 2 i ( b R ) for all i ≥ 1 , n + 1 − p + τ R = β Q 2 i +2 ( b R ) − β Q 2 i +1 ( b R ) for all i ≥ 1 . The ring Q b eing complete intersection, the sequenc e of Betti num ber s of each finite Q -mo dule is ev ent ually either strictly increasing or constan t, s ee [ 7 , 8.1] or [ 5 , 9.2.1(5)]. Thus, the left-hand sides of the equalities in ( 3.6.3 ) are either b oth po sitive or b oth equal to zer o. This is just a r e wording of the des ir ed c o nclusion.  The pro of of the next result, with its use of a DG mo dule structure on a minimal P -free r esolution o f a dualizing co mplex for b R , presents indep endent interest. Lemma 3.7. If c = 3 and R is not Gor enst ein, then l ≥ r + 1 holds. Pr o of. There is nothing to prov e for R in T , as then r = 0 ; see Remark 3.2 . By the same remark, rings in B hav e p = q = 1 and r = 2 . Case (b) in Lemma 3.6 then cannot hold, as it implies n = 0 , a nd case (a) g ives l ≥ 3 = r + 1. Rings R in H ( p, q ) have r = q , see Remar k 3.2 , and Lemma 3.6 g ives l ≥ q + 1. F or the rest of the pro o f we as s ume that R is in G ( r ). Thus, its Koszul homolog y algebra A has the form A = B ⋉ W , where B is a Poincar´ e duality k -algebra with rank k B 1 = r = rank k B 2 , rank k B 3 = 1 , B 1 · B 1 = 0 12 L. L. A VRAMOV and W is a grade d B -mo dule with B + W = 0. F or every gr aded B -mo dule N , set N ′ = Hom k ( N , Σ 3 k ) and endow this graded vector space with the natural B - mo dule structure describ ed in 1.5 . Cho ose β ∈ ( B ′ ) 0 with K er( β ) = B 6 2 . As B has P o inc a r´ e duality , the homo- morphism o f left graded B -mo dules α : B → B ′ with α (1) = β is bijective; thus, (3.7.1) A ′ = B β ⊕ W ′ and α : B ∼ = B β as gr aded B -mo dules, wher e B act on A -mo dules throug h the inclusio n B ⊆ A . As we may a ssume that R is complete, we fix a Cohen presentation R ∼ = P /I , a minimal resolution F of R as a P -mo dule, a DG P -alg ebra structures on F ; s ee 1.2 . Set F ′ = Hom P ( F, Σ 3 P ) and turn F ′ int o a DG F -module, as in 1.5 . Using ( 1.7.1 ) to identify the gra ded algebra s F ⊗ P k and A , w e get isomorphisms F ′ ⊗ P k ∼ = Hom P ( F, k ⊗ P Σ 3 P ) ∼ = Hom k ( F ⊗ P k , Σ 3 k ) = A ′ of graded A -mo dules. Cho ose ξ ∈ F ′ 0 , so that these ma ps send ξ ⊗ 1 to ( β , 0) ∈ A ′ ; see ( 3.7.1 ). The morphism φ : F → F ′ of left DG F -mo dules with φ (1) = ξ satisfies (3.7.2) ( φ ⊗ P k ) | B = α and ( φ ⊗ P k ) | W = 0 . Let Y denote the ma pping cone of φ . W e hav e H i ( F ) = 0 for i ≥ 1 by c hoice, and H i ( F ′ ) = E x t 3 − i P ( R, P ) = 0 for i ≥ 2 b ecause h ≤ 1 holds by L emma 3.5 (3). The exact sequences H i − 1 ( F ) → H i ( Y ) → H i ( F ′ ) now yield H i ( Y ) = 0 for i ≥ 2. Note tha t Y is a b ounded complex of finite free P -mo dules. If y ∈ Y i is an element with ∂ ( y ) = z / ∈ p Y i − 1 , then form a sub complex of Y as follows: Z = 0 → P y ∂ | P y − − − → P z → 0 Since Z is contractible and splits off a s a direct summand of Y , the natura l mor - phism Y → Y / Z is a homotopy equiv alence. Iteration pro duces a ho motopy equiv- alence Y → X , where X is a b ounded complex of finite free P -modules satisfying ∂ ( X ) ⊆ p X , (3.7.3) H i ( X ) ∼ = H i ( Y ) = 0 for i ≥ 2 , (3.7.4) X i ⊗ P k ∼ = H i ( Y ⊗ P k ) for i ∈ Z . (3.7.5) The co nstruction o f Y gives a n isomo r phism of c omplexes o f k -vector spaces Y ⊗ P k ∼ =                    0 W 3 W 2 W 1 ⊕ ⊕ ⊕ B 3 α 3 ( ( P P P P P P P B 2 α 2 ( ( P P P P P P P B 1 α 1 ( ( P P P P P P P B 0 α 0 ( ( P P P P P P P ⊕ ⊕ B 3 β B 2 β B 1 β B 0 β ⊕ ⊕ ⊕ W ′ 2 W ′ 1 W ′ 0 0 where in view of ( 3.7.1 ) and ( 3.7 .2 ) a ll maps not r epresented by arrows a re equal to zero and ea ch α i is bijective. Now ( 3 .7.5 ) yie lds isomorphis ms of vector spaces (3.7.6) X i ⊗ P k ∼ =      W ′ i for i = 0 , 1 , W ′ 2 ⊕ W 1 for i = 2 , W i − 1 for i = 3 , 4 . LOCAL RINGS OF EMBEDDING CODEPTH 3 13 The following eq ualities c o me from the definitions o f W ′ and W , ( 1.2.2 ) and ( 1.2.5 ): rank k W ′ 0 = r ank k W 3 = r ank k A 3 − rank k B 3 = n − 1 , rank k W ′ 1 = r ank k W 2 = r ank k A 2 − rank k B 2 = l + n − r , rank k W ′ 2 = r ank k W 1 = r ank k A 1 − rank k B 1 = l + 1 − r . As a r esult, w e now know that the complex X has the following form: X = 0 − → P n − 1 ∂ 4 − − → P l + n − r ∂ 3 − − → P 2( l +1 − r ) ∂ 2 − − → P l + n − r ∂ 1 − − → P n − 1 − → 0 The inclusion B 1 ⊆ A 1 yield r ≤ l + 1. W e finish the pro of by showing that if r = l + 1, then R is Gor enstein, and that r = l is not p oss ible. If r = l + 1, then X 2 = 0 , so the map ∂ 4 : P n − 1 → P n − 1 is bijective. In v iew of ( 3.7.3 ), this forc e s n = 1, hence X = 0. F r om ( 3.7 .6 ) we g et W = 0, so A has Poincar´ e duality , and hence R is Gorenstein by [ 8 , Thm.]; s ee also [ 16 , 3.4.5]. Assume now r = l . By ( 3.7.3 ) and ( 3.7.4 ), ∂ 2 ( X 2 ) has a minimal free res olution 0 → P n − 1 → P n → P 2 → 0 Since ∂ 2 ( X 2 ) is tor s ion-free, it is isomor phic to an ideal of P minimally generated by t wo elemen ts. Such ideals have pro jective dimensio n one, see Lemma 3.5 (2), hence n = 1. As W 1 6 = 0 , the algebra A do es not hav e Poincar´ e dualit y , so the r ing R is not Gorenstein; see 1.4 .2 . Thus, parts (3 ) and (4) of Lemma 3 .5 imply h = 1; that is, dim R = d + 1. A r esult of F o xby [ 18 , 3 .7] (for equicharacteris tic R ) and Rob erts [ 2 8 ] (in general) now gives µ d +1 R ≥ 2. This inequality , the exact sequence 0 → E xt d +1 R ( k , R ) → A 2 δ 2 − → Hom k ( A 1 , A 3 ) of [ 8 , Pr op. 1], see also [ 16 , 3.4.6], the equality ( 1.2 .5 ) , and our assumption yield 2 ≤ µ d +1 R = l + n − r = 1 . W e hav e o btained a contradiction, and this finishes the pr o of of the lemma.  Pr o of of The or em 3.1 . F o r the v alues of c , p , q , and r , see Remar k 3.2 . Lemma 3.5 (3) yields h ≤ 2, with strict inequality when R is not in H (0 , 0). F or l and n we argue one class at a time. (Class S ) . W e hav e l ≥ 2 − h b y Lemma 3.5 (1) a nd n = 0 by ( 1.2.3 ). (Class T ) . Lemma 3.5 (1) gives l ≥ 3 − h . Rings in T hav e q = 0, s o ca se (b) in Lemma 3.6 implies l = 1 ; since c = 3, this is r ule d out b y L e mma 3.5 (2). Thus, the inequalities (a) of Le mma 3.6 hold, a nd they g ive n ≥ 2. (Class B ) . Lemma 3.5 (4) gives n ≥ 2 − h ≥ 1, w hile Lemma 3.7 yields l ≥ r + 1 = 3. By 3.4.2 , the class B co ntains no ring with h = 0 and l = 3, so l ≥ 4 − h holds. (Class G ( r )) . H ere p = 0 , so case (b) in Lemma 3.6 gives n = − 1, which is absurd. Thu s, case (a) holds, whe nce l ≥ 3. By 3.4.2 , in G ( r ) there a re no r ings with h = 0 and l = 3 , hence l ≥ 4 − h holds. So do es l ≥ r + 1, by Lemma 3.7 . (Class H ( p, q )) . Parts (1) and (3) of Lemma 3.5 give l ≥ max { 3 − h, 2 } . By definition, A = ( C ⊗ k D ) ⋉ W with C = k ⋉ ( Σ k p ⊕ Σ 2 k q ), D = k ⋉ Σ k , and C + W = 0 = D + W . The relations A 1 ) C 1 ∼ = C 1 ⊗ k D 1 = A 1 · A 1 imply l ≥ p , while A 3 ⊇ A 1 · A 2 = C 2 ⊗ k D 1 ∼ = C 2 yield n ≥ q . O n the o ther ha nd, from Lemma 3.6 we obtain the inequalities l ≥ q + 1 and n ≥ p − 1.  