aa r X i v : . [ m a t h . A C ] A p r FUNDAMENTAL RESULTS ON s -CLOSURES WILLIAM D. TAYLOR
Abstract.
This paper establishes the fundamental properties of the s -closures, a recently introduced familyof closure operations on ideals of rings of positive characteristic. The behavior of the s -closure of homo-geneous ideals in graded rings is studied, and criteria are given for when the s -closure of an ideal can bedescribed exactly in terms of its tight closure and rational powers. Sufficient conditions are established forthe weak s -closure to equal to the s -closure. A generalization of the Brian¸con-Skoda theorem is given whichcompares any two different s -closures applied to powers of the same ideal. Introduction
In [Tay18], the author introduced a family of closure operations on the ideals of noetherian rings of positivecharacteristic which lie between and interpolate between integral closure and tight closure of those ideals.For a real number s ≥
1, the weak s -closure of an ideal I in a ring R is the set of x ∈ R such that thereexists c ∈ R , not in any minimal prime, such that cx q ∈ I ⌈ sq ⌉ + I [ q ] for all sufficiently high powers q of thecharacteristic of R . We denote the weak s -closure of I by I { s } . The s -closure of I is the ideal obtained byapplying the weak s -closure repeatedly until the ideal stabilizes.The s -closures are related to the s -multiplicity function, which similarly interpolates between the Hilbert-Samuel and Hilbert-Kunz multiplicities of an ideal. The s -multiplicity of an m -primary ideal I in a localring ( R, m ) is e s ( I ) = lim q →∞ λ (cid:0) R/ ( I ⌈ sq ⌉ + I [ q ] ) (cid:1) q d H s ( d ) , where H s ( d ) is a normalizing factor depending only on s and the Krull dimension d of R .The strongest result on the subject of s -closures in [Tay18] is Theorem 4.6, which states that if I and J are m -primary ideals of a ring R and I { s } = J { s } , then e s ( I ) = e s ( J ). The same theorem gives a partialconverse: if R is an F -finite complete domain, and I ⊆ J , and e s ( I ) = e s ( J ), then I { s } = J { s } . Furthermore,in this case the weak s -closure is the s -closure, i.e. I { s } = I cl s . In this paper, we show in Theorem 4.7 thatthe domain hypothesis for the converse direction may be weakened to an unmixed hypothesis.This paper’s purpose is to develop significantly more of the theory of s -closures, particularly to establishthe results that will be essential to further study. The three main goals of this paper are to understand thestructure of the s -closure in the graded case, identify situations in which I { s } = I cl s , and to compare the s -closures for different values of s using a generalization of the Brian¸con-Skoda theorem.Here we record those results in the paper we believe will be most relevant to future work. In some casesthe statement of the full theorem is slightly stronger but more technical. Lemma (4.9) . If R is a ring of characteristic p > , I ⊆ R is an ideal, and ≤ t < s , then (cid:0) I { t } (cid:1) { s } = I { t } . Theorem (4.6) . Let R be a ring of characteristic p > , I ⊆ R an ideal, and s ≥ . For any x ∈ R , x ∈ I { s } if and only if x ∈ ( IR/ p ) { s } for all p ∈ Min R . Theorem (2.9, 3.4) . Let R be a ring of characteristic p > , I ⊆ R an ideal, and s ≥ a rational number.We have that I ∗ + I s ⊆ I { s } , where I s is the s th rational power of I . Furthermore, equality holds if I is amonomial ideal in a polynomial or semigroup ring over a field. Theorem (3.1, 3.2, 3.3) . Let R be an N -graded ring of characteristic p > , I ⊆ R a homogeneous ideal, x ∈ R a homogeneous element, and s ≥ . (1) I { s } and I cl s are homogeneous ideals. (2) If all generators of I have degree at least d and x ∈ I { s } \ I ∗ , then deg x ≥ sd . (3) If ( R, m ) is graded local, I is m -primary and generated in degree at most d , and deg x ≥ sd , then x ∈ I { s } . Theorem (4.16) . For the following classes of ideals, I { s } = I cl s . (1) Monomial ideals in polynomial rings, or more generally affine semigroup rings, over a field (2)
