Ghione, Franco

Gulliksen, Tor H.

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SOME REDUCTION FORMULAS FOR THE , POINCARE SERIES OF MODULES. Franco Ghione and Tor H. Gulliksen. Introduction. In this paper we shall investigate the Poincare series of a finitely generated module M over a local (noetherian) ring R,~, that is the power series k being the residue field of R • P~(t) will be called the Poincare series of the ring R • Although not much is known about P~(t) in general, there is evidence to belieYe that this power series always represents a rational function. This is of course the case if R is a regular local ring, in which case P~(t) is a polynomial of degree rot exceeding the global dimen-sion of R • Recently it has been shown that ~(t) is a rational function if R is a local complete intersection, Gulliksen [3]. In the present paper we shall establish the rationality of ~(t) also in the case where R is a Golod ring (see section 2 for the definition). It is known that if R is a factor of a regular ring by two relations, or if R has imbedding dimension less than or equal to two, then R is either a complete inter-section or a Golod ring. Hence the rationality of P~(t) is established in those cases. This generalizes results of Shamash [5] and Scheja [6] who worked with the case M = k • In theorem 5 in section 3 we give the following reduction formula: 2 -Let y be a regular element in 1+1 • Let C'f.., be an ideal in R and put R' : = R/yvY, • If 00,1 = 0 then (i) p~' (t) = P~(t)[1 ~ t(a.(t)- 1 )J-1 where a(t) = P~;m(t). In section 4 we give applications of theorem 5 and the results of section 3. Several examples are worked out. In particular it is shown that if k is a field and oz.. is an ideal generated by monomials in the ring A= k[[X1 ,x2 ~x3 JJ , then the Poincare series of A/01/ is rational. In the last section we remark that in order to prove the rationality of ~(t) for all R and M , it suffices to prove R the rationality of PR/~(t) for all local rings R of dimension zero. Notations. If N = ll p>O Np is a graded R-module where each homogeneous component Np is a free R-module of finite rank, we let XR(N) or simply x(N) denote the power series The term "R-algebra" will be used. in the sense of Tate [7]. By an augmented R-algebra F we will mean an R-algebra F with a surjective augmentation map F ~ Rj1q which is a homomorphism of R-algebras, Recall that the Kosz.ul complex generated over R by a minimal set of generators for f1f is an R-algebra which up to (a non-canonical) isomorphism depends only of the ring R • Thus we shall talk about the KosZul complex of R • 7: - _.I -1. On r1assey operations. Let F be an augmented R-algebra with a trivial Massey operation y , and let S be the set of cycles associated with y. For the definitions and details the reader is referred to Gullik-sen [2]. Recall that S represents a minimal set of generators for the kernel of the map H(F) ~ R/~- induced by the augmenta-tion on F • y is a function with values in F , defined on the set of finite sequences of elements in S such that y(z) = z for z E S • By means of F and y it is possible to construct an R-free resolution of R/~ . We will briefly recall the con-struction: To each cycle z in S select a symbol u of degree one more than the degree of z • Let N = II qNq be the free graded R-module generated by the set of selected symbols u o Let T = TR(N) be the tensor algebra generated over R by N • Put X := F ~ T By means of the canonical map F - F ® T , sending f to f ® 1 , F will be considered as a submodule of X 5 We will now extend the differential d on F to a differential on X (also denoted by d) in the following way: It suffices to define d on a set generators for the R-module X • If f is a homogeneous element in F of degree deg f and if u 1 9 ••• ,un (n ~ 1) are selected symbols corresponding to the cycles z 1 , ••• ,zn in S we put One can show that d2 = 0 and that X is an R-free resolution of k. Cf. [2]. Moreover, if F is minimal in the sense that - ~- -dF c 114 F , and if Imy c 111 F then X is a minimal resolution. DEFINITION. The resolution X constructed above will be called the Golod extension of the couple (F,y) and will be