14 L. L. A VRAMOV Pr o of of Cor ol lary 3.3 . When l = q + 1 the v alues of q and b ounds for l in Theor em 3.1 show that R is in H ( p, q ). Lemma 3.6 now g ives n = p − 1, so (i) implies (iii). If (iii) holds, then we hav e a string l ≥ p = n + 1 ≥ q + 1 = l , where the inequalities co me fr o m Theor e m 3.1 , the fir st equality is given by L e mma 3.6 , and the sec ond one holds by hypothesis. W e get l = p a nd n = q , which is (ii). Assuming that (ii) holds, we see from Theorem 3.1 that R is in H ( p, q ). The description of A in 1.3 then yields A ∼ = C ⊗ k D with D = k ⋉ Σ k . In particular , if a is a non-zero element in 1 ⊗ k D 1 , then A is fre e as a graded mo dule over its subalgebra gener ated by a . Now [ 4 , 3.4 ] shows that (iv) holds. When (iv) holds T or P i ( P / J, P /z P ) = 0 for i ≥ 1, so we have isomor phisms A ∼ = T or P ( P / ( J + z P ) , k ) ∼ = T or P ( P / J, k ) ⊗ k T or P ( P / z P, k ) ∼ = T or P ( P / J, k ) ⊗ k ( k ⋉ Σ k ) of gr aded k -algebra s. They imply T or P 2 ( P / J, k ) ⊗ k Σ k ∼ = A 3 and p d P ( P / J ) = 2 the la tter b ecause A i = 0 holds for i > 3 b y ( 1.2.3 ). W e get a string of equalities q = rank k A 3 = rank k T or P 2 ( P / J, k ) = rank k ( J / p J ) − 1 = l − 1 , where the third one comes from ( 1.2 .5 ) and ( 1.2.4 ). Thus, (iv) implies (i).  T o complete the cla ssification of rings R with dim R − depth R = 3 along the lines of 1.3 and the r esults in this sec tio n, one needs to determine for those ring s al l the restrictions satis fied by the inv ariants in 1.1 . This leads to : Question 3.8 . Whic h s extuples ( h, l , n, p, q , r ), allow ed by Theorem 3.1 , Corollary 3.3 , or the r e s ults cited in 3 .4 , ar e realized b y so me lo cal ring R with c = 3 ? The lis t of av a ilable answers is no t lo ng and runs as follows. 3.9. Let ( P, p , k ) b e a reg ular lo cal with dim P = e ≥ 3 and x 1 , . . . , x e a minimal set of generato rs of p . W e describ e rings R = P /I with c = 3 by sp ecifying I . 3.9.1. The rings admitted by 1.4.2 , 3.4.1 , 3.4.2 , and 3.4 .3 are realiz ed by ideals I constructed in [ 15 , 6.2], [ 2 , 7.7 ], [ 3 , Rem. (1), p.171], and [ 13 , 3 .4, 3.6], resp ectively . 3.9.2. The following sextuples ( h, l, n, p, q , r ) with l = q + 1 a re realized: (a) (0 , 2 , 1 , 3 , 1 , 3) by I = ( x 2 1 , x 2 2 , x 2 3 ). (b) (0 , l , l − 1 , l , l − 1 , l − 1) b y I = ( x 1 , x 2 ) l − 1 + ( x 2 3 ) for each l ≥ 3. (c) (1 , l , l − 1 , l , l − 1 , l − 1) b y x 1 ( x 1 , x 2 ) l − 1 + ( x 2 3 ) for each l ≥ 2. There ar e no other sex tuples with l = q + 1, b y Co rollar y 3.3 a nd Lemma 3.5 (3). 3.9.3. Every s e xtuple (2 , l , n, 0 , 0 , 0) with l ≥ 2 a nd n ≥ 1 is realized when k = C . Indeed, for each such pair ( l, n ) W eyman [ 35 ] shows that P = C [ [ x 1 , . . . , x e ] ] contains a n ideal J with ra nk k T or P 1 ( P / J, k ) = l + 1 a nd ra nk k T or P 3 ( P / J, k ) = n . On the other hand, if w is a P -reg ular element, then P /J with J = w I realizes the sextuple (2 , l, n, 0 , 0 , 0), since for each i ≥ 1 there are isomor phisms o f vector spaces T or P i ( P / I , k ) ∼ = T or P i − 1 ( I , k ) ∼ = T or P i − 1 ( wI , k ) ∼ = T or P i ( P / J, k ) , and Shamash [ 30 , Thm. (3), p. 467] shows that P / w I is Golo d; se e a lso [ 5 , 5 .2.5]. There ar e no other sex tuples with h = 2 , by Lemma 3.5 (3). The o nly known examples in G ( r ) are the Gorenstein rings. W e pr o p ose: LOCAL RINGS OF EMBEDDING CODEPTH 3 15 Conje ctur e 3.10 . If R is in G ( r ) for some r ≥ 2, then R is Gorenstein. Lemma 3.7 is a first step tow ards a verification of this statement. If proved in full, it will eliminate a n entire family from the cla ssification in Theorem 3.1 . Another elusive cla ss is B , for which the only examples are those in [ 13 ]. Rings in T app ear in several situations, a nd the families H ( p, q ) s eem to b e ubiquito us. 4. Bass numbers The following theorem is the third main res ult of this pap er . Theorem 4.1. L et ( R, m , k ) b e a lo c al ring, and set e = edim R and d = depth R . When e − d ≤ 3 and R is not Gor enstein ther e is r e al numb er γ R > 1 , such t hat (4.1.1) µ d + i R ≥ γ R µ d + i − 1 R holds for every i ≥ 