ON THE REGULARITY OF THE RADICAL OF IDEALS

Let I = 〈f1, . . . , fk〉 be an ideal generated by homogeneous forms fi of degree di for i = 1, . . . , k in the Z-graded ring K[x0, x1, . . . , xn] where Char(K) = 0. It is well-known that there is an integer e(I) called Noether exponent defined as e(I) = min{μ : √ I μ ⊂ I}. In this paper, we estimate the regularity reg (√ I e(I) ) in terms of reg (I) and e(I) in certain cases. Mathematics Subject Classification (2010): 13D02, 13D45, 14B15


Introduction
In Chapter 14 of [15] Mumford introduced the concept of regularity for a coherent sheaf F on projective space P n : F is p-regular if, for all i ≥ 1 we have vanishing for the twists H i (P n , F(k)) = 0, for all k + i = p.This in turn implies the stronger condition of vanishing for k + i ≥ p. Regularity was investigated later by several people, notably Bayer and Mumford [1], Bayer and Stillman [2], Eisenbud and Goto [9], and Ooishi [16].Let R = K[x 0 , ..., x n ] be the polynomial algebra in n + 1 variables over a field K, graded in the usual way.
If M is a finitely generated graded R-module, then the local cohomology groups H i m (M ) with respect to the ideal m = (x 0 , ..., x n ) are graded in a natural way and we say that M is p-regular if If F is the coherent sheaf on P n associated with M in the usual way, we have which shows the compatibility of these definitions.
An important result in this theory is the following: suppose K is a field and where deg(e α,i ) ≤ p + i for all i ≥ 0. Hence, one can obtain the regularity of an ideal from the degree twists of its minimal free graded resolution, and vice versa.
In this short note we raise the general question: Is there any way to compare the regularity of an ideal I with that of its radical √ I?
Here is an example: Let I be a monomial ideal in a polynomial ring R = Then the primary decomposition of I has the form I = ∩ s i=1 p di i , where each associated prime p i ∈ Ass(R/I) is generated by a subset of the the elements {x 1 , . . ., x n }.Derksen and Sidman [8] proved that reg (∩ d i=1 I i ) ≤ d for any set of ideals I 1 , • • • , I d generated by linear forms, which had been conjectured previously by Sturmfels.Therefore, reg ( √ I) = reg (∩ s i=1 p i ) ≤ s, where s is the number of associated primes of R/I.Moreover, if I is a square-free monomial ideal, then by [Remark 15, [11]] the primary decomposition of I has the shape I = ∩ s i=1 p di i with p i ⊂ p j .Thus, if I is a square-free monomial ideal, then reg ( √ I) = s where s is the number of associated primes of R/I.

Many studies have been done to understand the relationship between reg (I)
and reg ( √ I).The paper by Ravi [17] proved that reg ( non-singular curve in P 3 .But this is not true in general, for example, Chardin-D'Cruz [6] considered the family of complete intersection ideals Then for all m, n ≥ 1, Hence regularity of the radical may be much larger than the regularity of the ideal itself.To our knowledge, there is not yet a complete general answer to this question. It is well-known that there is an integer e(I) = min{µ : There are recent papers by Kollár [14], Jelonek [12], Sombra [18] and others on effective versions of the Nullstellensatz that give quite good bounds on e(I).An interesting question is to estimate the regularity of √ I e(I) in terms of reg (I) and e(I).
In this paper, our focus is to provide bound for the reg ( √ I e(I) ) in terms of reg (I) and e(I).Our approach is to study dim R/I, the Krull dimension of R/I.
In Section 2 we analyze the case when dim R/I ≤ 1.The case when dim R/I ≥ 2 is extremely complicated, hence we focus our attention to irreducible projective varieties of minimal degree in Section 3. Illustrative computational examples are provided.
2. The case for dim R/I ≤ 1 In this section, we study the case when the Krull dimension dim R/I ≤ 1.To do this, let us recall some useful definitions and results.
Recall basic definitions: An ideal I ⊂ R is called a primary ideal, if whenever An ideal I is a p-primary ideal, if √ I = p for a prime ideal p.Any ideal has an irredundant primary decomposition, where q i is a p i -primary ideal, and the p i 's are called associated primes of I.Moreover, if p i does not properly contain any other associated prime, then it is called a minimal associated primes of I.The non-minimal associated primes are called embedded associated primes.For example, I = x 2 , xy = x ∩ x 2 , y is a primary decomposition, where the minimal associated prime is x and the embedded associated prime is x, y .Definition 2.1.Let I ⊂ R = K[x 0 , . . ., x n ], be a graded ideal.Then the saturation of the ideal I is defined as The cohomology sequence will be: Therefore, we have the following: for all k ≥ p.Thus, I is p-regular.
Example 2.3.Let R = K[x, y, z, t], and I = x 2 , y, z, t .Then We also observe that ( √ I) Remark 2.4.I sat is the largest ideal that defines the same closed subscheme of P n as I does.Every closed subscheme of P n is defined by some homogeneous ideal, and there is a bijection between closed subschemes of P n and homogeneous saturated ideals.A radical ideal is saturated.Furthermore, given a homogeneous ideal I ⊂ R, the subscheme of P n defined by I is reduced if and only if I sat is a radical ideal.
Remark 2.5.Let I ⊂ R = K[x 0 , . . ., x n ] be a graded ideal such that the Krull dimension dim R/I = 1.Then I has an irredundant primary decomposition, I = (∩ s i=1 q i ) J, where the √ q i = p i correspond to the distinct points in P n , and √ J = m = x 0 , . . ., x n .
We claim that r ∈ ∩ s i=1 p i , otherwise, if r / ∈ p j for some j, then rx k i ∈ p j for i = 0, . . ., n implies that x k i ∈ p j for i = 0, . . ., n, hence m ⊂ p j , contradicting the condition that each p i for i = 1, . . ., s corresponds to point in P n , i.e., ht(p j ) = n−1.