denoted X = (F, y ,N) • THEOREM 1. Let R be a local ring with re8idue field k and let tTV be an ideal in R • Let M be a finitely generated R-module of finite projective dimension such that OL :H = 0 • Let F be an augmented R-algebra which is an R-free resolution of k • Put R 1 := R/Ol 9 F 1 := F/otF and assume that F 1 has a trivial Massey operation y • Then there exists a polynomial n(t) with integral coefficients such that ~ 1 ( t ) = TT ( t ) [ 1 - t ( p~ 1 ( t ) · -· 1 ) ] - 1 PROOF. Let X= (F 1 ,y,N) be the Golod extension of (F 1 ,y) • Then we have an identity of graded R-modules X= F 1 <fl X®N (1) We let Y be the complex whose underlying graded module is X® N , and whose differential is d ® 1 N , d being the differen-tial on X. (1) leads to an exact sequence of complexes 0 ---> F' _2_> X _i_> Y ---> 0 • (2) where cr is the canonical injection, and ~ is the canonical projection onto the second factor. Since ~M = 0 and M has finite projectj.ve dimension we have H ( H ® F 1 ) = H ( M ® F) = TorR ( M, k) = 0 p R' p R p for all p sufficiently large. Hence from (2) we obtain ..,. 5 -( 3) for all p sufficiently large. Hence we have for all p sufficiently large~ Thus there exists a polynomial n(t) with integral coefficients such that ~' (t) = x(H(M®X)® N) + n(t) = ~'(t)x(N) + n(t) This yields the desired formula since X ( N ) = t ( X ( H ( F ' ) ) - 1 ) = t ( P~ 1 ( t ) - 1 ) 2. Modules over Golod rings. We will first recall the definition of a Golod ring. Let R be a local ring with maximal ideal ~, and let K be the Koszul complex of R • R is called a Golod ring if the canonically augmented R-algebra K has a trivial Massey operation in the sense of Gulliksen [2]. This is equivalent to saying that all Massey operations on H(K) vanish in the sense of Golod [1]. The following result shows that Golod rings can be characterized entirely in terms of the Poincare series. The proposition is due to Golod, and the proof can be found in [1]. PROPOSITION 2. A local ring R,~ is a Golod ring if and only if R ( )n 2 3 n+ 1 -1 PR/H1( t) = 1 + t [1 - c 1 t - c2 t - ••• - en t ] where n = dim U1/~d 2 and c. = dim H. (K) l l for - 6 -Examples of Golod rings: I. The rings of the form k[[X 1 ,x2 JJ/~ (where k is a field) which are not complete intersections. That these rings are Golod rings follows from Satz 9 in Scheja [6] and proposition 2 above. IIft The rings A/y (JV where A is a regular local ring and y is a non-unit in A . Cf. Schamash [5] . III. A/(a1 ,a2 ) where A is regular, and a1 and a2 do not form a regular sequence. This is a special case of example PROPOSITION 3. Let R , 111 be a local ring and let y E 41( -711 2 be a regular element. Then R is a Golod ring if and only if R/yR is a Golod ring. k = R/it! • R II. PROOF~ Let K be the Koszul complex of R • Put Since the k-algebra H(K) and the Poincare series PR!fft ( t) are invariant under 1# -adic completion, Proposition 2 shows that there is no loss of generality assuming that R is complete. Hence by the Cohen structure theorem we may assume that R = A/dV where A is a regular ring and dZ is an ideal contained in the square of the maximal ideal of A • Let y' be an element of A that represents y in R • We have an isomorphism of k-vector-spaces. ( 1 ) Similarly the homology of the Koszul complex of R/yR is isomor-A/y'A( I ) y' phic to Tor. R yR,k • Since is regular on A and R , and since y'k = 0 we have a canonical isomorphism (2) - 7 ... It follows from Satz 1 in Scheja [6] that P~(t) = (1 + t) ~/yR(t) (3) Let AK denote the maximal in R/yR o We have I 2 -;- 2 dim #1 -111 = dim 111 Aft + 1 (4) The proposition now follows from prop. 2 using (1), (2), (3) and (4). THEOREM 4. Let R , -111 be a local Golod ring and let M be a finitely generated R-module. Then P~(t) is a rational function. PROOF. Let R and M be the #;" -adic completions of R and M • It can easily be shown that P~(t) = P~(t) , hence as in the proof of proposition 3 we may assume that R = A/a& where A is a regular local ring and ~ is an ideal which is contained in the square of the maximal ideal of A • Let K be the Koszul complex of A $ Then K' := K/mK is the Koszul complex of R • Since R is a Golod ring, K' has a trivial Massey operation. Since M has finite projective dimension over the regular ring A , theorem 2 gives that P~(t) is a rational function. 