1 , with two exc eptions for i = 2 : If ther e exists an isomorphism b R ∼ = P / ( wx, w y ) or (4.1.2) b R ∼ = P / ( wx, w y , z ) , (4.1.3) wher e ( P , p , k ) is an e - dimensional r e gular lo c al ring, w a P -r e gular element, x, y a P -r e gular se quenc e, and z a P / ( wx, w y ) -r e gular element in p 2 , then µ d +2 R = µ d +1 R = 2 . In p articular, when R is Cohen-Mac aulay the ine qu alities ( 4.1.1 ) hold for al l i . The theorem should b e viewed in the context of a n um b er of pr oblems raised in recent publications, sometimes under the hypothesis tha t R is Cohen- Macaulay . W e say that a sequence ( a i ) of real num b ers is said to hav e str ongly exp onential gr owth if β i ≥ a i ≥ α i hold for all i ≫ 0 for so me rea l n umbers β ≥ α > 1. Questions 4 .2 . Assume that ( R, m , k ) is a non-Go renstein lo cal ring. (1) Determine the num b er inf { j ∈ Z | µ d + i R > µ d + i − 1 R for all i ≥ j } . (See [ 17 , 1.3].) (2) Do es µ d +1 R > µ d R alwa ys hold? (See [ 24 , 2 .6].) (3) Do es µ i R ≥ 2 hold for a ll i > dim R ? (See [ 17 , 1 .7].) (4) Do es the se q uence ( µ i R ) have strong ly exp onential growth? (See [ 24 , p. 64 7].) All of these q ue s tions ar e op en in general. Here is a list of the known a nswers: R emark 4.3 . Assume that ( R , m , k ) is not Gor enstein. (1) An ineq uality µ d + i R > µ d + i − 1 R holds for i ≥ 1 in the following cases: (a) m 3 = 0 ; se e [ 17 , 5 .1]. (b) R ∼ = Q/ (0 : q ) for some Go renstein lo ca l ring ( Q, q , k ); see [ 17 , 6 .2]. (c) R ∼ = S × k T with S 6 = k 6 = T , exc ept when S is a discrete v a luation ring, and either edim T = 1 > dim T or edim T = 2 = dim T ; s ee [ 17 , 3.3]. (d) R is Golo d, exc ept when e − d = 2 and µ d R = 1; see [ 17 , 2.4 ]. In a ddition, µ d + i R > µ d + i − 1 R is k nown to hold in the fo llowing cases: (e) for i ≥ 3 if R is among the exceptions in (c) and (d); se e [ 17 , 2 .5, 3.2 ]. (f ) for i ≫ 0 if R is C o hen-Macaulay with e − d ≤ 3; see [ 24 , 1.1 ]. (2) holds whe n R is Cohen-Macaulay and is generically Gor enstein; see [ 24 , 2.3 ]. (3) holds whe n R is a domain, see [ 27 , p. 67 ], or is Cohen- Ma caulay; see [ 17 , 1.6 ]. (4) holds in case s (a) through (f ) of (1); see the references g iven ab ove. 16 L. L. A VRAMOV F or rings with e − d ≤ 3, The o rem 4.1 provides shar p answers to all the questions in 4.2 . The next rema rk shows that the theorem also implies 4.3 (1)(e). R emark 4.4 . Let S b e a dis crete v a luation r ing and ( T , t , k ) a lo cal ring . If edim T = 1 and t s = 0 6 = t s − 1 , then R = S × k T satisfies b R ∼ = P / ( wx, w s ), where ( P , p , k ) is a regular lo cal ring and { w , x } is a minimal generating set for p . If T is regular of dimension 2, then R = S × k T sa tisfies b R ∼ = P / ( wx, w y ), where ( P, p , k ) is a reg ular lo cal ring and { w , x, y } is a minimal gene r ating set for p . In prepa ration for the pro of of Theor em 4.1 , w e establish a technical re s ult where the h yp otheses a re made on the Bas s ser ies of R and the Poincar´ e se r ies of k , not on the ring R itself. The argument relies on general pro p erties o f P R k ( t ). Lemma 4.5. L et ( R , m , k ) b e a lo c al ring and let d , e , l , m , and p b e as in 1.1 . Assume ther e exist p olynomials f ( t ) and g ( t ) in Z [ t ] , su ch that (4.5.1) P R k ( t ) = (1 + t ) e − 1 g ( t ) and I R R ( t ) = t d · f ( t ) g ( t ) . (1) If P ∞ i =0 a i t i is t he T aylor exp ansion of ( f ( t ) − g ( t )) / (1 − t 2 ) , then µ d R = a 0 + 1 , µ d +1 R − µ d R = a 1 − 1 , µ d +2 R − µ d +1 R = a 2 + ( l − 1) a 0 . In c ase l ≥ 1 and a i is non-ne gative for i ≥ 1 the fol lowing ine qualities hold: µ d + i R − µ d + i − 1 R ≥ a i + ( l − 1) a i − 2 ≥ a