On the Gorenstein Property of Rees and Form Rings of Power of Ideals
Author(s): M. Herrmann, J. Ribbe, S. Zarzuela
Source: Transactions of the American Mathematical Society, Vol. 342, No. 2 (Apr., 1994), pp.
631-643
Published by: American Mathematical Society
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TRANSACTIONS OF THE 
AMERICAN MATHEMATICAL SOCIETY 
Volume 342, Number 2, April 1994 
ON THE GORENSTEIN PROPERTY OF REES AND 
FORM RINGS OF POWER OF IDEALS 
M. HERRMANN, J. RIBBE, AND S. ZARZUELA 
Dedicated to Professor H. Hironaka on occasion of his 60th birthday 
ABSTRACT. In this paper we determine the exponents n for which the Rees ring 
R(In) and the form ring grA (In) are Gorenstein rings, where I is a strongly 
Cohen-Macaulay ideal of linear type (including complete and almost complete 
intersections) or an m-primary ideal in a local ring A with maximal ideal m. 
Given an ideal I in a local ring (A, m) it is well known that the Cohen- 
Macaulayness of the Rees algebra R(I) implies the Cohen-Macaulayness of all 
Rees algebras R(In). The same is true for the form rings grA (I) and grA (In); 
see [3, (2.7.8) and (8.8.5)]. In this paper we show that, in contrast to the Cohen- 
Macaulay property, the Gorenstein property of R(In) and grA(In) only holds 
for special exponents n. If in particular grA(I) is Gorenstein and R(I) is 
Cohen-Macaulay it turns out that these special exponents are closely related to 
the a-invariant of the form ring grA(I). Mainly under this aspect we prove 
some results concerning the Gorenstein property of Rees and form rings of 
powers of 
(i) strongly Cohen-Macaulay ideals of linear type (including almost com- 
plete intersections) in Gorenstein rings, 
(ii) m-primary ideals in Cohen-Macaulay rings, and 
(iii) equimultiple prime ideals p, $ m in a generalized Cohen-Macaulay ring. 
Our investigations are essentially based on the explicit computation of the 
a-invariants of form (and Rees) rings (see ?2). For the above classes of ideals- 
using a structure theorem for the canonical module of R(I) in [7]-we can 
determine in ?3 the exponents n > 1 for which R(In) and grA (In) are Goren- 
stein rings (see in particular Theorem (3.5)). A more geometrical interpretation 
of the results in ?3 is the observation that in these situations the Gorenstein 
property of Proj(R(I)) can be deduced from the Gorenstein property of a cer- 
tain Veronesean subring R(In) of R(I). It might be an interesting question 
when the Gorenstein property of a blow up Proj(R(I)) is inherited to an ap- 
propriate Rees ring R(J) with Proj(R(J)) = Proj(R(I)) over SpecA. 
In ?4, Part I we characterize-for Cohen-Macaulay (or Gorenstein) rings 
A-the Gorenstein property of R(md-i), i = 1, 2, 3, by conditions on the 
Received by the editors June 30, 1991 and, in revised form, April 10, 1992. 
1991 Mathematics Subject Classification. Primary 13D03, 13H10, 13H15; Secondary 14B05, 
14B 1 5. 
? 1994 American Mathematical Society 
0002-9947/94 $1.00 + $.25 per page 
631 
632 M. HERRMANN, J. RIBBE, AND S. ZARZUELA 
reduction exponent of m (see Theorem (4.4)). In Part II we describe sufficient 
conditions on an equimultiple prime ideal , in A for being a complete inter- 
section in terms of blow up properties. In case that dim(A) > 2 ht(,) and A 
is a generalized Cohen-Macaulay ring, the Gorenstein property of R(p,ht(P)-1) 
implies that , is a complete intersection (ht(a) denotes the height of an ideal 
a). For a somewhat similar question for projective schemes we refer to the 
forthcoming paper [ 16]. 
Some of the results in ?4 were also proved independently by A. Ooishi [10, 11 ] 
by different methods. We would like to thank A. Ooishi for many stimulating 
discussions during the preparation of this work. 
1. PRELIMINARIES 
To study the relationship between the various graded rings associated to an 
ideal I in a (commutative) Noetherian ring A, we will also use the so-called 
approximation complexes of I (see [4, 5, and 6]). The situation is particularly 
good when these complexes are exact, a condition that is fulfilled for some 
important families of ideals. In these cases it is often possible to prove that 
those graded rings are Cohen-Macaulay or Gorenstein. One important point 
in this theory is that under the "good" situation the ideal I is of linear type, 
namely SymA (I) R(I). 