Theorem 2 . 2 .
Let R = K[x 0 , . . ., x n ], and I be a homogeneous graded ideal.If the Krull dimension dim R/I = 0, then e(I) = reg I, and reg ( √ I) e(I) = reg I. Proof.First, we note that if Krull dimension dim R/I = 0, then V(I) = ∅ in P n (K) (K = algebraic closure of K), and √ I = m = x 0 , . . ., x n , i.e., I is a graded m-primary ideal.It is well-known that if dim R/I = 0, then I i = R i for all i ≥ p if and only if reg I = p.By definition, e(I) = min{µ ∈ Z | ( √ I) µ = m µ ⊂ I}.Since m µ is generated by R µ we see that e(I) = min{µ ∈ Z | R µ = I µ } = reg I.Moreover, ( √ I e(I) ) µ = (m e(I) ) µ = R µ for µ ≥ e(I) = reg I, but ( √ I e(I) ) µ = (m e(I) ) µ = 0 for µ < e(I) = reg I. Therefore, reg ( √ I) e(I) = reg I.This completes the proof of the theorem.For convenience we recall the proof of the implication: dim R/I = 0, then I i = R i for all i ≥ p if and only if reg I = p.First note that m k ⊆ I for some positive integer k, and for any r ∈ R we have rm µ ⊆ I for some µ.Hence I sat = R.If dim R/I = 0, then H i m (R/I) = 0 for i ≥ 1.Consider the following exact sequence: 0 − −−− → I − −−− → R − −−− → R/I − −−− → 0.

Lemma 2 . 6 .
Let R = K[x 0 , . . ., x n ], and I a homogeneous graded ideal.If the Krull dimension dim R/I = 1, then √ I = I sat .Proof.Since I sat is the largest ideal that defines the same closed subscheme of P n as I does, both I sat and √ I define the same subset of P n (K), we have √ I ⊂ I sat because √ I is the largest ideal defining V(I).On the other hand r ∈ I sat implies that

Theorem 3 . 1 .
Therefore, r ∈ ∩ s i=1 p i = √ I. Thus, √ I = I sat as claimed.Assume a variety V ⊂ P n is irreducible.It is well-known that deg(V ) ≥ n − dim V + 1, and when deg(V ) = n − dim +1, we obtain the irreducible varieties of minimal degrees.It is known that the irreducible varieties of minimal degree are the rational normal scrolls, the quadric hypersurfaces and the cone over the Veronese surface in P 5 (see [Theorem 2, [19]]).Let I be an ideal such that V(I) is an irreducible variety in P n of minimal degree, and spanning P n .Then reg ( √ I) k = 2k.Proof.It is proven in[3] that the ideal I(V ) of an irreducible variety V of minimal degree has linear powers.By the Nullstellensatz, we are assuming that√ I = I(V )for a minimal variety V .As mentioned above, this shows that reg ( √ I) k = dk for all k, where d is the common degree of the generators of √ I. From the classification of varieties of minimal degree, we have d = 2.This shows in particular that reg ( √ I) = 2, which also follows from the results in[10].