Abstract ON THE STRUCTURE OF THE CANONICAL MODULE OF THE REES ÁLGEBRA AND THE ASSOCIATED GRADED RING OF AN IDEAL

ON THE STRUCTURE OF THE CANONICAL MODULE OF THE REES ÁLGEBRA AND THE ASSOCIATED GRADED RING OF AN IDEAL S. ZARZUELA Dedicated to the memory of Pere Menal In this note we give the description of a morphism related with the structure of the canonocal module of the Rees algebra R(I) of an ideal I in a local ring . As an application we obtain Ikeda's criteria for the Gorensteinness of R(I) and a result of IierzogSimis-Vasconcelos characterizing when the canonical module of R(I) has the expected form . Let (A, m) be a Noetherian local ring and I be an ideal of A . In [4] Ikeda characterized the Gorensteinness of the Rees algebra R(I) = ® In tn by means of the canonical module of A and the canonical module n>0 of the associated graded ring grA(I) = ® In/I" -F1 , under the assumpn>0 tions that R(I) is Cohen-Macaulay and grade(I) >_ 2 . If in particular A is Cohen-Macaulay this characterization means that if ht(I) >_ 2, the Rees algebra R(I) is Gorenstein if and only if the ground ring A is Gorenstein and the associated graded ring grA(I) is Gorenstein with a-invariant a(grA (I» = -2. On the other hand in [3] I-Ierzog-Simis-Vasconcelos investigated the canonical modules of R(I) and grA(I), wllere I is an ideal in a local Cohen-Macaulay ring (A, m) . They were specially interested in characterizing when the canonical module of the Rees algebra R(I) is isomorphic to the R(I)-subrnodule of the polynomial ring A[t] generated by 1 ; t ; . . . , tr% where rrt >_ 0, or to the ideal IR(I) . Then it is said that the canonical module of R(I) has the expected form, that occurs if in particular R(I) is Gorenstein .

Let (A, m) be a Noetherian local ring and I be an ideal of A. In [4] Ikeda characterized the Gorensteinness of the Rees algebra R(I) = ® In tn by means of the canonical module of A and the canonical module n>0 of the associated graded ring grA(I) = ® In/I" -F1 , under the assump-n>0 tions that R(I) is Cohen-Macaulay and grade(I) >_ 2. If in particular A is Cohen-Macaulay this characterization means that if ht(I) >_ 2, the Rees algebra R(I) is Gorenstein if and only if the ground ring A is Gorenstein and the associated graded ring grA(I) is Gorenstein with a-invariant a(grA (I» = -2. On the other hand in [3] I-Ierzog-Simis-Vasconcelos investigated the canonical modules of R(I) and grA(I), wllere I is an ideal in a local Cohen-Macaulay ring (A, m).They were specially interested in characterizing when the canonical module of the Rees algebra R(I) is isomorphic to the R(I)-subrnodule of the polynomial ring A[t] generated by 1 ; t; . . ., tr% where rrt >_ 0, or to the ideal IR(I) .Then it is said that the canonical module of R(I) has the expected form, that occurs if in particular R(I) is Gorenstein .
These results Nave been recently used to study the Gorensteinness of the Ress algebas and associated graded rings of powers of ideals, see [2] .
Our deal in this paper is to give a common point of view for both results.For this, and mainly inspired in [3], we first give the description of a morphism of graded R(I)-modules F : ® 2 (KR(I»n -> KR(I), where KR(I) is the canonical module of the Rees algebra R(I) .By means of this morphism F some information about the structure of KR(I) can be transfered to the canonical module of the associated graded ring grA (I), and conversely.This is done is section 1, proposition (1.1) .Then, as a main application, we obtain in section 2 the above mentioned results of Ikeda and Herzog-Simis-Vasconcelos .
The existence of a morphism with similar properties as F for multigraded Rees algebas has been obtained by H. Hiri (Helsinki) .