3. Reduction formulas for the Poincare series. THEOREM 5. Let R , 411 be a local ring, let y be a regular element in W, and let m be an ideal in R • Put R' = R/y OL and a ( t) = P~;oz.. ( t) • Then ( i) If M is an R-module such that (}t I1 = 0 , then ~' (t) = ~(t)[1- t(a(t) -1 )]-1 • - 8 -(ii) If ~I R and if M is an R-module of finite projective dimension such that yOt r-1 = 0 , then pR' (t) = n(t)[1- t(a(t)- 1 )J-1 M where n(t) is a polynomial with integral coefficients. PROOF. Let F ~ k be an augmented R-algebra which is an R-free resolution of k = R/-#1 • Put F' := F/ymF ~ Vle are going to construct a trivial Massey operation on F' s First we remark that there exists an element a' of degree one in F' such that every cycle z' in F' is homologue with a cycle of the form a'x' where x' is in oz.., F I . In fact let z E F represent a given cycle z' in F' Then we may write dz = yx where X E Oz,F • Since 0 = d2z = ydx and since y is regular in R we have dx = 0 • Let a E F1 be such that da = y . We have d(z- ax) ::: 0 Since F is acyclic, z -ax is a boundary. Hence z'- a'x' is a boundary in F' , where a'x' is the image of ax in F' • Now choose a basis for the k-vectorspace Ker(H(F') ~ k) • Let S be a set of cycles in F' representing that basis, and choose S such that the cycles z' in S have the form z' = a'x' where x' E OZ.F' . Since a' has degree 1, we have (a' )2 = 0 • Thus we can construct a trivial r1Iassey operation y on F' by putting y(z') = z' for z' E S y(z~, .•• ,z~) = 0 for n > 2; z~,.~.,z~ E S. Now (ii) follows from theorem 2 since we have - 9 -( 1 ) We are now going to prove (i). Let X= (F' ,y,N) be the Golod extension of the couple (F',y) • Assume that O.ZM = 0. Since Im y ~ o<F' the following diagram is commutative M®X -> M®F' $ (M®X) ® N M®X->M®F' ® (M®X) ®N Here the horizontal isomorphisms are induced by the identity (1) in the proof of theorem 2. The diagram yields H(M®X) ~ H(M®F') ® H(M®X) ® N (2) ,..., We have M ®F' = M ®F • Hence if (Jl. -:/. R the deSired formula in (i) follows from (1) and (2). If ~ = R, then M = 0 and in this case the formula in (i) is trivial. COROLLARY 6. Let 021 c ••• c OZr be a chain of ideals in a local ring R , #! (r 2: 1) • Let y1 , ••• ,yr be a sequence of elements in ~ such that y1 is regular in R (1 < i < r-1) yi+1 is regular on i i R : = R/ L: yhtJL.. k=1 --n • and for every i Let M be an R-module such that ~rM = 0 • Then we have r p~ (t) = where for In particular, if R is a local complete intersection or a Golod r ring, then P~ (t) represents a rational function. - 10 -PROOF. The formula \'Till be proved by induction on r ., For r = 1 the formula is valid by theorem 5. Now let i be an in-teger such that 1 ~ i < r • Put i-1 cl. (t) = (1 + t)i - t I: (1 + t)P a.. (t) f-1 l-p p=O By induction we may assume that we have for all R-modules Q such that ()"&.Q = 0 l • Now an R-module such that 02,. 1L = 0 l+ • We are going let to ( 1 ) remains valid if i is replaced by i+ 1 , and L show Q is replaced by L Since Ul. - {)(,i+1 we have ati (R/ mi+1 ) • '--' l -Qt.L, so (1) yields l and Since y1+1 is regular in Ri , theorem 5 gives i+1 p~ ( t) = Substituting (2) and (3) in (4) and using the identity we obtain the desired result. ( 1 ) be that = 0 (2) (3) (4) We shall now give a lemma which gives 0onditions implying the hypothesis in the previous corollary. With the notation of that corollary we have = -~ 11 -LEMMA 7. Let t1t1 c ... c (}c., be a sequence of ideals in R ~ - - r Let y1 , ••• ,yr be a regular sequence contained in the maximal ideal and assume that y1 , ••• ,yi+1 is a regular sequence for R/at1 for all i (1 ~ i ~ r-1) • Then yi+1 is Ri-regular for all i • PROOF. We will prove the proposition by induction on r , the number of ideals. For r = 1 there is nothing to prove. Now let r > 2 and 1 < i < r-1 • Let A be an element of R such that i A E i: yh 0'2,h • h=1 It suffices to show that Reading (1) modulo and using the induction hypothesis one obtains Hence we may write ( 1 ) y.R, 1 (2) where ah E Llt-h and a E R • From ( 1) we obtain AYi+1 E OlJ. • 1 