ON THE GORENSTEIN PROPERTY OF REES AND FORM RINGS

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(I") and the form ring grA(I") are Gorenstein rings, where / 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 / in a local ring (A, m) it is well known that the CohenMacaulayness of the Rees algebra R(I) implies the Cohen-Macaulayness of all
Rees algebras R(I"). The same is true for the form rings grA(I) and grA(I") ;
see [3, (2.7.8) and (8.8.5)]. In this paper we show that, in contrast to the CohenMacaulay property, the Gorenstein property of R(I") and grA(I") only holds
for special exponents n. If in particular gr^(/) 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 complete 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(I") and gcA(In) are Gorenstein 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 (-/?(/)) can be deduced from the Gorenstein property of a certain Veronesean subring R(In) of R(I). It might be an interesting question
when the Gorenstein property of a blow up Pmj(R(I)) is inherited to an appropriate Rees ring R(J) with Proj(R(J)) = Proj(i?(/)) over Spec A.
In §4, Part I we characterize—for Cohen-Macaulay (or Gorenstein) rings
A—the Gorenstein property of R(md~'), i = 1, 2, 3, by conditions on the
Received by the editors June 30, 1991 and, in revised form, April 10, 1992.
1991 MathematicsSubject Classification.Primary 13D03, 13H10, 13H15; Secondary 14B05,
14B15.
©1994 American Mathematical Society
0002-9947/94 $1.00+ $.25 per page
631
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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 p in A for being a complete intersection in terms of blow up properties. In case that dim(^4) > 2 ht(p) and A
is a generalized Cohen-Macaulay ring, the Gorenstein property of Ä(pht(p)_1)
implies that p 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 / in a (commutative) Noetherian ring A, we will also use the so-called
approximation complexes of / (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 / is of linear type,
namely Sym^(Z) ~ R(I).
Assume for simplicity that A is a local ring with maximal ideal m and
dim(^4) = d. Let / be an ideal of A and a := {a\,...,
a„) a system of
generators of /, and consider the Koszul complex K(a) of A w.r.t. a. If
we denote by S the polynomial ring A[X\, ... , X„] it is possible to get two
complexes of ¿'-modules:
Z(a) :0^&„-^-►
Zx -►Zo -> 0
and
Jt(a) :0^jrn-+->
Jfx -►Jfo -» 0,
where Zt = Z¡(K(a)) ®AS and Jt¡ = Hi(K(a)) ®AS.
The Zj(K(a)) denote the cycles of the Koszul complex K(a), and the H¡(K(a))
denote the Koszul homology. Both complexes can be taken as complexes of
graded modules over 5" with mappings of degree -1.
We list the main properties of these complexes:
(1) The homology of 3?(a) and Jt(a) is independent of the system of
generators a.
(2) Hoi2-ia)) = SymAiI).
(3) HoiJ?ia)) = SymAiI/I2).
(4) The following are equivalent:
(i) Jf(a) is acyclic,
(ii) 2"(a) is acyclic and / 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, p(Ip) < ht(p) (where p(-)
the minimal number of generators).
(ii) For any r > 0 and for any prime ideal p D I,
depthApiHriKia))p) > inf(r, ht(p//)).
Then ^ia)
is acyclic.
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denotes
REESAND FORM RINGS
633
It turns out that the condition (ii) above is independent of the system of
generators of /, and that it is fulfilled if / is strongly Cohen-Macaulay (sCM
for short), that is, if for any r > 0 the Koszul homology HriKia)) is zero or a
maximal Cohen-Macaulay ^//-module.
(6) Suppose:
(i) For any prime ideal p 2 /, p(Ip) < ht(p) + 1.
(ii) For any r > 0 and for any prime ideal p 3 /,
depthApiHriKia))p) > inf(r, ht(p//)) - 1.
Then Z[a) is acyclic.
(7) Assume that ht(/) > 1 and
(i) For any prime ideal p ¡5 /, p(Ip) < ht(p),
(ii) / is strongly Cohen-Macaulay.
