Nov 25, 1995 - deduce the generic perfection directly from the results of [6]. Thus our proof ... In Section 1 we prove the generic perfection of Hankel ideals.
Hankel matrices and Hankel ideals Junzo Watanabe November 25, 1995 Introduction In [6] Hochster and Eagon obtained some basic results on determinantal ideals of generic matrices. This was followed by [8], where Kutz proved the same results for symmetric matrices using the same proof techniques introduced in [6]. In the present paper we deal with determinantal ideals of what we call Hankel matrices. (We shall also say Hankel ideals. For definition see Section 1.) Hankel matrices appear in the theory of binary forms as symmetric matrices in the theory of quadratic forms. With an n-ary r-form F (i.e., a homogeneous polynomial of degree r in n variables), one may associate a homogeneous Artinian Gorenstein ring A(F ) in a canonical way, and when n = 2 and r is fixed, the Hilbert series of the ring A(F ) is determined by the rank of the Hankel matrix formed by the coefficients of F . This was the original motivation for the study of Hankel matrices in the hope that the analogy of Kutz [8] was possible. It turns out that much the same results hold with those of [6] and [8]; it should be noted, however, that the proof is different. In [6] and [8] they first prove that the ideals in consideration are prime, then this fact is used to prove the generic perfection. In the present case we can deduce the generic perfection directly from the results of [6]. Thus our proof is much simpler than those of [6] and [8], especially in that we do not need a “principal radical system of ideals,” which was an important tool for proving that a certain family of ideals consists entirely of radical ideals. This is the reason why we stay in the small class of ideals; namely we consider only those ideals of the form It (H), which is the ideal generated by the t + 1 × t + 1 minors of a Hankel matrix H. It will be shown among other things that, if 1
H is an r × s matrix, then It (H) depends only on t and r + s. Thus the class of ideals we consider has in fact only two parameters. In Section 1 we prove the generic perfection of Hankel ideals. In Section 2 we construct generic points (in the sense of [6]) for Hankel ideals, and in particular it is proved that they are prime. In Section 3 the normality is proved for the quotient rings by Hankel ideals provided that the coefficient ring is normal. One might expect that these rings (over a field k) appear as rings of invariants by algebraic groups as in the cases [6] and [8]. This is true if t = 1 and ch k = 0. In fact they are rings of invariants by finite cyclic groups put in the center of GL(2, k) acting on the two dimensional polynomial ring. But if t > 1, we do not know if this is true. Section 4 can be read independently of the previous sections, where we survey how the Hankel matrices and Hankel ideals arise from the theory of binary forms.
1
The main results
By a Hankel matrix over a commutative ring R, we mean a matrix of the form:
a0 a1 a2 ··· ··· am
a1 a2 a3 ··· ··· am+1
a2 a3 a4 ··· ··· am+2
a3 a4 a5 ··· ··· am+3
··· ··· ··· ··· ··· ···
an−m an−m+1 an−m+2 ··· ··· an
n We denote it by Hm or by Hm (a0 , a1 , · · · , an ) when we need to refer to n its size or the entries. Note that Hm or Hm (a0 , a1 , · · · , an ) is an m + 1 by n − m + 1 matrix. If H is any matrix over R, then by It (H) we denote the ideal generated by all the t + 1 by t + 1 minors of H. For a Hankel matrix H we call It (H) a Hankel ideal. Our main results are stated as follows: n be Theorem 1 Let R be a commutative Noetherian ring with unit. Let Hm a Hankel matrix over R. Let t be an integer, 0 ≤ t ≤ Min{m, n − m}. Then n n )) = n+1−2t, ) is n+1−2t. If grade (It (Hm the largest possible grade for It (Hm n then the ideal It (Hm ) is perfect.