Hence we have A E mi . Now (2) yields y.a E 1 oti i hence a E Ol· f so A E i: yhOlh • 1 h=1 We will end this section by giving an example where lemma 7 can be ~pplied. Let s be a local ring, let r > 1 be an integer and let ot1 c c (}/_,I be ideals in s Put R . - S[[y,, ••• ,yr]] and . . . • . --r put en. = 1 Ot ~R 1 for 1 < i < r • Then the sequences y,, ••• ,yr and 01'1 c . . . c OS. satisfy the hypothesis in corollary 6. -- 12 -4.. Examples. I. Let R be a local ring and let ~ be an ideal in R • II. Let M be an R-module such that O'ltM = 0 • Let be a regular sequence in R which is contained in -1# , and assume that y1 , ••• ,yr is also regular sequence on R/~. Put R' := R/(y1 , ••• ,yr)~. Then lemma 7 and corollary 6 yields the formula p~' ( t) = p~ ( t )[a: ( t) - a: ( t) f3 ( t) + s ( t) J - 1 where a:(t) = P~;~(t) and S ( t) = ~R/( ) ( t) = ( 1 +t) r. Y1, ••• ,yr In particular if R is a local complete intersection or a Golod ring, then R' RN (t) represents a rational function. Let A be a regular local ring of dimension n • Let r and s be integers such that 0 < s < r < n • Let Y1, ••• ,yr be a regular sequence in -'1112 and let u be an element in ~JH such that Y1, .... ,ys , u is a regular sequence. Put OL = ( Y 1 ' • • • ' Y s ' uy s+ 1 ' • • • ' uy r) • Then considering A/~ as a factor ring of the complete intersection A/(y1 , ••• ,ys) , one easily deduces the fol-lowing formula from theorem 5: III. Let k be a field and consider the following ring of formal powerseries A= k[[X1 , ••• ,Xn' Y1 , ••• ,Yr]] • Let CJL, I C 1 • • • c C1,. i r be a chain of ideals in IV. Put ~i - 13 -for 1 < i < r , and put r Ot = L: y. 0&. • i=1 l l Then for each A-module M such that au~ = 0 we have where et~(t) = ~;/02.-'q (t) • Clearly ~ja-u (t) is a rational function. In particular, if k' denotes the residue field of A/00 we get Let au be an ideal generated by monomials in the ring A = k[[X1 ,X2 ,x3]] , where k is a field. We shall also let k denote the residue field of A/fJL.· • We will show that pA/OZ-- ( t) k is rational. We may write ~ = 0(1 x, + {J1_;2 where ot1 and 00 2 are ideals in A and k[ [X2 ,x3]] respectively. Put R = k([X2 ,x3]J/a&2 • Then R is either a complete intersection or a Golod ring. See example I in section 2. By proposition 3 we see that the same holds for R[[X1]J • Hence P~[[X1]](t) is a rational function for every R[(X1]J-module M • Since A/OZ ~ (lr[[X2 ,x3]J/ov 2)[[X1 ]J/az..1x1 = R[[X1JJ/6L 1X1 it follows from theorem 5 that pAlau ( t) M is rational for every module M such that ot1M = 0 • In particular ~/&£-(t) is rational. - 14 -Using theorem 5 it is also possible to prove that the ring k[[X1, ••• ,Xn]]/(m1 ,m2 ,m3) has rational Poincare series, m1 pm21 m3 being monomials. 5. ·Reduction to the case of dimer.sion zero. PROPOSITION 8. The following statements are equivalent~ (i) P~/~ (t) is rational for every local ring R ,~ of dimen-sion zero. (ii) P~(t) is rational for every local ring R and every finitely generated R-module T>1 • PROOF. It suffices to prove (i) ~> (ii). Suppose that P~(t) is rational for every local ring of dimension zero. From theorem R 3.17 in Levin [4] one deduces that PR/~ is rational for every local ring R,1# By theorem 2 in [2] it then follows that ~(t) is rational for all R and all !JI • - 15 -References. 1. E.S. Golod: On the homology of some local rings. Soviet Math. Dokl~ 3 (1962). 2. T.H. Gulliksen: Massey operations and the Poincare series of certain local rings. J. Algebra 22(1972)o 3. T.H~ Gulliksen: A change of ring theorem with applications to Poincare series and intersection multi-plicity. To appear in Math. Scand. 4. 5. 6. 7. G. Levin: J. Shamash: G. Scheja: J. Tate: FRANCO GHIONE Local rings and Golod homomorphisms. To appear. The Poincare series of a local ring. J. Algebra 12 (1969) • .. Uber die Bettizahlen lokaler Ringe. Math. Ann. 155 (1964). Homology of noetherian rings and local rings. Illinois J. Math. 1 (1957). ISTITUTO DI MATEMATICA UNIVERSITA DI FERRARA FERRARA, ITALY TOR H. GULLIKSEN MATEMATISK INSTITUTT UNIVERSITETET I OSLO BLINDE~~, OSLO 3, NORWAY 

SOME REDUCTION FORMULAS FOR THE POINCARÉ SERIES OF MODULES.

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