Then gcA(I) and R(I) are Cohen-Macaulay. Moreover, if A is Gorenstein
then gr^(7) is Gorenstein.
There is an important connection between the theory of approximation complexes and the theory of ¿/-sequences:
(8) If / is generated by a ¿/-sequence, then •/#(«) is acyclic.
(9) If \A/m\ = oo and Jifia) is acyclic, then / can be generated by a
¿/-sequence (but the given a is not necessarily a ¿/-sequence!).
Finally we mention some relevant families of strongly Cohen-Macaulay ideals:
(a) Complete intersection ideals.
(b) Ideals I ç A such that p(I) - ht(/)+ 1 and A/1 is Cohen-Macaulay. In
particular, almost complete intersection (a.c.i) ideals / such that A/I is CohenMacaulay are sCM. We can say that an ideal / is a.c.i. if p(I) = ht(/) + 1 and
IAp is complete intersection for any prime ideal p e Mm(A/I).
(c) Suppose A is Gorenstein and I c A is an ideal such that p(I) = ht(/)+2.
If A/I is Cohen-Macaulay then / is sCM.
2. The ¿mnvariant
of graded modules
In this section we collect some important properties of the ¿z-invariants of
Rees and form rings.
Recall that for a positively graded Noetherian ring R = 0;>O-R, defined
over a local ring Ro and a Noetherian graded Ä-module G the ¿z-invariant of
G is defined as
aiG) = max{jeZ:HrMiG)j¿0},
where r = dim(G) and HrM(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 —0„>o Rn be a Noetherian graded algebra defined over a
local ring Rq and G —©„>0 G„ 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) < aiG/xG) - 1.
Moreover, if G is Cohen-Macaulay the equality holds.
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M. HERRMANN, J. RIEBE, AND S. ZARZUELA
634
Proof. Consider the exact sequence of graded modules
(1)
0^Gi-l)^G^G/xG-+0.
Putting r = dimn(G) and M —maximal homogeneous ideal of R, we get from
( 1) the long exact cohomology sequence
(2)
... -» Hr~\G)i - H£\G/xG),
- «£,((?),_, - HrM(G),- 0.
For i = a(G) + l we have
(3)
HrM(G)i-x¿0
and
HrM(G)i= 0.
Then (2) implies Hr^l(G/xG)¡ ¿ 0, i.e. aiG/xG) > i = a(G) + 1. This
proves the first part of the claim. The second part follows from (2) since now
H£\G) = 0.
In our context the importance of ¿z-invariants is a consequence of the following 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
H'M(grA(I))n= 0 for n¿-I,
i = 0, ... , d - 1,
and
HdM(%xA(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,
WMi&AiI)U * Hi{A) for i = 0, ... , d - 1.
(b) [8, Theorem (3.1)] If R(I) is Cohen-Macaulay and grade(7) > 2, then
R(I) is Gorenstein if and 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 ç A with a - a(grA(I)), the ¿z-invariant of the Rees algebra R(I)
is in many cases strictly determined. In particular if R(I) is Cohen-Macaulay
and ht(7) > 0, then a(R(I)) = -1. In this case ¿z(gr^(7))< -1. This can be
easily deduced from the proof of Proposition (2.1) in [8].
For several classes of ideals 7 the ¿z-invariant of the form ring gr^(7) 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
¿z-invariant a(gr^(7)) for a strongly Cohen-Macaulay ideal which is of linear
type. First we define the reduction exponent S(I) of an ideal 7 in a local ring
A with infinite residue field as
ô(I) :—min{« e 7Y| there exists a minimal reduction q of 7 s.t. In+l = ql"}.
Lemma (2.4). Let I be an m-primary ideal in a local ring (A, m). Then the
followinghold:
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REES AND FORM RINGS
635
(a) [13, Theorem and Theorem 2.1] If the reduction exponent of the maximal
ideal m of A is 0, 1 or 2, then gr^m) is Cohen-Macaulay iresp. Gorenstein)
iff A is so.
(b) If grA(I) is Cohen-Macaulay, then
a(grA(I)) = ô(I)-dim(A).