2
Moreover if Λ is a Noetherian integral domain, and R = Λ[a0 , a1 , · · · , an ], n where ai are indeterminates, and H = Hm = Hm (a0 , a1 , · · · , an ) then It (H) is a perfect prime ideal of grade n + 1 − 2t. If Λ is a normal ring then so is R/It (H). If m′ is another integer such that 0 ≤ t ≤ M in{m′ , n − m′′ }, then n n It (Hm ) = It (Hm ′ ). As in Hochster-Eagon [6] the essential part of Theorem 1 is the statements in the second paragraph, especially when Λ is the ring of integers or a finite prime field. When Λ is a field and t = Min{m, n − m}, the perfection of the generic Hankel ideals was implicitly proved in Eagon [2]. His proof is actually valid for any Noetherian ring Λ, which is important to us because we need it in order to prove that the ideals are prime if Λ is a domain. In this section we modify Eagon’s proof to establish the generic perfection of the Hankel ideals. The proof of all other statements of Theorem 1 will be given in the later sections. Proposition 1 Let R = Z[a0 , a1 , · · · , an ] be the polynomial ring over the integers. Let H = Ht (a0 , a1 , · · · , an ) be the Hankel matrix with t ≤ n − t and let I = It (H). Then I is generically perfect of grade n + 1 − 2t. Proof. Let X = (xij ) be the generic t + 1 × n − t + 1 matrix. By theorem 1 and corollary 1 of [6], I = It (X) is a perfect ideal of Z[X] of grade n + 1 − 2t. Hence Z[X]/I is a Cohen-Macaulay ring of dimension 1 + (n + 2)t − t2 . Consider the set of linear forms: S = {xij − x(i−1)(j+1) |1 ≤ i ≤ n − t, 0 ≤ j ≤ t − 1} Note that R = Z[X]/(S) and R/I = Z[X]/It (X) + (S). Since #S = (n − t)t, it follows that dim R/I ≥ 1 + 2t. Now let T be the set of 2t elements {a0 , a1 , · · · , at−1 , an−t+1 , · · · , an−1 , an } . Notice that R/I + (T ) ∼ = t+1 Z[at , at+1 , · · · , an−t ]/(at , at+1 , · · · , an−t ) , which is finite flat over Z. That dim R/I + (T ) = 1 implies that dim R/I ≤ 1 + 2t , and that the set S ∪ T consists of a regular sequence both for Z[X] and for Z[X]/It (X). Thus I is a perfect ideal of R of grade n + 1 − 2t. That R/I is flat over Z follows easily from the fact that R/I + (T ) if Z-flat and T form a regular sequence. Remark By proposition 4 of Eagon-Northcott [3], Proposition 1 above proves the statements of the first paragraph of Theorem 1 3
Before ending this section we note the following, which will be used in the inductive argument in the next section. n Proposition 2 Let R be a commutative ring, and ai ∈ R. Let Hm = n−1 Hm (a0 , a1 , · · · , an ) and Hm = Hm (a0 , a1 , · · · , an−1 ). Then n n n n (0) t Hm = Hn−m (Here t H denotes transpose.) Hence It (Hm ) = It (Hn−m ). n−1 n n ) for 0 < t ≤ M in{m, n − m}. ) ⊂ It−1 (Hm−1 ) ⊂ It−1 (Hm (1) It (Hm n n−1 n ) for 0 < t ≤ M in{m, n − m}. ) ⊂ It−1 (Hm (2) It (Hm ) ⊂ It−1 (Hm n−1 Proof. (0) is obvious. For (1) note that the matrix Hm−1 is obtained n from Hm by omitting the last row. Hence by expanding various minors in n It (Hm ) along the last row we immediately have the first inclusion. The second inclusion is obvious. To prove (2) replace m by n − m in each term, then use (0) and (1).
2
The generic points of the Hankel ideals
We introduce some notation that is kept fixed throughout this section. Let t ≥ 0 be an integer, and let Λ be a Noetherian integral domain with unit. Let Q = Λ[x0 , x1 , · · · , xt−1 , y0 , y1 , · · · , yt−1 ] be the polynomial ring over Λ in the 2t ∑ variables. For i = 0, 1, 2, 3, . . . , define xt+i inductively by xt+i = t−1 j=0 xi+j yj . n n Let Gm be the Hankel matrix: Gm = Hm (x0 , x1 , · · · , xn ) for m ≥ t and n − m ≥ t. Let X = Ht−1 (x0 , x1 , · · · , x2t−2 ) and X ′ = Ht−1 (x0 , x1 , · · · , x2t−1 ). Finally let Y =
0 0 0 1 0 0 0 1 0 ··· ··· ··· ··· ··· ··· 0 0 0