Principal ideals (3)
Powers of R + , where R is an N -graded ring generated in degree 1 over R and R + is generated byall elements of positive degree. Theorem (5.1) . Let R be a ring, ≤ t < s , and I an ideal of R . If r ≥ ( µ ( I ) − s − t ) t ( s − , then for all n ∈ N , ( I n + r ) { t } ⊆ ( I n ) { s } . An outline of this paper is as follows. In section 2, we give the basic definitions and results on powers ofideals that we use throughout the paper. We also record some results about rational powers of ideals, andprove a characterization of them which is particularly relevant to us. In section 3, we consider the s -closureof homogeneous ideals in graded rings. We obtain degree conditions which can be used in some cases tocheck the membership or non-membership of a homogeneous element in the s -closure of a homogeneousideal. Section 4 considers the question of when I { s } = I cl s , and gives some sufficient conditions on I forequality to hold. Section 5 includes our generalization of the Brian¸con-Skoda theorem which compares anytwo s -closures. 2. Preliminaries
Throughout this paper, all rings R are assumed to be commutative and noetherian, and the notation R ◦ indicates the set of all elements of R not in any minimal prime ideal. For an ideal I , we use µ ( I ) for theminimal number of generators of I . When we work with a ring of positive characteristic p , the symbols q and q ′ stand for positive integer powers of p . For an ideal I in a ring of characteristic p >
0, the ideal I [ q ] = ( f q | f ∈ I ) is called the q th Frobenius power of I , and is generated as an ideal by the q th powers ofany set of generators of I .We are interested in the relationships between ordinary and Frobenius powers of ideals. In particular, werely on the following result. Lemma 2.1. If R is a ring of characteristic p > , h ≥ is a real number, I is an ideal of R , and q is apower of p , then I ⌈ h ⌉ ⊆ (cid:0) I [ q ] (cid:1) ⌊ h/q − µ ( I )+1 ⌋ ⊆ (cid:0) I [ q ] (cid:1) ⌈ h/q − µ ( I ) ⌉ Proof.
Let x , . . . , x µ ( I ) be a set of generators for I . For any generator x of I ⌈ h ⌉ , there exist a i , b i ∈ N suchthat b i < q , P a i q + b i = ⌈ h ⌉ , and x = µ ( I ) Y i =1 x qa i + b i i = µ ( I ) Y i =1 ( x qi ) a i · µ ( I ) Y i =1 x b i i ∈ (cid:16) I [ q ] (cid:17) P i a i . Furthermore, µ ( I ) X i =1 a i = µ ( I ) X i =1 qa i + b i − b i q = ⌈ h ⌉ q − X i b i q ≥ ⌈ h ⌉ q − µ ( I ) q − q > hq − µ ( I )Therefore, since P i a i is an integer, P i a i ≥ ⌊ h/q − µ ( I ) + 1 ⌋ .The last containment is implied by the fact that ⌈ α ⌉ ≤ ⌊ α + 1 ⌋ for all real α . (cid:3) Mixed Powers and s -Closure. Given a ring R and ideal I , the integral closure I of I is the set of all x ∈ R such that there exists c ∈ R ◦ such that cx n ∈ I n for infinitely many positive integers n , or equivalentlyall sufficiently large integers n [HS06, Corollary 6.8.12]. When R has characteristic p >
0, the tight closure I ∗ of I is the set of all x ∈ R such that there exists c ∈ R ◦ such that cx q ∈ I [ q ] for all q ≫
0. The similaritybetween these two descriptions suggests a method of interpolating between the two closures. We begin byconsidering a set of ideals which interpolate between ordinary powers and Frobenius powers of an ideal.
Definition 2.2.