Statement (b) follows from (2.1) using the fact that the initial forms of the
generators of a minimal reduction of 7 (which is a parameter ideal in A) form
a regular sequence in gr^ (7).
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 p(Ip) < ht(p) for all prime
ideals p D 7. Then a(grA(I)) = - ht(7).
Proof. Put 5 = ht(7) = grade(7), and p(I) = s + t. Note that the homology
H¡(K.) of the Koszul complex K. of A with respect to some minimal system of
generators of 7 is zero for i > t. Moreover grA(I) is CM and the ^#-complex
is exact by §1, (5) and (7), which implies in particular that 7 is of linear type.
The .^-complex gives a resolution of gr^ (7) :
(i)
o^jt,^—►^ri^^r0-gr^(/)-»o,
where J(, = H¡(K.) ® A[X\, ... , Xs+t], dim(^)
= d + t and Ji, is CM
over A\X\, ... , Xs+t]. The idea is now to compute ¿z := a(grA(I)) via the
¿z-invariants a(Ji{) : Applying Lemma (2.1) exactly (s + t) times we see that
(2)
a{Jti) = a(H,(K.)) - p(I) = -p(I),
since (1) is considered as a sequence of A[X\, ... , Xs+i]-modules. Note that by
construction the morphism y>, are of degree 1. Therefore we get the following
exact sequences for the cokernels 3>¡ of cp,+\
0 ^ &i+i(-l) ^ J?i ^ @i-+0
with morphisms of degree zero.
Since G := gr^(7) and Ji, are CM, we get for the local cohomology with
respect to the maximal homogeneous ideal M of A/I\X\, ... , Xs+t]
(3)
HÍ(G)j ~ 77¿+1(^);-i ^ ••• - Hb+t-x&t-i)j-t+i.
Moreover we have the exact sequence
(4)
0 - Hi+'-^&t-új-M
- HdM+t(3lt)j-t- HdM+t(A-x)j-t+i -
Note that
(5)
HdM+t(®t)j-tciHdM+'(jrt)j-t.
Case 1. j > -h, where h = ht(7), i.e. j - t > -p(i) =: -n. Therefore
HM+'(^t)j-t = 0, since a(Jft) = -n . But by (3), (4) and (5) we know that
HdM(G)j ~ HdM+t-\2,-X)}-,+x c HdM+t(j?,)j-t = 0,
hence a(G) < -h .
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636
M. HERRMANN,J. RIBBE,AND S. ZARZUELA
Case 2. j = -h . Then (3) implies
HdM(G)_h ~ HdM+t-\2,-X)_h_t+l
c HdM+t(J/,)-n ¿0,
hence a(G) = - ht(7).
Remark. We can apply (2.5) in particular to an almost complete intersection
ideal 7, if A/I is a Cohen-Macaulay ring (in this case 7 is strongly CohenMacaulay). It is shown in [12] that the formula for the ¿z-invariant in (2.5) holds
for any (possibly non-Cohen-Macaulay) almost complete intersection ideal 7.
Next we describe the relationship between the ¿z-invariants a(g\~A(I)) and
¿z(gr^(7")). 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 gr^(7").
Proposition (2.6). Let I be an ideal in a local ring A and assume that grA(I)
is Cohen-Macaulay. Then
a(grA(r)) = [a(gxA(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 & {0, ... , n - 1}. For every i G {I, ... , n} there is an exact sequence of
Ä(7")-modules:
0 _* /«-'+> &Á'i») - 7"-¡'gr^(7") -» grAiI)in - i)« -» 0.
Let N be the maximal homogeneous ideal of R(I") = jR(7)(">. Then (see [3,
Proposition (47.5)]):
H'NigrAiI)in - *)(">)~ HUèrAiI)in
- i))(B).
Hence for every j e Z and i e {I, ... , n} there is an exact sequence
0 _» HÍ(I-M
gTAiI"))j- HdNiI»-'gTAiI»))j- HftiffjUtynj+n-i - 0.