··· ··· ··· ··· ··· ···
0 y0 0 y1 0 y2 · ··· · ··· 1 yt−1
. Note that X, X ′ and Y are t × t square matrices. Proposition 3 With the above notation, 4
(1) XY i = Ht−1 (xi , x1+i , · · · , x2t−2+i ). In particular XY = X ′ and Y = X −1 X ′ . (That det X ̸= 0 can be seen easily.) (2) Fix m so that Gnm have the same number of rows for ∀n. If ei is the i-th column of any Gnm for n − m ≥ i, then (ei ei+1 · · · ei+t−1 )Y = (ei+1 ei+2 · · · ei+t ), where (ej ej+1 · · ·) denotes the matrix consisting of ej ’s as columns. Hence (e0 e1 · · · et−1 )Y i = (ei ei+1 · · · ei+t−1 ). (3) The 2t elements x0 , x1 , · · · , x2t−1 are algebraically independent over Λ. Moreover Λ[x0 , x1 , · · · , x2t−1 ][∆−1 ] = Q[∆−1 ] where ∆ = det X. (4) Rank Gnm = t for any m, n such that m ≥ t − 1 and n − m ≥ t − 1. (5) Let fi be the i-th column of X, i = 0, 1, · · · , t − 1, and ft the last column of X ′ . Put ∆i = (−1)i det (f0 f1 · · · fˆi · · · ft ) . Then yi = −∆i /∆t . (Note that ∆t = det X.) Proof. (1) and (2) are immediate from the definition of X, X ′ , Y , etc. (3) Put Q′ = Λ[x0 , x1 , · · · , x2t−1 ]. Then X and X ′ are defined over Q′ . Hence Y is defined over Q′ [∆−1 ]. This shows that Q′ [∆−1 ] = Q[∆−1 ] and that x0 , x1 , · · · , x2t−1 are algebraically independent over Λ. (4) As was shown in (2) any column of Gnm is a linear combination of the first t columns. So the assertion is immediate. (5) Since rank (f0 , f1 , · · · , ft ) = t, ker[(f0 f1 · · · ft ) : Qt+1 → Qt ] is one dimensional over the quotient field of Q. Notice that both (∆0 ∆1 · · · ∆t ) and (y0 y1 · · · yt−1 1) are in this kernel passed to the quotient field. Hence the assertion follows. Proposition 4 Let Λ, Q and xi be as in Proposition 2. Let R = Λ[a0 , a1 , · · · , an ] be the polynomial ring over Λ, and let ϕ : R → Q be the homomorphism defined by ϕ(ai ) = xi . Then ker ϕ = It (Ht (a0 , a1 , · · · , an )). Proof. Put H = Ht (a0 , a1 , · · · , an ) and I = It (H). If n ≤ 2t − 1, then I = 0, and the assertion is trivial by Proposition 3 (3). Henceforth we assume n ≥ 2t. First we prove that ker ϕ = rad I. Since Gnt = Ht (x0 , x1 , · · · , xn ) has rank t (Proposition 2(4)), and since ker ϕ is prime, rad I ⊂ ker ϕ. Let P be an associated prime of I. We want to show that P = ker ϕ. Let K ¯ be the quotient field of R/P . Let “ ¯ ” denote reduction mod P , and let H 5
¯ has rank at most t, if e¯i is the i-th row of H ¯ , then be H mod P . Since H ∑t there are elements c¯i ∈ K such that i=0 c¯i e¯i = 0. We claim that c¯t ̸= 0. ¯ are linearly dependent, Assume c¯t = 0. This means that the first t rows of H ¯ which does not involve the last row is 0. Thus hence any t × t minors of H) P contains It−1 (Ht−1 (a0 , a1 , · · · , an−1 )), of which grade is n − 2t + 2, by Proposition 1. (Notice that the grade of this ideal is the same whether it is considered in Λ[a0 , a1 , · · · , an−1 ] or in R.) But again by Proposition 1, I is perfect of grade n+1−2t, and since a perfect ideal is grade unmixed, it follows that grade P = n + 1 − 2t, which is a contradiction. Thus ct ̸= 0 and we can ∑t−1 ∑ assume ct = −1. This means that e¯t = j=0 c¯j e¯j , hence a ¯t+i = t−1 ¯i+j , j=0 c¯j a for i = 0, 1, 2, · · · , n − t. Now we define a homomorphism from Q to K by sending xi → a¯i , and yi → c¯i , for i = 0, 1, · · · , t − 1. This map induces a surjection Qn := Λ[x0 , · · · , xn ] −→ R/P , because of the way xj ’s were defined. Thus we have proved that rad I = ker ϕ. Our next objective is to prove that ker ϕ is a unique associated prime of I. For this it suffices to prove that I has no embedded primes. Let Z be the prime ring of Λ (i.e., either the ring of integers or a finite prime field) and consider the diagram: u