Let R be a ring of characteristic p > s ≥ I an ideal of R . For any q , the ( s, q ) mixed power of I is I ( s,q ) = I ⌈ sq ⌉ + I [ q ] . UNDAMENTAL RESULTS ON s -CLOSURES 3 Note that I (1 ,q ) = I q , and that if s ≥ µ ( I ), then I ( s,q ) = I [ q ] . Furthermore, we have that if s > t , then I ( s,q ) ⊆ I ( t,q ) . Therefore, the ideals I ( s,q ) form a decreasing family of ideals parameterized by s . In [Tay18],the author used the mixed powers defined above to construct a family of closures which lie between integralclosure and tight closure. Definition 2.3. [Tay18, Definition 4.1] Let R be a ring of characteristic p > s ≥ I an ideal of R . The weak s -closure of I , denoted I { s } , is the set of all x ∈ R such that there exists c ∈ R such that for all q ≫ cx q ∈ I ( s,q ) .It is easy to see that I { s } is an ideal containing I , and that if I ⊆ J then I { s } ⊆ J { s } , but it is notclear that the weak s -closure is idempotent. Thus, to construct a true closure operation, we apply the weak s -closure repeatedly. Definition 2.4. [Tay18, Definition 4.3] Let R be a ring of characteristic p > s ≥
1, and I an ideal of R .The s -closure of I , denoted I cl s , is the ideal at which the following increasing chain of ideals stabilizes: I ⊆ I { s } ⊆ (cid:16) I { s } (cid:17) { s } ⊆ (cid:18)(cid:16) I { s } (cid:17) { s } (cid:19) { s } · · · . It is not known whether I { s } = I cl s for all s and ideals I . The condition that I { s } = I cl s is explored inSection 4.Since s > t implies I ( s,q ) ⊆ I ( t,q ) , we have that if s > t , then I { s } ⊆ I { t } and I cl s ⊆ I cl t . Moreover, since I [ q ] ⊆ I ( s,q ) ⊆ I q for all ideals I , s ≥ q , we have that I ∗ ⊆ I { s } ⊆ I cl s ⊆ I for all s and I .Furthermore, when s is very small or very large, some of the containments above become equalities. Theorem 2.5. If R is a ring of characteristic p > and I an ideal of R , then the following hold. (1) I { } = I cl = I . (2) If either s ≥ µ ( I ) or s > µ ( J ) , where J is a reduction of I , then I { s } = I ∗ . In particular, if R islocal with infinite residue field and s > dim R , then I { s } = I cl s = I ∗ .Proof. (1) If x ∈ I { } , then there exists c ∈ R ◦ such that cx q ∈ I (1 ,q ) = I q for all q ≫
0, and hence x ∈ I . If x ∈ I , then there exists c ∈ R ◦ such that cx n ∈ I n for all n ≫
0, and hence cx q ∈ I q = I (1 ,q ) for all q ≫ x ∈ I { } . Therefore I { } = I , and since this holds for all ideals I , we have that weak 1-closure andtight closure are the same operation, hence weak 1-closure is idempotent, i.e. I { } = I cl .(2) If s ≥ µ ( I ) and x ∈ I { s } , then there exists c ∈ R ◦ such that cx q ∈ I ( s,q ) = I [ q ] for all q ≫
0, andtherefore x ∈ I ∗ .Suppose s > µ ( J ), where J is a reduction of I with reduction number w and s > µ ( J ). Since I ∗ ⊆ I { s } we need only show that I { s } ⊆ I ∗ . Let x ∈ I { s } and let c ∈ R ◦ such that cx q ∈ I ( s,q ) for q ≫
0. Now, forlarge enough q , we have that wq ≤ s − µ ( J ), and so by Lemma 2.1, cx q ∈ I ( s,q ) = I ⌈ sq ⌉ + I [ q ] ⊆ J ⌈ sq − w ⌉ + I [ q ] ⊆ (cid:16) J [ q ] (cid:17) ⌊ s − w/q − µ ( J )+1 ⌋ + I [ q ] ⊆ J [ q ] + I [ q ] = I [ q ] . Therefore x ∈ I ∗ .If R is local with infinite residue field, then every ideal has a reduction generated by at most dim R elements. Hence in this case, if s > dim R , then weak s -closure and tight closure are the same operation,and so in particular weak s -closure is idempotent. Hence for any ideal I , I cl s = I { s } = I ∗ . (cid:3) Rational Powers.