First consider these sequences for j > I + I . Since nj + n - i > n(l + I) +
n - i > nl + r = a, we have 77^(gryí(7))„J+n_¡ = 0 for each i € {1,...,«}.
Using the above sequences it follows inductively that 77^(7"_,gr^(7"))7 = 0
for i = 0,... , n. In particular Hf¡(gxÁ(In))j = 0 for any y > / + 1, i.e.
¿z(gr/4(7'1))< /. To finish the proof, consider the cohomology sequences from
above in degree j = I. Since HdMigrAiI))nl+r /Owe
get 77#(7rgryl(7',))/ ±
0 and then successively 77$(7r-i:gr/1(7"))/ ^ 0 for A: = 1, ... , r. Hence
Hf>(BtA(I"))i¿0. Q.E.D.
Corollary (2.7). Let I be an ideal of height > 1 in a local ring A . If gr^(7) is
Cohen-Macaulay and R(In) is Cohen-Macaulay for some n e N, then i?(7) is
Cohen-Macaulay.
Proof. First, note that
-l>a(grA(P))
= [a(gxA(I))/n]
by(2.2)(a)
by (2.6).
Hence ¿z(gr^(7)) 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).
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637
REES AND FORM RINGS
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(V) are Gorenstein. Since R(Ist) = R(F)W =
£(7')(i) and since R(I')(-l)
and R(I'){-1) are canonical modules of R(F)
and R(I') (see (2.3) for the correct shifting degree -1) we know that both,
Ä(J*)(-1)W and Ä(/')(-l)(i)
are canonical modules of R(I"). Hence they
must be isomorphic, and comparing their homogeneous parts of degree j > 1
we see that the ideals /*W-1) and /'W-1) are isomorphic as /i-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 I' are isomorphic, then s = t.
Proof. We may assume that A is a local ring with maximal ideal m. The isomorphism 7s ~ I' induces isomorphisms Ijs/mljs ~ IJ'/mIJt for all numbers
j. Now, there is a polynomial P = Yl'iZoaixi e Q\x] of degree / - 1 (where /
denotes the analytic spread of 7) such that P(i) = X(Il/ml1) for i » 0 (A denotes the length). From P(sj) = P(tj) for j » 0 we get a¡-is'~l = ¿z/_ií/_1.
Since / > ht(7) > 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
grAiI) is Gorenstein iff (1, t)~a~2i-l) is a canonical module of R(I), where
a := ¿z(gr^(7)) and (1, t)m denotes the RiI)-submodule of the polynomial ring
A[t] which is generated by I, t, ... , tm in case m>0 or (1, t)~l = 7J?(7) 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, i)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(-l) by K, we get
ro
Kj = (l,t)m(-l)]
if y < o,
= \ A
if l<j<m+l,
{ JJ-(m+\) if;>m
+ 2.
Now recall that the Veronesean AT'm+1)is a canonical module of the Veronesean
R(I)(m+l) = R(Im+l). We get
ro
(K(m+% = Kj(m+l)= I A
{ jj(m+\)-(m+\)
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if j"< o,
ifj=l,
if / > 2 .
638
M. HERRMANN, J. RIBBE, AND S. ZARZUELA
Hence R(Im+l)(-l)
= 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 grAiI) is Gorenstein. Then the following hold for n e N :
(a) 7?(7") is Gorenstein iff R(I) ¿s Cohen-Macaulay and n = -a(grA(I)) - 1.
(b) If R(I) is Cohen-Macaulay, then grA(In) is Gorenstein iff -a(gTA(I)) = 1
mod in).
Proof, (a) Put ¿z= ¿z(gr/j(7)). Since gr^(7) is Gorenstein and R(I) is CohenMacaulay 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~{) is Gorenstein, by (3.4). This is by
(3.1) the only power of I which has a Gorenstein Rees algebra.
(b) Put b = ¿z(gr^(7")). We can assume that n > 2. Note that R(In)
and gr^(7") are Cohen-Macaulay rings and that K^ is a canonical module
of Ä(/)(") ~ R(In). Hence by (3.3) pA(In) is Gorenstein if and only if the
i?(7")-module L := (1, t)-b-2(-l)
is isomorphic to K™ . 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 re{0,...,n-l}.