Assume for simplicity that A is a local ring with maximal ideal m and 
dim(A) = d. Let I be an ideal of A and a := {a,, ... , an} a system of 
generators of I, and consider the Koszul complex K(a) of A w.r.t. a. If 
we denote by S the polynomial ring A[X1, ..., X,] it is possible to get two 
complexes of S-modules: 
(a) : O -4 Y, * , * *o 0? 
and 
A(a) : O A n > 00 0 , 
where Zi = Zi(K(a)) ?AS and Xti =Hi(K(a)) ?A S. 
The Zi(K(a)) denote the cycles of the Koszul complex K(a), and the Hi(K(a)) 
denote the Koszul homology. Both complexes can be taken as complexes of 
graded modules over S with mappings of degree -1 . 
We list the main properties of these complexes: 
(1) The homology of Y(a) and f(a) is independent of the system of 
generators a. 
(2) Ho(Y(a)) = SymA (I). 
(3) Ho(1/((a)) = SymA (I/I2). 
(4) The following are equivalent: 
(i) f(a) is acyclic, 
(ii) Y(a) is acyclic and I is of linear type. 
Now assume that A is Cohen-Macaulay. Then the following properties (5) 
to (9) hold: 
(5) Suppose: 
(i) For any prime ideal p D I, u(Ip) < ht(p) (where ,u(-) denotes 
the minimal number of generators). 
(ii) For any r > 0 and for any prime ideal p D I 
depthAp (Hr (K(a) )p) > inf(r, ht(p/I)) 
Then X(a) is acyclic. 
REES AND FORM RINGS 633 
It turns out that the condition (ii) above is independent of the system of 
generators of I, and that it is fulfilled if I is strongly Cohen-Macaulay (sCM 
for short), that is, if for any r > 0 the Koszul homology Hr(K(a)) is zero or a 
maximal Cohen-Macaulay A/I-module. 
(6) Suppose: 
(i) For any prime ideal p D I, u(Ip) < ht(p) + 1 . 
(ii) For any r > 0 and for any prime ideal p D I, 
depthAp(Hr(K(a))p) > inf(r, ht(p/I)) - 1. 
Then Y(a) is acyclic. 
(7) Assume that ht(I) > 1 and 
(i) For any prime ideal p D I, u(Ip) < ht(,), 
(ii) I is strongly Cohen-Macaulay. 
Then grA(I) and R(I) are Cohen-Macaulay. Moreover, if A is Gorenstein 
then grA (I) is Gorenstein. 
There is an important connection between the theory of approximation com- 
plexes and the theory of d-sequences: 
(8) If I is generated by a d-sequence, then X(a) is acyclic. 
(9) If IA/ml = oo and X(a) is acyclic, then I can be generated by a 
d-sequence (but the given a is not necessarily a d-sequence!). 
Finally we mention some relevant families of strongly Cohen-Macaulay ideals: 
(a) Complete intersection ideals. 
(b) Ideals I C A such that ,u(I) = ht(I) + 1 and A/I is Cohen-Macaulay. In 
particular, almost complete intersection (a.c.i) ideals I such that A/I is Cohen- 
Macaulay are sCM. We can say that an ideal I is a.c.i. if u(I) = ht(I) + 1 and 
IAP is complete intersection for any prime ideal p e Min(A/I). 
(c) Suppose A is Gorenstein and I c A is an ideal such that ,u(I) = ht(I)+2 . 
If A/I is Cohen-Macaulay then I is sCM. 
2. THE a-INVARIANT OF GRADED MODULES 
In this section we collect some important properties of the a-invariants of 
Rees and form rings. 
Recall that for a positively graded Noetherian ring R = ej>0 R1 defined 
over a local ring Ro and a Noetherian graded R-module G the a-invariant of 
G is defined as 
a(G) = max{j E Z: H4(G)j $ O}, 
where r = dim(G) and H4(G) is the rth local cohomology w.r.t. the maximal 
homogeneous ideal M of R. Note that R is Gorenstein if and only if R 
is Cohen-Macaulay and the canonical module of R is KR - R(a(R)), see [3, 
Chapter VII]. 
We start with an easy but useful lemma. For completeness we sketch a proof. 
Lemma (2.1). Let R = en>o R, be a Noetherian graded algebra defined over a 
local ring Ro and G = en>O Gn a Noetherian graded R-module. Let x E R, 
be a regular element on G. Then we get for the a-invariants of G and G/xG 
a(G) < a(G/xG) - 1. 
Moreover, if G is Cohen-Macaulay the equality holds. 
634 M. HERRMANN, J. RIBBE, AND S. ZARZUELA 
Proof. Consider the exact sequence of graded modules 
(1) 0 G(-1) x G lG/xG-O. 