. The main result
We shall use the book [1] as a reference for unexplained results and terminology.Let R = ® Rn be a Noetherian graded ring defined over n>o a local ring Ro .If L is a finitely generated graded R-module then the Krull dimension of L, dim(L), satisfies dim(L) = sup {iI Hn,(L) A 0}, where HÑ (L) are the ith graded local cohomology modules of L with respect to N, the maximal homogeneous ideal of R. The a-invariant of L is then defined by a(L) := sup { ni (HNm1Ll (L)) n ~0} Since HÑ`n(L) (L) is an artinian graded R-module, a(L) is a well defined integer .
Assume that R has a canonical module, KR .Passing to the completion if necessary and by local duality one has that a(R) = -inf {ni (KR)n 0 0} , and R is Gorenstein if and only if R is Cohen-Macaulay and KR -R(a(R)) .Proposition .1) .A . and .
Proof. .where n and v are the obvious restriction maps.Furthermore Homs (S, Ks) may be identified with Ks by the map that in degree n is given by 0 Ws Hom s (S + , Ks) O> KA 0 0 ---> Ks ~> Hom s (IS, KS) -°> KG 0.
Define w := -r7r.KA is a graded S-module reduced to degree 0 while (KS)n = 0 for any n < 0, hence is an isomorphism and is also an isomorphism of degree 1 .Hence we may define Moreover, by definition of n 7r : Ks -® Hom s (S+, Ks)n n>1 w : Ks -> ® Hom s (IS, Ks ),, n>2 F := w -iu : (Ks)+ -Ks, a morphism of graded S-modules of degree -1 .Since Q is injective F is injective too, proving (i).
Set a := -a(grA(I)) .It is clear that QI( K,), is an isomorphism for any n < a, hence I'I(Ks) .+i: (Ks)n+l -(Ks)n is an isomorphism for any n < a -1, that is (ii).
Therefore s Q = (ts) F(P) for any s E IS.Now assume that (ts)a = (ts) F(0) for any s E IS, where a E Ws.Then (ts)(a -F(Q)) = 0 for any s E IS and in particular (0 : (F(P)a)) D S+ , with ht(S+) = 1 .This implies that a = F(O) since Ks is a Cohen-Macaulay S-module with depth(Ks) = dim(S) .`' Furthermore for any element a E Ks and any integer r _> 1 we have that s(t'ry) a = (ts)(tr -ly) a for any s E IS and y E Ir .By (iii) F((Cy)cx) = (C-l y)a, showing (iv) .(ii) Let (A, m) be a Cohen-Macaulay ring and I a strongly Cohen-Macaulay ideal in A (that is, an ideal such that for any r >_ 0 the Koszul homology H,(K(á)) is zero or a maximal Cohen-Macaulay A/I-module, where a is any system of generators of I) .Assume that the minimal number of generators p,(Ip) < ht(p) for all prime ideals p I? I. Then a(grA(I)) = -ht(I), see [2, (2 .5)] .
As we have already commented in the introduction, Ikeda proved in [4] that if grade(I) > 2 and R(I) is Gorenstein the a-invariant a(grA(I)) = -2 .This is a particular case of the following result that we may obtain from the proof of proposition (1.1) .

Corollary (1.4) . Let (A, m) be a local ring and I C A a non principal ideal of A such that I-
Proof: We use the same notation as in the proof of proposition (1.1) .First we show that a(G) <_ -2 .For this it is enough to see that o,j(Ks), is an isomorphism, that is, that any f E Hom s (IS, Ks)1 is given by the product by an element a E (Ks)1 .And this follows immediately from the fact that (Ks)1 -A (S is Gorenstein), I generates IS and HoMA(I, A) = I-1 = A. Now by proposition (1.1) we have (KG)_2 -((Ks)11r((Ks)2)) gá 0 since I' is injective and (Ks)2 -I is not principal .
Throughout this section we sliall use the saíne notation as in the preceding one .Let I be an ideal of A such that the associated graded ring 97-A(I) has a canonical module .First we ask when there exists a finitely generated A-module P such that g7-p(I) := ® I"P/I'Z+1P 9L>_0 is isornorphic to KgrA(1) (r), where r is sorne integer (see [6] for this question when I is the maximal ideal of A) .Observe that in any case r = -a(grA(I)) .Suppose that 97-A(I) is Cohen-Macaulay (then A is Cohen-Macaulay too) .By [1, (36 .11)]KgrA(~) is a graded grA (I)-module of finite injective dimension with rkk (Extig"A([)(k, I~9TA (I))) = 1 if i = dirn (g'r'A(I)) ; and 0 otherwise, where k is the residue field of A. By standard methods it is easy to show that if grp(I) -KgIA(1) (r) then P is an A-module of finite injective dimension with rkti (Ext;1 (k, P)) = 1 if i = dim(A), and 0 otherwise .Hence P is a canonical module of A.