Hence we prove the following claim.
Claim. L ~ KM iff r = n - 1 .
Assume that r = n - 1. Then we get for each j e Z :
(0
Lj = (1, t)-b-2(-l)j =1 A
if;'<o,
if 1 < j < -b- 1,
{ jnU-l+b-2)
if j>-b,
and
{&%
= ((i,
tya-2(-i)^)j
{0
^
l(jn-\)+(bn+n+l)
= ((i,
/)-*"-"-%-!,
if jn - 1 < 0,
if 0 < jn - 1 < -bn -n-l,
jf jn_i>_bn_n^
(0
if 7 < 0,
= <¿
if 1 < 7 < -b - 1,
[ jnu+¿>+i) if y > -¿> - 1 +■1/«, i.e., if ; > -ô.
Hence L ~ K^ . For the converse assume that L ~ A^("'; in particular L_¿ ~
K_bn = /sTr_a, i.e. I" ~ 7r+1. Since ht(7) > 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 CohenMacaulay ideal satisfying p(Ip) < ht(p) for all prime ideals p D 7. Assume that
h := ht(7) > 2. Then:
(a) R(I") is Gorensteiniff n = h - 1,
(b) grA(In) is Gorenstein iff h = 1 mod (n).
Proof. Use (3.5) together with (2.5) and §1(7).
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REES AND FORM RINGS
639
Proposition (3.7). Let iA, m) be a d-dimensional Gorenstein local ring with
reduction exponent ¿(m) < 2 and ¿/ > 3. Then the following hold for n £ N :
(a) gr^m") is Gorenstein iff d - <5(m)= 1 mod («).
(b) Rim") is Gorenstein iff n = d - ¿(m) - 1.
In particular: If A is regular, then Rimd~l) is Gorenstein, and if A is a
quadratic hypersurface, then Rimd~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)
0 - /»-W grAiI") -» 7"-/gr4(7") - &A{I){n - i){n) - 0
one can show the following: If 7 is an ideal primary to the maximal ideal m of
a local ring (A, m), such that gtA(I) and gr^(7") are Gorenstein rings for some
n , then a(%rA(I)) = -1 mod («) (see [12]; the primary property of 7 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
7 is an ideal satisfying the conditions in (3.6), then by localizing at a minimal
prime of 7 we reach the described situation (i.e. the primary case). Since
moreover a(grA(I)) = -ht(7) = a{grAp(Ip)) for p e AsshA(A/I) (by (2.5)), the
implication "gr^(7") Gorenstein =* h = 1 mod («) " follows. Finally (under
the assumptions of (3.6)) the Gorenstein property of Ril") forces gr/4(7") to
be Gorenstein too, by (2.2); hence h = 1 mod («), as we have just pointed out.
Furthermore -2 = aigrAiI")) = [-h/n] by (2.2), (2.6) and (2.5). Now it is
easy to see that the two relations h = 1 mod («) 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 gr^(m") are
Gorenstein rings, see [12].
Remark (3.9). The assumption d > 3 in Proposition (3.7) was only used to get
that Ril) 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 Ril"), where 7 is m-primary, determines the structure of the ideal 7 via
the reduction exponent ¿(7). As a corollary we obtain a characterization of
regular ¿/-dimensional local rings iA, m) by the Gorensteinness of i?(m</~1),
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
iA, m), and q = (x\, ... , x¡) a minimal reduction of I. Assume that grA(I")
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M. HERRMANN, J. RIBBE, AND S. ZARZUELA
640
is Cohen-Macaulay for some natural number n. Put a := a(grA(I")). Then
jna+n+d g q ¡f moreover /?(/«) ¿j Gorenstein and d > 2, then Id~" ç 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~l) is Gorenstein, then I is a parameter ideal.