Putting r = dimR(G) and M = maximal homogeneous ideal of R, we get from 
(1) the long exact cohomology sequence 
(2) -* - HMl'(G/xG)1 -* -* H (G) -O0. 
For i = a(G) + 1 we have 
(3) HM(G)j1 $ 0 and HM(G) = 0. 
Then (2) implies HM-'(G/xG)j 7 0, i.e. a(G/xG) > i = a(G) + 1. This 
proves the first part of the claim. The second part follows from (2) since now 
HM (G) = 0. 
In our context the importance of a-invariants is a consequence of the follow- 
ing characterizations of the Cohen-Macaulay and Gorenstein property of Rees 
algebras. 
Proposition (2.2). Let I be an ideal of positive height in a d-dimensional local 
ring (A, m). 
(a) [15, Theorem 1.1] The Rees algebra R(I) is a Cohen-Macaulay ring if 
and only if 
HM(grA(I))n =O for n 7& -1, i=O .., d - 1, 
and 
Hmd(grA(I))n = 0 for n > 0 (i.e. a(grA(I)) < 0), 
where M denotes the maximal homogeneous ideal of R(I). In this case, 
HM(gr (I)) - I H. (A) for i = 0, d - 1. (b) [8, Theorem (3.1)] If R(I) is Cohen-Macaulay and grade(I) > 2, then 
R(I) is Gorenstein ifand only if a(grA(I)) = -2 and the rings A and grA(I) are 
quasi-Gorenstein (i.e. the canonical modules of the specific rings are isomorphic 
to the suitably shifted rings). 
Remark (2.3). While for any nonpositive integer a there is a local ring A and 
an ideal I C A with a = a(grA(I)), the a-invariant of the Rees algebra R(I) 
is in many cases strictly determined. In particular if R(I) is Cohen-Macaulay 
and ht(I) > 0, then a(R(I)) = -1. In this case a(grA(I)) < -1. This can be 
easily deduced from the proof of Proposition (2.1) in [8]. 
For several classes of ideals I the a-invariant of the form ring grA (I) can be 
explicitly computed. In the following Lemma (2.4) we recall some well-known 
facts concerning m-primary ideals, and in Proposition (2.5) we determine the 
a-invariant a(grA(I)) for a strongly Cohen-Macaulay ideal which is of linear 
type. First we define the reduction exponent J (I) of an ideal I in a local ring 
A with infinite residue field as 
.5(I) := min{n E NI there exists a minimal reduction q of I s.t. In+, = qnl}. 
Lemma (2.4). Let I be an m-primary ideal in a local ring (A, m). Then the 
following hold: 
REES AND FORM RINGS 635 
(a) [ 1 3, Theorem and Theorem 2.1 ] If the reduction exponent of the maximal 
ideal m of A is 0, 1 or 2, then grA(m) is Cohen-Macaulay (resp. Gorenstein) 
iff A is so. 
(b) If grA(I) is Cohen-Macaulay, then 
a(grA(I)) = J (I) - dim(A). 
Statement (b) follows from (2.1) using the fact that the initial forms of the 
generators of a minimal reduction of I (which is a parameter ideal in A) form 
a regular sequence in grA(I). 
Proposition (2.5). Let (A, m) be a Cohen-Macaulay ring of dimension d. Let 
I be a strongly Cohen-Macaulay ideal in A. Assume ,u(I4) < ht(,) for all prime 
ideals p D I. Then a(grA(I)) = - ht(I). 
Proof. Put s = ht(I) = grade(I), and ,u(I) = s + t. Note that the homology 
Hi (K.) of the Koszul complex K. of A with respect to some minimal system of 
generators of I is zero for i > t. Moreover grA(I) is CM and the X-complex 
is exact by ? 1, (5) and (7), which implies in particular that I is of linear type. 
The X4-complex gives a resolution of grA (I): 
(1) - t * .. 4 , grA(I)0, 
where Xi = Hi(K.) ? A[XI, Xs+t]], dim(g) = d + t and Z is CM 
over A[X1, ..., Xs+t]. The idea is now to compute a := a(grA(I)) via the 
a-invariants a(g2): Applying Lemma (2.1) exactly (s + t) times we see that 
(2) a(XZ) = a(Hi(K.)) - A(I) -= (I), 
since (1) is considered as a sequence of A[Xi, ... , Xs+t]-modules. Note that by 
construction the morphism gi are of degree 1 . Therefore we get the following 
exact sequences for the cokernels i2 of ji+1 
? - i+ (-1 ) IX At -4 i -?0 
with morphisms of degree zero. 