The notion of rational powers of ideals is related to that of s -closure. In particular,rational powers will be used to describe the s -closures of certain kinds of ideals in graded rings in Theorem 3.4.The presentation here is based on [HS06, Section 10.5]. Definition 2.6.
Let R be a ring, I ⊆ R an ideal, and α ∈ Q ≥ . The α th rational power of I , denoted I α ,is the set of all x ∈ R such that x b ∈ I a , where α = ab , a, b ∈ N .The ideal I α does not depend on the choice of representation of α as a fraction. The most importantproperty of rational powers that we will use is the following. Theorem 2.7. ( [HS06, Propositon 10.5.5] ) Let R be a ring and I ⊆ R an ideal of positive height. Thereexists e ∈ N such that for all α ∈ Q ≥ , I α = I ⌈ αe ⌉ /e . WILLIAM D. TAYLOR
We can use Theorem 2.7 to give an alternate description of I α which simultaneously relates to the s -closureand doesn’t depend on the representation of α . Lemma 2.8.
Let R be a ring of any characteristic, I an ideal of positive height, and α ∈ Q ≥ . (1) If x ∈ I α , then there exists c ∈ R ◦ such that for all n ≫ N , cx n ∈ I ⌈ αn ⌉ . (2) If there exists c ∈ R ◦ and m ∈ N > such that for infinitely many n , cx m n ∈ I ⌈ αm n ⌉ , then x ∈ I α .Proof. (1) Suppose α = ab with a, b ∈ N and x ∈ I α , so that x b ∈ I a . Therefore there exists c ′ ∈ R ◦ suchthat for all k ≫ c ′ x bk ∈ I ak . Let c ′′ ∈ I a ∩ R ◦ and n ≫
0. We have that c ′ c ′′ x b ⌊ n/b ⌋ ∈ c ′′ I a ⌊ n/b ⌋ ⊆ I a ⌊ n/b ⌋ + a ⊆ I ⌈ na/b ⌉ = I ⌈ nα ⌉ So, setting c = c ′ c ′′ and noting that x n ∈ ( x b ⌊ n/b ⌋ ), we’re done.(2) Suppose that there exists c ∈ R ◦ and m ∈ N > such that for infinitely many n , cx m n ∈ I ⌈ αm n ⌉ . Let e ∈ N such that for any β ∈ Q ≥ , I β = I ⌈ βe ⌉ /e . Choose a, k ∈ N such that am k < α and (cid:6) am k e (cid:7) = ⌈ αe ⌉ . Now,for infinitely many n ≥ k , we have that c (cid:16) x m k (cid:17) m n − k = cx m n ∈ I ⌈ αm n ⌉ ⊆ I ⌈ ( a/m k ) m n ⌉ = I am n − k . Therefore, x m k ∈ I a , and so x ∈ I a/m k = I α . (cid:3) This description of the rational powers gives us another bound for the s -closure of an ideal. Theorem 2.9.