Proof of Lemma (4.1). Clearly, J := (x", ... , x%) is a minimal reduction of
I" , and the ideal J* generated by the initial forms (x")* in grA(I") is a complete intersection in the Cohen-Macaulay ring gr^(7"). Since G := grAiI")/J*
is artinian and a(G) = a + d, we get Ga+(¡+l = 0. That means
jn(a+d+l)
_ jn(a+d) j q j
Since Xy,... ,x¿ form a regular sequence we get
jna+n+d
_ jn(a+d+\)—d(n—l)
(- „
If moreover R(I") is Gorenstein, then ¿z= -2 by (2.2)(b), hence Id~" ç 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 i?(7A_1) is Gorenstein.
Proof. By Proposition (3.6) the "only if part is already clear. For the converse
let ¿7 be a minimal reduction of 7. Since 7 is equimultiple q is generated by
a regular sequence. On the other hand, by Corollary (4.2) IAP — qAp for all
p e AsshiA/I), thus we have 7 = q since Ass(7/¿¡r)ç Ass(A/q) = Assh(A/q) =
Assh(^/7) and hence Ass(7/¿/) is empty.
Theorem (4.4). Given a d-dimensional local Gorenstein ring iA, m) and an
integer ¿€{1,2,3}.
Then for d - i > 1 we have
Rimd~') is Gorenstein iff Sim) = i - 1.
Proof. If S(m) = i-1 then R(m") is Gorenstein for n —¿/-¿(m)-1
= d—i by
(3.7). Conversely, assume that Rimd~l) is Gorenstein and let ¿7 be a minimal
reduction of m. By (4.1), we know that m' ç ¿7. Hence, for i G {1, 2} we
get m' = qm'~l and m'"1 ^ qm'~2 by (3.1) and (3.7), i.e. <5(m)= i - I for
these two cases. For i — 3 assume that T^rn''-3) is Gorenstein. Then m3 ç q,
where ¿7 is a minimal reduction of m. In particular m3 ç ¿7m, and since A is
Gorenstein m3 = ¿7m2by [13, Proposition (3.3) and Theorem (3.4)]. From this
we get f5(m)= 2 since ô(m) $ {0, 1} again by (3.1) and (3.7).
Corollary (4.5). Let iA ,m) be a d-dimensional Cohen-Macaulay local ring.
Then:
(a) T^nV*-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(^4) = e(A) + d - 2,
where ¿/ > 2 for (a) , ¿/ > 3 for (b), and ¿/ > 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 [11, Proposition (4.7)]. In the following we prove
a somewhat similar result, which contains the "if part" of Ooishi's result.
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REES AND FORM RINGS
641
Proposition (4.6). Let (A ,m) be a hypersurface with d = dim(^) > 2. Then
R(m") is Gorenstein iff d > e and n = d - e.
Proof. Since A is a hypersurface, gr^(m) is Gorenstein. We put a —a(grA(m)).
Then R(m") is Gorenstein <£•R(m) is Cohen-Macaulay and
n = -a - 1 (by Theorem (3.5))
■&a < 0 and n- -a-l
<$ e < d and n = d - e
&n = d-e
(since ¿z= ô(m) -d —e-l-d)
(> 1).
If gr^(m) is Cohen-Macaulay, but A is not a hypersurface, then ô(m) <
e - 2 . The next statement describes the case ô(m) = e - 2.
Proposition (4.7). Let (A, m) be a d-dimensional local Cohen-Macaulay ring
with ¿/ > 2. Assume that
(i) R(m) is Gorenstein,
(ii) ô(m) = e-2.
Then e = d = 4.
Proof. From assumption (i) we get by [8, Corollary (3.6)(2)] that S(m) - d-2,
hence d = e by assumption (ii). Then [2, Theorem (4.1)(iii)] implies that
¿/ = 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 ¿zprime 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~x) 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/m
prime ideal of height h>2 such that A/p and Ap are Cohen-Macaulay.
Assume that
(i) R(ph~l) is Gorenstein,
(ii) p/p2 is a free A/p-module.
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a
M. HERRMANN, J. RIBBE, AND S. ZARZUELA
642
Then A is a Gorenstein-domain and p is a complete intersection.