Since G := grA(I) and Aj are CM, we get for the local cohomology with 
respect to the maximal homogeneous ideal M of A/I[X1, .,Xs+t 
(3) HMd(G)j - Hmd+l (-12)j-1 I * Hmd+"_2t_ 1 )-t I= . 
Moreover we have the exact sequence 
(4) 0-+ HM+t l(it_j)j_t+- H-+H(2t)j_t H+t(HM t_)j-t+l. 
Note that 
(S) H~~~~~M (t)j_t - HM tX)j_t X 
Case 1. j > -h, where h = ht(I), i.e. j - t > -,u(i) =: -n. Therefore 
Hd+t(ot)j_t = 0, since a(ot) = -n. But by (3), (4) and (5) we know that 
HdH(G)j j Hmd+t'? t - )j-t+l c Hd+t(At)j_t = 0, 
hence a(G) < -h. 
636 M. HERRMANN, J. RIBBE, AND S. ZARZUELA 
Case 2. j = -h. Then (3) implies 
(G)_h Hd+t l(O)-h-t+ i H t(t4)-n $ 0, 
hence a(G) =-ht(I). 
Remark. We can apply (2.5) in particular to an almost complete intersection 
ideal I, if A/I is a Cohen-Macaulay ring (in this case I is strongly Cohen- 
Macaulay). It is shown in [12] that the formula for the a-invariant in (2.5) holds 
for any (possibly non-Cohen-Macaulay) almost complete intersection ideal I. 
Next we describe the relationship between the a-invariants a(grA(I)) and 
a(grA(In)). This result is a generalization of [9, Corollary 4] and it will be used 
in ?3 for the characterization of the Gorenstein property of R(In) and grA(In) . 
Proposition (2.6). Let I be an ideal in a local ring A and assume that grA(I) 
is Cohen-Macaulay. Then 
a(grA(In)) = [a(grA(I))/n], 
where [] denotes the smallest integral part. 
Proof. Put d = dim(A), a = a(grA(I)), I = [a/n] and write a = In + r with 
r E {0, ..., n - 1}. For every i E {1, ..., n} there is an exact sequence of 
R(In)-modules: 
O + In-i+1 grA(In) InfigrA(In) -+ grA(I)(n - i)(n) - 0. 
Let N be the maximal homogeneous ideal of R(In) - R(I)(n). Then (see [3, 
Proposition (47.5)]): 
H~(grA(I)(n - i)(n)) HM (grA(I)(n - 
Hence for every j E Z and i E {1, ... , n} there is an exact sequence 
0 -~ Hf$(In-i+1 grA(In))j -) Hd (In i grA(In))j Hd (grA(I))fnj+n-i 0 ? . 
First consider these sequences for j > 1 + 1. Since nj + n - i > n(l + 1) + 
n - i > nl?+ r = a, we have Hd (grA(I))nj+n-i = 0 for each i E {1, ..., n}. 
Using the above sequences it follows inductively that Hd(In-i grA(In))j = 0 
for i = 0, ..., n. In particular HNd(grA(In))j = 0 for any j > 1 + 1, i.e. 
a(grA(In)) < 1. To finish the proof, consider the cohomology sequences from 
above in degree j = 1. Since Hmd(grA(I))nl+r $ 0 we get Hd(IrgrA(In)), $ 
0 and then successively Hdf(Ir-k gr (In)), $ 0 for k= 1,..., r. Hence 
HNd(grA(In)), $ 0. Q.E.D. 
Corollary (2.7). Let I be an ideal of height > 1 in a local ring A. If grA(I) is 
Cohen-Macaulay and R(In) is Cohen-Macaulay for some n E N, then R(I) is 
Cohen-Macaulay. 
Proof. First, note that 
- > a (grA(In)) by (2.2)(a) 
= [a(grA(I))/n] by (2.6). 
Hence a(grA(I)) is negative. Finally, by (2.2)(a), R(I) is Cohen-Macaulay. 
Remark. Note that in general the Cohen-Macaulayness of R(In) does not imply 
this property for R(I) . 
REES AND FORM RINGS 637 
3. POWERS OF STRONGLY COHEN-MACAULAY IDEALS 
We start with the following general observation. 
Proposition (3.1). Let I be an ideal of height > 2 in a Noetherian local ring 
A. Then at most one power of I has a Gorenstein Rees algebra. 
Proof. Assume that R(Is) and R(It) are Gorenstein. Since R(Ist) = R(Is)(t) - 
R(It)(s) and since R(Is)(-1) and R(It)(-1) are canonical modules of R(Is) 
and R(It) (see (2.3) for the correct shifting degree -1) we know that both, 
R(Is)(-1)(t) and R(It)(-1)(s) are canonical modules of R(Ist). Hence they 
must be isomorphic, and comparing their homogeneous parts of degree j > 1 
we see that the ideals Is(tj-1) and It(sj-1) are isomorphic as A-modules. By 
the following Lemma (3.2) we get s = t. 