i for i ≥ 2 . (2) If P ∞ i =0 b i t i is the T aylor exp ansion of f ( t )(1 + t 3 ) s / (1 − t 2 ) 2 , wher e s is an inte ger satisfying 0 ≤ s ≤ m − p , then µ d R = b 0 , µ d +1 R − µ d R = b 1 , µ d +2 R − µ d +1 R = b 2 + ( l − 2) b 0 . In c ase l ≥ 2 and b i is non-ne gative for i ≥ 1 the fol lowing ine qualities hold: µ d + i R − µ d + i − 1 R ≥ b i + ( l − 2) b i − 2 ≥ b i for i ≥ 2 . R emark. One has m − p = rank k A 2 − rank k ( A 1 ) 2 = r ank k ( A 2 / ( A 1 ) 2 ) ≥ 0 . Pr o of. Recall that the Poincar´ e ser ies of k ca n b e written a s a pro duct (4.5.2) P R k ( t ) = (1 + t ) e (1 + t 3 ) m − p (1 − t 2 ) l +1 · Q ∞ i =2 (1 + t 2 i +1 ) ε 2 i +1 Q ∞ i =1 (1 − t 2 i +2 ) ε 2 i +2 with non- negative integers ε j ≥ 0 ; se e [ 22 , 3 .1.2(ii), 3.1 .3] or [ 5 , 7.1.4 , 7.1.5 ]. T o compare consecutive Bass num b ers, we will use the identit y (4.5.3) X i ∈ Z ( µ d + i R − µ d + i − 1 R ) t i = (1 − t ) I R R ( t ) t d . LOCAL RINGS OF EMBEDDING CODEPTH 3 17 (1) In view of ( 4.5.2 ), fo r j ≥ 0 there exist non-negative integers c j , such that P R k ( t ) = (1 + t ) e (1 − t 2 ) l +1  1 + ∞ X j =3 c j t j  . F ormulas ( 4.5.3 ) and ( 4.5.1 ) give equalities X i ∈ Z ( µ d + i R − µ d + i − 1 R ) t i =  1 + f ( t ) − g ( t ) g ( t )  (1 − t ) = 1 − t + f ( t ) − g ( t ) 1 − t 2 1 (1 − t 2 ) l − 1  1 + ∞ X j =3 c j t j  = 1 − t +  ∞ X i =0 a i t i  1 + ( l − 1) t 2 + ∞ X j =3 d j t j  with d j ≥ 0 for j ≥ 3. They yield µ d R = a 0 + 1 and the expr e ssions for µ d + i R − µ d + i − 1 R when i = 1 , 2. In case l ≥ 1, and a i ≥ 0 ho lds for i ≥ 1, we get the following relations, wher e < denotes a co efficient wise inequality of formal power series X i ∈ Z ( µ d + i R − µ d + i − 1 R ) t i < 1 − t +  ∞ X i =0 a i t i   1 + ( l − 1) t 2  = a 0 + ( a 1 − 1) t + ∞ X i =2  a i + ( l − 1) a i − 2  t i . They imply the des ired lower b ounds for µ d + i R − µ d + i − 1 R when i ≥ 2. (2) In view of ( 4.5.2 ), we can write P R k ( t ) in the fo rm P R k ( t ) = (1 + t ) e (1 + t 3 ) s (1 − t 2 ) l +1  1 + ∞ X j =3 c j t j  with non- negative integers c j . F ormulas ( 4.5.3 ) a nd ( 4 .5.1 ) give equalities X i ∈ Z ( µ d + i R − µ d + i − 1 R ) t i = f ( t )(1 + t 3 ) s (1 − t 2 ) 2 1 (1 − t 2 ) l − 2  1 + ∞ X j =3 c j t j  =  ∞ X j =0 b j t j  1 + ( l − 2) t 2 + ∞ X j =3 d j t j  with d j ≥ 0 for j ≥ 3. They yield µ d = b 0 , and the expressio ns for µ d + i R − µ d + i − 1 R when i = 1 , 2. When l ≥ 2, and b i ≥ 0 holds for i ≥ 1, we als o hav e X i ∈ Z ( µ d + i R − µ d + i − 1 R ) t i <  ∞ X i =0 b i t i   1 + ( l − 2) t 2  = b 0 + b 1 t + ∞ X i =2  b i + ( l − 2) b i − 2  t i . The desired low er bounds for µ d + i R − µ d + i − 1 R when i ≥ 2 follow from here.  The nex t lemma is the ma jor step towards the pro of of Theorem 4.1 . 18 L. L. A VRAMOV Lemma 4.6. If ( R, m , k ) is a non-Gor en s t ein lo c al ring with e − d ≤ 3 , then µ d + i R ≥ µ d + i − 1 R + 1 holds for i ≥ 1 , unless i = 2 and b R is describ e d by ( 4.1.2 ) or ( 4.1.3 ) , and t hen µ d +2 R = µ d +1 R = 2 . Pr o of. Once ag ain, there are several different cases to consider . (Class S ) . Theorem 2.1 gives (1 − t − l t 2 ) I R R ( t ) = t d ( l + t − t 2 ), hence µ d R = l , µ d +1 R − µ d R = 1 , µ d +2 R − µ d +1 R = l 2 − 1 , µ d + i R − µ d + i − 1 R = lµ d + i − 2 R ≥ 2 for i ≥ 3 . W e get µ d + i R ≥ µ d + i − 1 R + 1 for all i ≥ 1 , except when i = 1 a nd l = 1, and then µ d +2 R = µ d +1 R = 2. F urthermore, l = 1 implies ( 4.1.2 ) by Lemma 3.5 (2). F or the res t of the pro of we assume c = 3 a nd let f ( t ) and g ( t ) b e the po lynomials from Theor em 2.1 , satisfying P R k ( t ) = (1 + t ) e − 1 /f ( t ) and I S S ( t ) = t d f ( t ) /g ( t ). (Class T ) . T he v a lue of f ( t ) from Theor