R = Λ[a0 , · · · , an ]/It (H) −→ S := Λ0 [a0 , · · · , an ]/It (H) ↑ ↑v Z[a0 , · · · , an ]/It (H) −→ T := Z0 [a0 , · · · , an ]/It (H) where ( )0 denotes quotient field, and all the maps are natural maps. By theorem 2 of [9] applied to the map v, we see that S has no embedded primes because T does not by Proposition 1. Again by Proposition 1 R is flat over Λ, hence u is the localization by a set of non zero divisors. Thus u−1 (Ass(S)) = Ass(R), which shows that I in R has no embedded primes as claimed. Now we prove that I is a prime ideal. Let d = det Ht−1 (a0 , a1 , · · · , a2t−2 ), and ∆ = det Ht−1 (x0 , x1 , · · · , x2t−2 ). By Proposition 3(3) Qn [∆−1 ] = Q[∆−1 ], from which one sees easily that natural surjection R/I[d−1 ] −→ Qn [∆−1 ] can be inverted. Since ϕ(d) = ∆ ̸= 0, d is a non zero divisor of R/I. Thus it follows that I is a prime ideal. Proposition 5 Let R be any commutative ring with 1, and let a0 , a1 , · · · , an be arbitrary elements of R. Then It (Hm (a0 , a1 , · · · , an )) = It (Ht (a0 , a1 , · · · , an )) 6
for any integers t and m such that t ≤ m and t ≤ n − m. Proof. We may assume that R = Z[a0 , a1 , · · · , an ] and t > 0 . Put Jm = It (Hm (a0 , a1 , · · · , an )). Since the matrix Hm (x0 , x1 , · · · , xn ) has rank t by Proposition 3(4), Jm ⊂ ker ϕ = Jt , where ϕ is the map defined in Proposition 4. Since Jm and Jt are generated by homogeneous polynomials of degree t + 1, it suffices to prove that Jm has as many generators as Jt . we can ′ ′′ prove this by induction on n. Put Hm = Hm (a0 , a1 , · · · , an−1 ) and Hm = ′ Hm (a0 , a1 , · · · , an ). Then induction hypothesis implies It (Hm ) = It (Ht′ ) ′′ ′ ′′ and It (Hm ) = It (Ht′′ ). Consider Vm := Jm /It (Hm ) + It (Hm ). Vt is generated by forms which involve both a0 and an . More precisely a generator f ∈ Vt is of the form f = a0 an f ′ + (terms of degree at most 1 with respect to a0 and an ), where f ′ ∈ Ht−2 (a1 , · · · , an−1 ). (If t = 1, we understand f ′ = 1.) Corresponding to f we may find an element h ∈ Vm which has the term a0 an f ′ . Clearly these elements constitute a part of a minimal generating set of Vm . Thus we have completed the proof.
3
Normality
Proposition 6 Let R = Λ[a0 , a1 , . . . , an ] be the polynomial ring over a normal ring, and I = It (Ht (a0 , a1 , . . . , an )) the Hankel ideal. Then R/I is a normal ring. Proof. First we treat the case where Λ is a field. We may assume n ≥ 2t ≥ 0. Put A = R/I and H = Ht (a0 , a1 , . . . , an ). Since A is a Cohen-Macaulay ring, it suffices to prove that if P is a prime ideal of A of height one, then AP is regular. Let J = It−1 (Ht−1 (a0 , a1 , . . . , an−1 )). Then by Proposition 1 and Proposition 2, ht J/I = 1. By Proposition 4 it is a prime ideal. Suppose that P is a height one prime ideal of A different from J/I. Then there is a t × t minor D of H such that D ̸∈ P and D ∈ J. Write D = det (zij ) with zij ∈ a0 , a1 , . . . , an . Note that none of zij is an since D ∈ J. Now we let zij∗ = aℓ+1 if zij = aℓ . By Proposition 4 we have the isomorphism ϕ : R/I → Λ[x0 , x1 , . . . , xn ] = Qn ⊂ Λ[x0 , x1 , . . . , xn−1 , y0 , y1 , . . . , yn−1 ] Let Y be the matrix defined in Section 3. Then we have t Y (ϕ(zij )) = (ϕ(zij∗ )). (Note that (zij ) consists of some t columns and the first t consecutive rows 7
of H.) Since AP is a localization of Qn [D−1 ] = Q[D−1 ] it is regular. It remains to prove that if P = J/I, then AP is regular. Notice that we have a symmetry ai ↔ an−i in H. Thus the proof is complete in the case Λ is a field. For the general case we can employ the argument of [6] (proof of Corollary 3, specifically the last paragraph of p.1054). The argument is precisely the same and we leave the verification to the reader.