Lemma (3.2). Let I be an ideal of height > 2 in a Noetherian ring A. If two 
powers Is and It are isomorphic, then s = t. 
Proof. We may assume that A is a local ring with maximal ideal m. The iso- 
morphism Is _ It induces isomorphisms Iis/mIjs ' Ijt/mIIjt for all numbers 
j. Now, there is a polynomial P = Z'I_ aiX1 E Q[X] of degree 1 - 1 (where / 
denotes the analytic spread of I) such that P(i) = t(Ii/mIi) for i > 0 (A de- 
notes the length). From P(sj) = P(tj) for j > 0 we get a1_1s1_1 = a,_ t1-1 
Since l > ht(I) > 2, we get s = t. 
For proving our main theorem, Theorem (3.5), we also need the following 
structure theorem for the canonical module of the Rees algebra. It comes from 
Corollary (2.5) in [7], if one takes the correct gradings in the proof given there. 
Proposition (3.3) [7, Corollary (2.5)]. Let I be a proper ideal in a local ring A of 
dimension d. Assume that A is Gorenstein and R(I) is Cohen-Macaulay. Then 
grA(I) is Gorenstein if (1, t)-a-2(_1) is a canonical module of R(I), where 
a := a(grA(I)) and (1, t)m denotes the R(I)-submodule of the polynomial ring 
A[t] which is generated by 1, t, ...,tm in case m > O or (1, t)-1 = IR(I) in 
case m=-1. 
Next we need the following lemma: 
Lemma (3.4). Assume that A is Gorenstein and R(I) is Cohen-Macaulay. If 
(1, t)m(-l) is a canonical module of R(I) for some integer m > -1, then 
R(Im+l) is Gorenstein. 
Proof. Note that for m = -1 there is nothing to prove. Hence we may assume 
m > 0. Then, denoting (1, t)m(-1) by K, we get 
0 ? if j < 0, 
Kj = (1, t)m(-1)j = A if 1 < j < m + 1, 
Ij-(m+1) if j > m + 2. 
Now recall that the Veronesean K(m+l) is a canonical module of the Veronesean 
R(I)(m+l) = R(Im+l). We get 
0 O if j < 0, 
(K(m+1))j= Kj(m+l) = if j 1, 
t Ij(m+')-(m+') if j > 2. 
638 M. HERRMANN, J. RIBBE, AND S. ZARZUELA 
Hence R(Im+l)(-1) = K(m+l), which proves the claim. 
Here is the main theorem: 
Theorem (3.5). Let I be an ideal of height > 2 in the local ring A. Assume 
that grA(I) is Gorenstein. Then the following holdfor n E N: 
(a) R(In) is Gorenstein iff R(I) is Cohen-Macaulay and n = -a(grA(I)) - 1. 
(b) If R(I) is Cohen-Macaulay, then gr (In) is Gorenstein iff -a(gr,(I)) 1 
mod (n). 
Proof. (a) Put a = a(grA(I)). Since grA(I) is Gorenstein and R(I) is Cohen- 
Macaulay by (2.7), we know by (3.3) that K := (1, t)-a-2(_l) is a canonical 
module of R(I). It follows that R(I-a-1) is Gorenstein, by (3.4). This is by 
(3.1) the only power of I which has a Gorenstein Rees algebra. 
(b) Put b = a(grA(In)). We can assume that n > 2. Note that R(In) 
and grA(In) are Cohen-Macaulay rings and that K(n) is a canonical module 
of R(I)(n) _ R(In). Hence by (3.3) grA(In) is Gorenstein if and only if the 
R(In)-module L := (1, t)-b-2(_1) is isomorphic to K(n) . Therefore, to finish 
the proof, we have to show that this statement holds iff -a _ 1 mod (n) . First 
we note that b = [a/n] by (2.6), i.e. a = bn + r with r E {O,., n - 1}. 
Hence we prove the following claim. 
Claim. LK(n) iff r= n- 1. 
Assume that r = n - 1. Then we get for each j E Z: 
0 ? if j < O0 
Lj = (1, t)-b-2(1). A if 1 < ? <-b-i, 
j n(j- l+b-2) if j > -b, 
and 
(K(n))j =((I1, t)-a-2(_l)(n%) = ((I1 t)-bn-n-i j- 
JO if jn - 1 < 0, 
= A ifO <jn- 1 <-bn-n- 1, 
I(In-l)+(bn+n+l) if in - 1 > -bn - n, 
f0 ifj < O, 
= < A if 1 <j<-b-i, 
jn(j+b+l) if j>-b-1+ l/n, i.e., if?>-b. 