e m 2.1 provides the first equa lit y b elow: f ( t ) (1 − t 2 ) 2 = n + l t − 2 t 2 − t 3 + t 4 (1 − t 2 ) 2 = ( n − 2 t 2 + t 4 ) ∞ X j =0 ( j + 1) t 2 j + ( lt − t 3 ) ∞ X j =0 ( j + 1) t 2 j = n + ∞ X j =0  ( l − 1) j + l  t 2 j +1 + ∞ X j =1 ( n − 1)( j + 1 ) t 2 j Theorem 3.1 g ives l , n ≥ 2 , so Lemma 4.5 (2) applies with s = 0 and yields µ d +1 R − µ d R = l − 1 ≥ 1 , µ d +2 j R − µ d +2 j − 1 R ≥ ( n − 1 )( j + 1 ) ≥ 2 for j ≥ 1 , µ d +2 j +1 R − µ d +2 j R ≥ ( l − 1 ) j + l ≥ 2 for j ≥ 1 . (Classes B , G ( r ), and H (0 , 0 )) . Theorem 3 .1 provides uniform expre s sions, f ( t ) = n + ( l − r ) t − ( r − 1) t 2 + ( p − 1 ) t 3 + q t 4 and g ( t ) = 1 − t − l t 2 − ( n − p ) t 3 + q t 4 , for the p olynomials that app ear in T heo rem 2.1 . Using them, we obtain f ( t ) − g ( t ) 1 − t 2 = ( n − 1)(1 + t 3 ) + ( l + 1 − r )( t + t 2 ) 1 − t 2 = ( n − 1 ) + ( l + 1 − r ) t + ( l + n − r )  ∞ X j =2 t j  . LOCAL RINGS OF EMBEDDING CODEPTH 3 19 Lemma 3.7 gives l + n − r ≥ l + 1 − r ≥ 2, so the series above has non-negative co efficients. Th us, Lemma 4.5 (1) applies a nd yields µ d +1 R − µ d R = l − r ≥ 1 , µ d + i R − µ d + i − 1 R ≥ l + n − r ≥ 2 for i ≥ 2 . (Class H ( p, q ) with p + q ≥ 1) . Theorem 2.1 g ives f ( t )(1 + t 3 ) (1 − t 2 ) 2 = n + ( l − q ) t − pt 2 − t 3 + t 4 (1 − t 2 ) 2 (1 + t 3 ) = n + ( l − q ) t + (2 n − p ) t 2 + ∞ X i =3 b i t i , where for i ≥ 3 the n umbers b i are defined by the for mulas b 2 j +1 = ( l − q + n − p ) j + l + p − q − 2 for j ≥ 1 , b 2 j = ( l + n − p − q ) j − l + n + q + 1 for j ≥ 2 . By Theorem 3.1 , we hav e l − q ≥ 1, n − p ≥ − 1, and n ≥ 1 , hence 2 n − p = n + ( n − p ) ≥ n − 1 ≥ 0 , b 2 j +1 ≥ b 3 = n − 2 + 2( l − q ) ≥ n ≥ 1 for j ≥ 1 , b 2 j ≥ b 4 = n + 2 ( n − p ) + ( l − q ) ≥ n ≥ 1 for j ≥ 2 . Thu s, Lemma 4.5 (2) a pplies with s = 1. With the pr eceding inequa lities, it gives µ d +1 R − µ d R = l − q ≥ 1 , µ d +2 R − µ d +1 R = 2 n − p + ( l − 2) n = l n − p ≥ l ( n − 1) ≥ 0 , µ d + i R − µ d + i − 1 R ≥ b i ≥ 1 for i ≥ 3 . W e conclude that µ d + i R ≥ µ d + i − 1 R + 1 holds for all i ≥ 1 , except when i = 2 and l n − p = l ( n − 1 ) = 0. T o finish the pro o f, we unrav e l this sp ecia l case. The last tw o equalities for ce n = 1 and l = p . Now Theor em 3.1 gives the inequalities in the string 2 = n + 1 ≥ p = l ≥ 2, whence l = p = n + 1 = 2. Thus, we hav e shown that condition (iii) in Co r ollary 3.3 holds with n = p − 1 = 1. F rom condition (ii) in that cor ollary we g et q = n = 1, so the for mulas above yield µ d +2 R = µ d +1 R = µ d R + l − q = n + l − q = 2 . On the other hand condition (iv) gives a n isomorphism b R ∼ = P / ( J + z R ), where ( P, p , k ) is a reg ular lo ca l ring, J is a n ideal of P con tained in p 2 and minimally generated by 2 elements, and z is an element of p 2 that is regular on P /J . Since R is not complete in tersection, neither is P / J , whic h means that J = ( w x, wy ) for some non-zero element w in p a nd P -regula r s equence x, y . Thus, ( 4.1.3 ) holds.  Pr o of of The or em 4.1 . W e may a s sume tha t R is complete. A construction of F oxb y , see [ 18 , 3.10], then yields a finite R -mo dule N , such that (4.7.1) µ d + i R = β R i ( N ) for all i ≥ dim R − d . By [ 4 , 1 .4 a nd 1.6], when c ≤ 3 the Betti sequence of every finite R -mo dule either has strongly expo nential gr owth or is even tually constant. Since R is not Gorenstein, Lemma 4.6 rules o ut the se cond c a se for the mo dule N in ( 4.7.1 ). Thu s, β R i ( N ) ≥ α i holds for some r eal num be r α > 1 and all i ≫ 0. 