4
Binary forms and Hankel ideals
Let k be a field of characteristic 0. Let x1 , x2 , . . . , xn be indeterminates, and let Xi = ∂x∂ i . Further let O = k[X1 , X2 , . . . , Xn ] and R = k[x0 , x1 , . . . , xn ]. We regard O as a commutative polynomial ring in the variables X1 , X2 , . . . , Xn and R as an O-module. Besides R itself is a polynomial ring, of course. Let Rr and Or be the homogenous parts of degree r of R and O respectively. An element of Rr is called an n-ary r-form. It is convenient to write a form using divided powers. Namely, put x(p) = (1/p!)xp . The set (1)
(r ) (r )
{x1 1 x2 2 · · · xn(rn ) |
∑
ri = r}
is a basis of Rr . Thus F ∈ R is written as F =
∑
(r ) (r )
n) ar1 r2 ···rn x1 1 x2 2 · · · x(r n .
Note that (1) is the dual basis of (2)
{X1r1 X2r2 · · · Xnrn |
∑
ri = r}. Let G = GL(n, k) be the
general linear group acting on R as automorphisms of k-algebra induced by the linear transformation of the variables. It then induces, for each r, the representation ρr : G → GL(Rr ). On the other hand, since O1 is naturally the dual space of R1 , G acts on O1 as the contragradient of G, and it induces, from the ring structure of O, the representation ρ∗r : G → GL(Or ). It is easy to see that ρr and ρ∗r are contragradient to each other with the dual bases (1) and (2). ⊕ ∗ As an O-module R can be identified with Homgr OP , the k (O, k) = graded dual of O. Thus R is actually the injective envelope of k in the category of graded O-modules. (cf. Goto-Watanabe [4], (1.2.10), (1.2.11)) The theory of Matlis duality reveals that there is a bijective correspondence 8
between the following two objects: (I) The graded isomorphism types of homogeneous Artinian Gorenstein rings ⊕ of embedding dimension at most n and of rank r. (Here rank of A = Ai means the maximal r such that Ar ̸= 0. (II) The linear equivalence classes of n-ary r-forms. In fact let F ∈ Rr . Then OF = O/ann F is a Gorenstein ring described in (I). Moreover if F ′ = F g for some g ∈ GL(R1 ), then OF = O/ann F and OF ′ = O/ann F ′ are isomorphic under the contragradient action of g. Conversely if A is a Gorenstein ring with embedding dimension not greater than n, then there is a natural surjection O → A → 0, which yields 0 → HomO (A, R) → HomO (O, R) = R Since HomO (A, R) is generated by one element, say F the ring A determines a form F = F (A) of R, whose degree r is evidently the rank of A. One sees easily that if A and A′ are isomorphic by a grade preserving homomorphism then it lifts to g ∈ GL(O1 ), and F (A) is transformed to F (A′ ) by g ∗ ∈ GL(R1 ). In the above argument the condition ch k = 0 is not essential. In the general case we start with the polynomial ring O = k[X1 , X2 , · · · , Xn ] with the indeterminates X1 , X2 , · · · , Xn . O has the structure of coalgebra with ⊗ ⊗ ⊗ comultiplication O → O O defined by Xi → Xi 1 + 1 Xi . This induces the structure of a commutative algebra on R = Homgr k (O, k) called the divided power algebra on O1∗ . (For details we refer to BuchsbaumEisenbud [1] Section 1.) As in the case ch k = 0 Rr is spanned by the set (1), and R is the injective envelope of k. For a homogeneous Artinian ⊕ ring A = Ai over A0 = k, the sequence hi = dimk Ai , i = 0, 1, 2, ..., r, where r is the rank of A, is called the h-vector of A. It is well known that if A is Gorenstein then A has a symmetric h-vector. Now we consider the h-vectors of Gorenstein rings of embedding dimension 2 in connection with binary forms. Let O = k[X1 , X2 ], J ⊂ O a homogeneous Gorenstein ideal of height 2, and A = O/J. It is well known that J is generated by two elements. Suppose J = (f1 , f2 ), deg fi = di , i = 1, 2. It is easy to see that r := rankA = d1 + d2 − 2, and if d = Min {d1 , d2 }, then the h-vector of A is given by 9
i+1
hi =
i = 0, 1, · · · , d − 1, d i = d, d + 1, · · · , r − d, r − i + 1 i = r − d + 1, · · · , r.