Hence L - K(n) . For the converse assume that L - K(n); in particular L-b 
Kbn = Kr-a, i.e. In - Ir+1 . Since ht(I) > 2 it follows n = r + 1 by (3.2). 
This proves the claim and statement (b) of the theorem. 
As immediate consequences of (3.5) we get the following propositions. 
Proposition (3.6). Let A be a Gorenstein local ring and I a strongly Cohen- 
Macaulay ideal satisfying (Ip) < ht(p) for all prime ideals p B I. Assume that 
h := ht(I) > 2. Then: 
(a) R(In) is Gorenstein iff n = h - 1, 
(b) grA(In) is Gorenstein iff h _ 1 mod (n). 
Proof. Use (3.5) together with (2.5) and ?1(7). 
REES AND FORM RINGS 639 
Proposition (3.7). Let (A, m) be a d-dimensional Gorenstein local ring with 
reduction exponent c(m) < 2 and d > 3. Then the following hold for n E N: 
(a) grA(Mn) is Gorenstein iff d - 3(m) - 1 mod (n). 
(b) R(mn) is Gorenstein iff n = d - b(m) - 1. 
In particular: If A is regular, then R(md-1) is Gorenstein, and if A is a 
quadratic hypersurface, then R(md-2) is Gorenstein. 
Proof. Use (3.5) together with (2.4)(a) and (b). 
Remark (3.8). (a) We stated Propositions (3.6) and (3.7) as a direct consequence 
of Theorem (3.5). This theorem was shown by using the structure theorem (3.3). 
In the following we indicate another method for proving the "only if' parts in 
(3.6) and (3.7), which does not depend on (3.3). 
Proof idea. Using the sequences in the proof of (2.6) 
O In-i+1 grA(In) , In-i grA(In) - grA(I)(n -i)(n) 0 O 
one can show the following: If I is an ideal primary to the maximal ideal m of 
a local ring (A, m), such that grA(I) and grA (In) are Gorenstein rings for some 
n, then a(grA(I)) _ -1 mod (n) (see [12]; the primary property of I is used 
in order to know that the homogeneous parts of the graded local cohomology 
modules of the modules in the above sequences are of finite length). Hence, if 
I is an ideal satisfying the conditions in (3.6), then by localizing at a minimal 
prime of I we reach the described situation (i.e. the primary case). Since 
moreover a(grA (I)) = - ht(I) = a(grA (Ip)) for p E AsshA (A/I) (by (2.5)), the 
implication "grA(In) Gorenstein =X h =1 mod (n)" follows. Finally (under 
the assumptions of (3.6)) the Gorenstein property of R(In) forces grA (In) to 
be Gorenstein too, by (2.2); hence h 1 mod (n), as we have just pointed out. 
Furthermore -2 = a(grA(In)) = [-h/n] by (2.2), (2.6) and (2.5). Now it is 
easy to see that the two relations h _ 1 mod (n) and [-h/n] = -2 have the 
unique common solution n = h - 1. 
Similar arguments can be used to prove the "only if' parts in (3.7). 
(b) In (3.7), also the "if' parts can be shown without using (3.3). If A is 
regular or a quadratic hypersurface, it comes out by elementary computations 
that for the special numbers n mentioned in (3.7) the form rings grA (mn) are 
Gorenstein rings, see [12]. 
Remark (3.9). The assumption d > 3 in Proposition (3.7) was only used to get 
that R(I) is Cohen-Macaulay. Then one could apply Theorem (3.5)(b). From 
the arguments given in (3.8)(a) it is easy to see that Proposition (3.7) is also 
true for d = 2. 
4. POWERS OF EQUIMULTIPLE IDEALS 
Part I: m-primary ideals. Next we show to which extent the Gorensteinness 
of R(In), where I is m-primary, determines the structure of the ideal I via 
the reduction exponent 5(I). As a corollary we obtain a characterization of 
regular d-dimensional local rings (A, m) by the Gorensteinness of R(mid-l), 
which is a generalization of a result of Goto-Shimoda [1, Proposition 4.8]. First 
we prove the Key-lemma: 
Key-lemma (4.1). Let I be an m-primary ideal in the d-dimensional local ring 
(A, m), and q = (x1, ... , xl) a minimal reduction of I. Assume that grA(In) 
640 M. HERRMANN, J. RIBBE, AND S. ZARZUELA 
is Cohen-Macaulay for some natural number n. Put a := a(grA(In)). Then 
Ina+n+d C q. If moreover R(In) is Gorenstein and d > 2, then Id-n C q, 
hence n < d . 