20 L. L. A VRAMOV The series P R N ( t ) conv er ges in a cir c le of r adius ρ > 0, s ee [ 5 , 4.1.5 ]. As ρ is equal to lim sup i { 1 / i p β i R ( N ) } we get 0 < ρ ≤ 1 /α < 1 . Fix a re a l num b er β sa tisfying (4.7.2) 1 /ρ > β > 1 . Sun [ 31 , 1.2(c)] prov ed that there is an integer f , such that (4.7.3) β R i ( N ) ≥ β β R i − 1 ( N ) holds fo r all i ≥ f + 1 . Set j = max { 3 , dim R − d, f } and define r eal num be r s γ ′ and γ ′′ by the formulas γ ′ = min  β , µ d +1 R /µ d R , min { µ d + i R /µ d + i − 1 R } 3 6 i 6 j  , γ ′′ = min  γ ′ , µ d +2 R /µ d +1 R  . In vie w of ( 4.7.1 ) a nd ( 4 .7.3 ), the following ine q ualities then hold: µ d + i R ≥ ( γ ′ µ d + i − 1 for i = 1 and i ≥ 3 , γ ′′ µ d + i − 1 for i ≥ 1 . F rom ( 4.7.2 ) a nd Le mma 4.6 w e see tha t γ ′′ > 1 holds unless b R satisfies ( 4.1.2 ) or ( 4.1.3 ), els e γ ′ > 1 holds and µ d +2 R = µ d +1 R = 2 . This is the desir ed result.  Appendix A. Graded a lgebras Here k denotes a field and B a graded k -algebra that is graded-commutativ e , has B 0 = k and B i = 0 for i < 0, and rank k B is finite; set B + = B > 1 . In a ddition, M and N deno te finitely generated graded B -mo dules; we s et H M ( t ) = X i ∈ Z rank k M i t i . W e treat B as a DG alge bra a nd M , N as DG B -mo dules, a ll with zero differ- ent ials. As a conseq uence, T or B i ( M , N ) and Ext i B ( M , N ) ar e formed a s in 1.6 . The k -spaces T or B i ( M , N ) and Ext i B ( M , N ) are finite for each i ∈ Z and zero for i ≪ 0 , so Poinc ar´ e series P B M ( t ) and Bass series I N B are defined; see ( 1.6.1 ) and ( 1.6.2 ). Here we assemble a collection o f such s e ries, used in the b o dy of the pa p e r. Their computations r ely on analogs o f results concerning finite mo dules ov er lo cal ring s. A.1. F or the gr aded B -mo dules Σ s N and N ∗ = Hom k ( N , k ), see 1.5 , one ha s P B Σ s N ( t ) = t s · P B N ( t ) and I Σ s N B ( t ) = t − s · I N B . (A.1.1) P B N ∗ ( t ) = I N B ( t ) and I N ∗ B ( t ) = P B N ( t ) . (A.1.2) An exa ct sequence 0 → N → G → M → 0 with G free a nd N ⊆ B + G yields (A.1.3) P B N ( t ) = t − 1 · ( P B M ( t ) − H M /B + M ( t )) . If B = C ⊗ k D and M = T ⊗ k U , where T is a graded C -mo dule a nd U a graded D -mo dule, then the K ¨ unneth F ormula g ives (A.1.4) P B M ( t ) = P C T ( t ) · P D U ( t ) a nd I M B = I T C · I U D . The formu las ab ov e suffice to compute P B k ( t ) and/or I B B ( t ) in some simple cases. LOCAL RINGS OF EMBEDDING CODEPTH 3 21 Example A.2. If B = k ⋉ W for some graded k -vector space W 6 = 0, then: P B k ( t ) = 1 1 − t · H W ( t ) . (A.2.1) I B B ( t ) P B k ( t ) = H W ( t − 1 ) − t . (A.2.2) Indeed, B 2 + = 0 implies P B B + ( t ) = H W ( t ) · P B k ( t ), s o ( A.2.1 ) follows fr om ( A.1.3 ). As B + ( B ∗ ) = ( B ∗ ) > 0 , any lifting to B ∗ of some basis of the k - s pace ( B + ) ∗ mini- mally generates B ∗ ov er B . Thus, there is an exact sequence o f gra ded B -mo dules 0 → U → B ⊗ k ( B + ) ∗ → B ∗ → 0 with U ⊆ B + ⊗ k ( B + ) ∗ , hence B + U = 0. F ro m this and ( A.1.2 ) w e obta in I B B ( t ) = P B B ∗ ( t ) = H U ( t ) · t · P B k ( t ) + H W ( t − 1 ) , bec ause H ( B + ) ∗ ( t ) = H W ( t − 1 ). Since H B ( t ) = H W ( t ) + 1, the sequence a lso g ives H U ( t ) = ( H W ( t ) + 1) · H W ( t − 1 ) − ( H W ( t − 1 ) + 1) = H W ( t ) · H W ( t − 1 ) − 1 , bec ause H B ∗ ( t ) = H B ( t − 1 ). F ro m the last tw o formulas and ( A.2.1 ), we get I B B ( t ) P B k ( t ) = t ·  H W ( t ) · H W ( t − 1 ) − 1  + H W ( t − 1 ) ·  1 − t · H W ( t )  = H W ( t − 1 ) − t . Example A.3. If B = V k V , where V i = 0 fo r all even i , then there is a n equality (A.3.1) P B k ( t ) = Y i ∈ Z 1 (1 − t i +1 ) rank k V i . Indeed, set c = rank k V . When c = 1 the iso morphism V k Σ i k ∼ = k ⋉ Σ i k a nd ( A.2.1 ) give the desire d expressio n. F or c ≥ 2 it is o btained by induction, using the isomorphism V k ( V ′ ⊕ V ′′ ) ∼ = V k V ′ ⊗ k V k V ′′ and ( A.1.4 ). Example A.4. When B has Poincar´ e dua lity in degree s there is a n equality (A.4.1) I B B ( t ) = t − s . Indeed, the condition on B mea ns that the B i → Hom k ( B s − i , B s ), induced by the pro ducts B i × B s − i → B s , are bijective for all i ∈ Z . This implies an isomorphism B ∗ ∼ = Σ − s B of graded B -mo dules, so ( A.1.1 ) and ( A.1.2 ) give I B B ( t ) = P B B ∗ ( t ) = P B Σ − s B ( t ) = t − s · P B B ( t ) = t − s . The next result is a n analog of a theor em of Gulliksen; see [ 20 , Thm. 2]. The original pr o of, or the one for [ 23 , Cor. 2], carries ov er essentially without changes. A.5. If B = C ⋉ W for some g raded k -algebr a C and gra ded C -mo dule W , then (A.5.1) 1 P B k ( t ) = 1 P C k ( t ) − t · P C W ( t ) P C k ( t ) . Example A.6. If B = C ⋉ Σ s ( C ∗ ) with C = k ⋉ W , then the following hold: P B k ( t ) = 1 1 − t · H W ( t ) − t s +1 · H W ( t − 1 ) + t s +2 . (A.6.1 ) I B B ( t ) P B k ( t ) = 1 − t · H W ( t ) − t s +1 · H W ( t − 1 ) + t s +2 t s . (A.6.2 ) 22 L. L. A VRAMOV Indeed, the is omorphism of g raded B -mo dules Σ s ( C ∗ ) ∼ = ( Σ − s C ) ∗ and ( A.1.2 ) give P C Σ s ( C ∗ ) ( t ) = I Σ − s C C ( t ) = t s · I C C ( t ). Now ( A.5.1 ), ( A.2.1 ), and ( A.2.2 ) yield 1 P B k ( t ) = 1 P C k ( t ) − t · t s · I C C ( t ) P C k ( t ) = 1 − t · H W ( t ) − t s +1 · H W ( t − 1 ) + t s +2 . Since B ha s Poincar´ e duality in degr ee s , ( A.6.1 ) and ( A.4.1 ) imply ( A.6.2 ). The following analog of a re s ult of Lesco t, see [ 25 , 1.8 (2 )], can b e pr ov ed along the lines o f the orig inal argument, but subtle c hanges a r e needed. Instead of going int o thos e details, we refer to [ 10 ] for a dir ect pr o of covering b oth cases. A.7. If B + 6 = 0, then the following equa lity holds : (A.7.1) I B B ( t ) P B k ( t ) = I B + B ( t ) P B k ( t ) − t . In the last two ex a mples we adapt the arg umen ts for [ 25 , 3 .2(1) and 1.9]. Example A.8. If B = C ⋉ W for so me gr aded k -algebra C with C + 6 = 0 and graded C -mo dule W with C + W = 0, then the following equality holds: (A.8.1) I B B ( t ) P B k ( t ) = I C C ( t ) P C k ( t ) + H W ( t − 1 ) . Indeed, C is an alg e br a retra ct o f B . The pro of of [ 23 , Thm. 1] transfers verb atim and giv e s P B N ( t ) /P B k ( t ) = P C N ( t ) /P C k ( t ) for eac h graded C -mo dule N , wiewed as a B - mo dule via the natural homomorphism B → C . By ( A.1.2 ), this implies that I N B ( t ) /P B k ( t ) = I N C ( t ) /P C k ( t ) holds as w ell. Since B + = C + ⊕ W as graded B - mo dules, using the pr e c eding equality and ( A.7.1 ) (twice) we obtain I B B ( t ) P B k ( t ) = I C + B ( t ) P B k ( t ) + I W B ( t ) P B k ( t ) − t = I C + C ( t ) P C k ( t ) + H W ( t − 1 ) − t = I C C ( t ) P C k ( t ) + H W ( t − 1 ) . Example A.9. If B = E /E > s , where E is a graded k -a lgebra that has Poincar´ e duality in degree s , then the following equa lity holds : (A.9.1) I B B ( t ) P B k ( t ) = t − s − 1 ·  1 − 1 P B k ( t )  − t . Indeed, set ( − ) ′ = Hom k ( − , Σ s k ). Applying the functor ( − ) ′ to the exact sequence 0 → B + → B → k → 0 w e get ( B + ) ′ ∼ = B ′ /B ′ > s as g raded B -mo dules. Since E ∼ = E ′ as grade d E -mo dules, ( − ) ′ applied to 0 → Σ s k → E → B → 0 gives B ′ ∼ = E + , hence B ′ /B ′ > s ∼ = E + /E > s ∼ = B + , a nd thus B + ∼ = ( B + ) ′ ∼ = ( Σ − s B + ) ∗ . Now fro m for mulas ( A.7.1 ), ( A.1.2 ), and ( A.1.3 ) w e obta in I B B ( t ) P B k ( t ) + t = I ( Σ − s B + ) ∗ B P B k ( t ) = P B Σ − s B + ( t ) P B k ( t ) = t − s · t − 1 · P B k ( t ) − 1 P B k ( t ) . All the lab eled formulas in this a ppe ndix a re used in c o mputations in Sectio n 2 . 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