Proof of '(4.10) and (4.11). 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(yi) >
dim(^4/p) + 1, i.e. depth(^) > d - h + I > h + 1 because d > 2/z by assumption. Hence A ~ KA satisfies Serre's condition Sf,+i, where KA is the
canonical module of A . Then by [14, Satz 3.2.3] we know that Hlm(A)= 0 for
d-(h + l) + 2<i<d,
therefore A is Cohen-Macaulay.
Example [8, Example (2)]. Let A = K[[XX, X2, Xi,Yx,Y2,Yi,
K is a field of characteristic 2 and
Y4]]/J, where
J = (XxYx +X2Y2 + X3Y3, Y2, Y2, Y2, Y2,YlY4, Y2Y4, Y3Y4,
Y\Y2- X3Î4, Y2Yt,- X\Y4, Y\Y$- X2Y4).
A is generalized Cohen-Macaulay. Let p be the maximal ideal of A . Then
h =: ht(p) = l(p) = 3 and d = 3?2h,
R(p) is Gorenstein, i.e. R(p2) 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).
References
1. S. Goto and Y. Shimoda, On the Rees algebras of Cohen-Macaulay rings, Commutative
Algebra: Analytical Methods (R. N. Draper, ed.), Lecture Notes in Pure and Appl. Math.,
vol. 68, Dekker, New York and Basel, 1982, pp. 201-231.
2. M. Herrmann and S. Ikeda, On the Gorenstein property of Rees algebras, Manuscripta Math.
59(1987), 471-490.
3. M. Herrmann, S. Ikeda, and U. Orbanz, Equimultiplicity and blowing up, Springer-Verlag,
Berlin and Heidelberg, 1988.
4. J. Herzog, A. Simis, and W. V. Vasconcelos, Koszul homology and blowing up rings, Commutative Algebra, Proc. Trento Conf. (S. Greco and G. Valla, eds.), Lecture Notes in Pure
and Appl. Math., vol 84, Dekker, New York and Basel 1983,pp. 79-169.
5. _,
Approximation complexes of blowing up rings, J. Algebra 74 (1982), 466-493.
6. _,
Approximation complexes of blowing up rings. II, J. Algebra 82 (1983), 53-83.
7. _,
On the canonical module of the Rees algebra and the associated graded ring of an
ideal, J. Algebra105 (1987), 285-302.
8. S. Ikeda, On the Gorensteinness of Rees algebras over local rings, Nagoya Math. J. 102
(1986), 135-154.
9. T. Marley, The coefficients of the Hubert polynomial and the reduction number of an ideal,
J. London Math. Soc. (2) 40 (1989), 1-8.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
REES AND FORM RINGS
643
10. A. Ooishi, Stable ideals in Gorenstein local rings, J. Pure Appl. Algebra 69 ( 1990), 185-191.
11. _,
On the Gorenstein property of the associated graded ring and the Rees algebra of an
ideal, preprint, 1990.
12. J. Ribbe, Thesis, Universität zu Köln, 1991.
13. J. Sally, Tangent cones at Gorenstein singularities, Compositio Math. (2) 40 (1980), 167-
175.
14. P. Schenzel, Dualisierende Komplexe in der lokalen Algebra und Buchsbaum-Ringe, Lecture
Notes in Math., vol. 907, Springer-Verlag, 1982.
15. N. V. Trung and S. Ikeda, When is the Rees algebra Cohen-Macaulay1, Comm. Algebra 17
(1989), 2893-2922.
16. A. Lascu and M. Fiorentini, Linkage among subcanonical and quasicomplete intersection
projective schemes, preprint, 1991.
(M. Herrmann and J. Ribbe) Mathematisches
Institut
der Universität
zu Köln, Weyer-
tal 86-90, D-50931 Köln, Germany
(S. Zarzuela) Departament d'Àlgebra i Geometría,
Via 585, E-08007 Barcelona, Spain
E-mail address : ribbeflmi. uni-koeln. de
Universität
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de Barcelona,
Gran