Corollary (4.2). Let I be an m-primary ideal in a Cohen-Macaulay ring A with 
dimension d > 2. If R(Id-1) is Gorenstein, then I is a parameter ideal. 
Proof of Lemma (4.1). Clearly, J I(, ..., xd ) is a minimal reduction of 
In, and the ideal J* generated by the initial forms (xl )* in grA (In) is a com- 
plete intersection in the Cohen-Macaulay ring grA (In). Since G grA(In)/J* 
is artinian and a(G) = a + d, we get Ga+d+l = 0. That means 
1n(a+d+l) = In(a+d)J c j. 
Since xl, ..., Xd form a regular sequence we get 
ina+n+d = In(a+d+1)-d(n-1) c q 
If moreover R(In) is Gorenstein, then a = -2 by (2.2)(b), hence Id-n c q. 
As a consequence we obtain 
Proposition (4.3). An equimultiple ideal I of height h > 2 in a Gorenstein local 
ring A is a complete intersection if and only if R(1h-1) is Gorenstein. 
Proof. By Proposition (3.6) the "only if' part is already clear. For the converse 
let q be a minimal reduction of I. Since I is equimultiple q is generated by 
a regular sequence. On the other hand, by Corollary (4.2) IAp = qAp for all 
p E Assh(A/I), thus we have I = q since Ass(I/q) C Ass(A/q) = Assh(A/q) = 
Assh(A/I) and hence Ass(I/q) is empty. 
Theorem (4.4). Given a d-dimensional local Gorenstein ring (A, m) and an 
integer i E { 1, 2, 3}. Then for d - i > 1 we have 
R(md i) is Gorenstein iff 3 (m) = i - 1. 
Proof. If 3(m) = i - 1 then R(mn) is Gorenstein for n = d - 3(m) - 1 = d - i by 
(3.7). Conversely, assume that R(md-i) is Gorenstein and let q be a minimal 
reduction of m. By (4.1), we know that mi c q. Hence, for i E {1, 2} we 
get i = qmi-1 and mi-I =, qmi-2 by (3.1) and (3.7), i.e. 3(m) = i - 1 for 
these two cases. For i = 3 assume that R(md-3) is Gorenstein. Then m3 C q 
where q is a minimal reduction of m. In particular m3 C qm, and since A is 
Gorenstein m3 = qm2 by [13, Proposition (3.3) and Theorem (3.4)]. From this 
we get 3(m) = 2 since 3 (m) ? {0, 1 } again by (3.1) and (3.7). 
Corollary (4.5). Let (A, m) be a d-dimensional Cohen-Macaulay local ring. 
Then: 
(a) R(md-1) is Gorenstein iff A is regular. 
(b) R(md-2) is Gorenstein iff A is a hypersurface with e(A) = 2. 
(c) R(md-3) is Gorenstein iff A is Gorenstein and emb(A) = e(A) + d - 2, 
where d > 2 for (a), d > 3 for (b), and d > 4 for (c). 
Remark. Corollary (4.5) was also proved by A. Ooishi [11, Proposition (4.6)]. 
Moreover he could show that if e = e(A) < d, R(md-e) is Gorenstein if and 
only if A is a hypersurface [1 1, Proposition (4.7)]. In the following we prove 
a somewhat similar result, which contains the "if part" of Ooishi's result. 
REES AND FORM RINGS 641 
Proposition (4.6). Let (A, m) be a hypersurface with d = dim(A) > 2. Then 
R(mn) is Gorenstein iff d > e and n = d - e. 
Proof. Since A is a hypersurface, grA(m) is Gorenstein. We put a = a(grA (M)). 
Then R(mn) is Gorenstein 4> R(m) is Cohen-Macaulay and 
n = -a - 1 (by Theorem (3.5)) 
X a < 0 and n = -a - 1 
,# e<dandn=d-e (sincea=3(m))-d=e-1-d) 
,#n=d-e (> 1). 
If grA(m) is Cohen-Macaulay, but A is not a hypersurface, then 3(m) < 
e - 2. The next statement describes the case 3(m) = e - 2. 
Proposition (4.7). Let (A, m) be a d-dimensional local Cohen-Macaulay ring 
with d > 2. Assume that 
(i) R(m) is Gorenstein, 
(ii) 3(m)= e - 2. 
Then e=d=4. 