Let R be a ring of characteristic p > and I ⊆ R an ideal of positive height. If s ≥ is arational number then I ∗ + I s ⊆ I { s } .Proof. That I ∗ ⊆ I { s } is already known. Suppose x ∈ I s . By Lemma 2.8, there exists c ∈ R ◦ such that forall q , cx q ∈ I ⌈ sq ⌉ ⊆ I ( s,q ) . Therefore x ∈ I { s } . (cid:3) Graded Rings
Here we establish the essential facts about the s -closure of homogeneous ideals in graded rings. Throughoutthis section, for a semigroup G , a G -graded ring R = L g ∈ G R g , and x ∈ R , we write x g for the homogeneouscomponent of x lying in R g . If I ⊆ R is an ideal, we write [ I ] g for I ∩ R g . If g = ( g , g , . . . , g n ) ∈ Z n , thenwe write k g k ∞ = max {| g i | | i = 1 , . . . , n } . We begin with an expected result. Theorem 3.1. If R is a Z n -graded ring of characteristic p > , I is a homogeneous ideal of R , and s ≥ ,then I { s } and I cl s are homogeneous ideals. Furthermore, if R is a domain and x ∈ I { s } , there exists ahomogeneous element c such that cx q ∈ I ( s,q ) for all q ≫ .Proof. Let x = P j ∈ Z n x j ∈ I { s } and c = P i ∈ Z n c i ∈ R ◦ such that cx q ∈ I ( s,q ) for all q ≫
0. Let i ∗ = max {k i − i ′ k ∞ | c i , c i ′ = 0 } . If c i x qj = 0 and c i ′ x qj ′ = 0 have the same degree, then i + qj = i ′ + qj ′ ,and so i − i ′ = q ( j ′ − j ). If in addition q > i ∗ , we must have that j = j ′ and i = i ′ . Therefore each nonzerohomogeneous component of cx q is c i x qj for some i, j . Since I is homogeneous, so is I ( s,q ) , and therefore for q ≫
0, we have that c i x qj ∈ I ( s,q ) . Hence, for each j , cx qj ∈ I ( s,q ) , which shows that each x j ∈ I { s } . If R is adomain, then for each nonzero c i , c i x q ∈ I ( s,q ) for q ≫
0, which shows the last statement.Since I { s } is homogeneous, so is (cid:0) I { s } (cid:1) { s } , and each time we take the weak s -closure we preserve homo-geneity. After a finite number of steps we will reach I cl s , which shows that I cl s is homogeneous. (cid:3) Our primary goal will be to establish necessary and sufficient degree conditions for a homogeneous ringelement to belong to the s -closure of an ideal based on the degrees of its generators. Theorem 3.2.
Let I be a homogeneous ideal in an N n -graded ring R . If x ∈ I { s } \ I ∗ is a homogeneouselement, then deg x ≥ sδ , where δ = ( δ , . . . , δ n ) ∈ N n and δ i = min { d i | deg f = ( d , . . . , d n ) , = f ∈ I } .Proof. Let x ∈ I { s } , deg x = ( m , . . . , m n ), and assume that deg x sδ . Let c ∈ R ◦ be homogeneous suchthat for all q ≫ cx q ∈ I ( s,q ) . For any such q , there exist homogeneous y q ∈ I ⌈ sq ⌉ and z q ∈ I [ q ] , each ofthe same degree as cx q , such that cx q = y q + z q . Since y q ∈ I ⌈ sq ⌉ , if y q = 0 then deg y q ≥ δ ⌈ sq ⌉ , and sincedeg x sδ , for large enough q we have that deg( cx q ) = q · deg x + deg c δ ⌈ sq ⌉ . Therefore y q = 0 and cx q ∈ I [ q ] for all q ≫
0, and hence x ∈ I ∗ . (cid:3) UNDAMENTAL RESULTS ON s -CLOSURES 5 When the ideal we consider is primary to the homogeneous maximal ideal, we may conclude that allelements above a certain degree are in I { s } . Theorem 3.3.