. Define s(A) to be Max{dimk Ai }. Then s(A) = dimk A[ 1 r] . If r is fixed, 2 the maximal possible s(A) is dim O[ 1 r] = [ 12 r] + 1. If this is the case A is 2 called compressed by Iarrobino [7].) Let R = k[x1 , x2 ], O = k[X1 , X2 ], with ∑ (i) (r−i) Xi = ∂x∂ i , i = 1, 2. For F ∈ Rr , write F = ai x1 x2 , and consider the ring O/ann F . If Oi is the homogeneous part of O of degree i, then the h-vector of A is given by dim Oi F , i = 0, 1, ..., r. (Recall that a Gorenstein ring has a symmetric h-vector.) Notice that the condition dim Oi F = t is equivalent to rank Hi (a0 , a1 , · · · , an ) = t, where Hi are the Hankel matrices as in the previous sections. Now we regard the coefficients a0 , a2 , · · · , an as indeterminates, and consider the polynomial ring S = k[a0 , a1 , · · · , an ]. We let G = GL(2, k) act on S by linear transformation of the variables ∑ (i) (r−i) a0 , a1 , · · · , an in the way that the form F = ai x1 x2 is kept fixed. (So ∗ G −→ GL(S1 ) coincides with ρr : G → GL(Or ) described before.) Since the h-vectors of the rings of the form A = OF do not change under linear base change, it is clear that the Hankel ideals Ii (Ht (a0 , a1 , · · · , an )) are G-stable. If r is even and t = 21 r, then det Ht (a0 , a1 , · · · , an ) is a relative invariant classically known as the Hankel determinant of the binary r-forms. Spec(S) is the space of binary r-forms. Let Vt =Spec(S/It Ht (a0 , a1 , · · · , an )). Then V0 ⊂ V1 ⊂ V2 ⊂ · · · ⊂ V[ 1 r] 2
By Proposition 1 dimk Vt = 2t. Moreover Max{dim Oi F } = t if and only if F ∈ Vt − Vt−1 . In this case ann F in O is generated by two elements f1 , f2 such that degf1 +degf2 − 2 = r and Min{degf1 , deg f2 } = t.
10
Acknowledgements The results of this paper the author anounced at the fall annual meeting of the Japan Mathematical Society in 1985, and had prepared this paper mostly in the present form but did not publish it, as he came to believe that most of the results were implicitly contained in the works of Eagon and Hochster and seemed fairly well known. Now Professor Anthony Geramita and Professor Anthony Iarrobino recommended that it should be worth while publishing in the Queen’s Papers even in the orignal form as the proofs are direct and no handy references are available. The author would like to thank them very much. He also learned from them that the equality of the ideals (the last statement of Theorem 1) is proved in Lemma 2.3 in [10] by a different method. References [1 ] D.B.Buchsbaum and D.Eisenbud, Generic free resolution and a family of generically perfect ideals, Adv. in Math. 18 (1975) 245-301. [2 ] J.A.Eagon, Examples Cohen-Macaulay rings which are not Gorenstein, Math. Z. 109 (1969) 109-111. [3 ] J. A. Eagon and D. G. Northcott, Generically acyclic complexes and generically perfect ideals, Proc. Roy. Soc. A 299 (1967) 147-172. [4 ] S. Goto and K. Watanabe, On graded rings I, J.Math. Soc. Japan 30, no.2 (1978) 179-213. [5 ] M.Hochster, Generically perfect ideals are strongly genericly perfect, Proc. London Math. Soc. (3) 23 (1971) 477-488. [6 ] M.Hochster and J.A.Eagon, Cohen-Macaulay rings, invariant theory and the generic perfection of determinantal loci, Amer. J. Math. 93 (1971) 1020-1058. [7 ] A.Iarrobino, Compressed algebras: Artin algebras having given socle degrees and maximal length, Trans. Amer. Math. Soc. 285 no.1 (1984) 337-378.
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[8 ] R.E.Kutz, Cohen-Macaulay rings and ideal theory in rings of invairants of algebraic groups, Trans. Amer. Math. Soc. 194 (1974) 115-129. [9 ] R.Y.Sharp, The effect on associated prime ideals produced by an extension of the base fields, Math. Scand. 38 (1976) 43-52. [10 ] L. Gruson and C. Peskine, ”Courbes de l’espace projectif: vari`et`es de s`ecantes”, in Patrick Le Barz and Yves Hervier, editors, Enumerative Geometry and Classical Algebraic Geometry, Birkhauser, Boston (1982)1-32.
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