Proof. From assumption (i) we get by [8, Corollary (3.6)(2)] that 3 (m) = d - 2, 
hence d = e by assumption (ii). Then [2, Theorem (4.1) (iii)] implies that 
d = 4. 
Part II: Equimultiple prime ideals. First we mention two known results about 
the Gorenstein property of the Rees ring of equimultiple ideals of height 2. 
Proposition (4.8) [2, Theorem 2.6]. Let A be generalized Cohen-Macaulay ring 
of dimension > 4 and p an equimultiple prime ideal of height 2. If R(p) is 
Gorenstein, then A is a Gorenstein-domain and p is a complete intersection. 
Proposition (4.9) [2, Proposition 4.10]. Let A be a local ring of dimension > 3 
and p a prime ideal of height 2 such that 
(i) R(p) is Gorenstein, 
(ii) p/p2 is a free A/p-module. 
Then A is a Gorenstein-domain and p is a complete intersection. 
Remark. (a) Note that p in (4.9) is equimultiple. (b) The crucial step in the 
proofs of (4.8) and (4.9) was to show that A is Cohen-Macaulay. Then A 
was even Gorenstein by (2.2)(b) and p is a complete intersection by [8, (3.6)]. 
Hence R(p) was Cohen-Macaulay. Finally the domain property of A followed 
from the fact that Ap was regular; see [2, (2.1)]. 
In the following we give generalizations of (4.8) and (4.9). 
Proposition (4.10). Let A be a d-dimensional generalized Cohen-Macaulay ring 
and p an equimultiple prime ideal of height h > 2. If R(ph-1) is Gorenstein 
and d > 2h, then A is a Gorenstein-domain and p is a complete intersection. 
Proposition (4.11). Let (A, m) be a d-dimensional local ring and p i m a 
prime ideal of height h > 2 such that A/p and A. are Cohen-Macaulay. 
Assume that 
(i) R(ph-l) is Gorenstein, 
(ii) p/p2 is a free A/p-module. 
642 M. HERRMANN, J. RIBBE, AND S. ZARZUELA 
Then A is a Gorenstein-domain and p is a complete intersection. 
Proof of (4.10) and (4.1 1). For both propositions it is enough to show that A 
is Cohen-Macaulay. If this is known, A must be Gorenstein by (2.2)(b) and p 
is a complete intersection by (4.3). Since Ap is regular by (4.5) and R(p) is 
Cohen-Macaulay, A is again a domain. For the Cohen-Macaulayness of A in 
(4.11) we refer to the proof of (2.6) in [2]. 
The Cohen-Macaulay property of A in (4.10) we get as follows: From 
the assumptions we conclude by [3, Proposition (4.5.4)(ii)] that depth(A) > 
dim(A/p) + 1, i.e. depth(A) > d - h + 1 > h + 1 because d > 2h by as- 
sumption. Hence A - KA satisfies Serre's condition Sh+1, where KA is the 
canonical module of A . Then by [14, Satz 3.2.3] we know that Hl (A) = 0 for 
d - (h + 1) + 2 < i < d, therefore A is Cohen-Macaulay. 
Example [8, Example (2)]. Let A = K[[XI, X2, X3 , Y, Y2, Y3, Y4]]/J, where 
K is a field of characteristic 2 and 
J = (X1 Y1 + X2Y2 + X3Y3, Y, y, Y3, y Yy4, Y2Y4, Y3Y4, 
Y1 Y2 - X3 Y4, Y2 Y3 - X1 Y4, Y1 Y3 - X2 Y4). 
A is generalized Cohen-Macaulay. Let p be the maximal ideal of A. Then 
h =: ht(p) = /(p) = 3 and d = 3 4 2h, 
R(p) is Gorenstein, i.e. R(p 2) cannot be Gorenstein. Hence, two assumptions 
of (4.10) are not fulfilled, and we see that A is not Cohen-Macaulay (otherwise, 
A would be a hypersurface by (4.5), which is obviously not the case). 
ACKNOWLEDGMENTS 
The last author was supported by the DAAD (Germany) and DGICYT-grants 
BE 90-049 and PB 88-0224 (Spain). He received stimulating hospitality by the 
Mathematical Institute of the University of Cologne (Germany). 
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(M. Herrmann and J. Ribbe) MATHEMATISCHES INSTITUT DER UNIVERSITAT zu KOLN, WEYER- 
TAL 86-90, D-50931 KOLN, GERMANY 
(S. Zarzuela) DEPARTAMENT D'ALGEBRA I GEOMETRIA, UNIVERSITAT DE BARCELONA, GRAN 
VIA 585, E-08007 BARCELONA, SPAIN 
E-mail address: ribbeOmi.uni-koeln.de