Let k be a field of characteristic p > , ( R, m ) an N n -graded local finitely generated k -algebra, I an m -primary homogeneous ideal generated in degree at most δ , and s ≥ . If x ∈ R is a homogeneouselement, deg x = (0 , . . . , , and deg x ≥ sδ , then x ∈ I { s } .Proof. If ht I = 0, then since I is m -primary, R is a dimension 0 local ring. In this case, all elements of m are nilpotent, and so since deg x = (0 , . . . , x is nilpotent and hence in I { s } . Therefore, we may assumethat ht I > R is N -graded by “flattening” the grading on R . Precisely, for m ∈ N , let R m = L | g | = m R g , where the sum is taken over all degrees in the original grading whose sum of coordinatesis equal to m . Under this new grading, we still have that deg x ≥ sδ , so we may assume R is N -graded.Suppose that δ >
0. Let ∆ = deg x ≥ sδ , and let f , . . . , f m be a set of homogeneous generators of I withdeg f i ≤ δ . Since I is m -primary, we have that k [ f , . . . , f m ] ⊆ R is integral, and so there exists an equationof integral dependence for x δ over k [ f , . . . , f m ]:(1) (cid:0) x δ (cid:1) N + a (cid:0) x δ (cid:1) N − + · · · + a N − x δ + a N = 0 . we may choose the a i homogeneous, and so deg a i = ∆ δi for each i . Since each a i is a polynomial in the f i ,we have that a i ∈ I ∆ i for all i . Therefore x δ ∈ I ∆ , and so x ∈ I ∆ /δ . By Theorem 2.9, x ∈ I { ∆ /δ } ⊆ I { s } .Now suppose that δ = 0, so that I is generated by its degree 0 piece I ∩ R . In this case, for all n ∈ N wehave that I ∩ R n = ( I ∩ R ) R n . Since I is m -primary, ∞ > λ R ( R/I ) = X n ∈ N λ R (cid:18) R n I ∩ R n (cid:19) = X n ∈ N λ R (cid:18) R n ( I ∩ R ) R n (cid:19) Therefore, there exists N ∈ N such that if n ≥ N , ( I ∩ R ) R n = R n , and by Nakayama’s Lemma, R n = 0.Hence any homogeneous x ∈ R with deg x ≥ x ∈ I { s } . (cid:3) For certain kinds of ideals in graded rings we can describe the s -closure completely in terms of rationalpowers. Theorem 3.4.
Let R be a ring of characteristic p > which is G -graded for some semigroup G , I ⊆ R ahomogeneous ideal of positive height, and s ≥ rational. If, for every q ≫ and g ∈ G , either [ I ⌈ sq ⌉ ] g ⊆ [ I [ q ] ] g or [ I [ q ] ] g ⊆ [ I ⌈ sq ⌉ ] g , then I { s } = I ∗ + I s .Proof. By Theorem 2.9, I ∗ + I s ⊆ I { s } . Let x ∈ I { s } be homogeneous, and let c ∈ R ◦ be homogeneous suchthat for all q ≫ cx q ∈ I ( s,q ) = I ⌈ sq ⌉ + I [ q ] . For all sufficiently large q , since cx q is a homogeneous element,we have that cx q ∈ I ⌈ sq ⌉ or cx q ∈ I [ q ] . If x / ∈ I ∗ , then for infinitely many q , cx q / ∈ I [ q ] . Hence for infinitelymany q , cx q ∈ I ⌈ sq ⌉ . By Lemma 2.8, x ∈ I s . Hence x is in either I ∗ or I s , so x ∈ I ∗ + I s . (cid:3) Situations where we might apply Theorem 3.4 include homogeneous ideals in rings with monomial-likegradings, in which each graded piece is a 1-dimensional vector space over k . Examples of these includemonomial ideals in polynomial rings and toric rings.4. When is The Weak s -closure a Closure? In this section we consider a collection of conditions an ideal I may have relating to its various s -closures.In particular, we are concerned with when the s -closure is an honest closure itself, a property we refer toas ( ID s ) below. Before defining the properties, we look at an example to show that, a priori , there may beinfinitely many distinct s -closures for a given ideal. More precisely, this example shows that it is possible forthe quotient of two s -closures to have infinite length. The example is based on [EY17, Example 2.2] Example 4.1.
Let R be a domain of characteristic p > I ⊆ R an ideal, and s > t ≥ I { s } = I { t } . Let S = R [ X ], J = IS , and z ∈ I { t } \ I { s } . We have that J { t } J { s } ⊇ J { s } + SzJ { s } ∼ = SzJ { s } ∩ Sz ∼ = S (cid:0) J { s } : S z (cid:1) WILLIAM D. TAYLOR
We claim that zX n / ∈ J { s } for any n ∈ N . If there were such an n , then since R [ X ] is naturally N -graded,there would be an element rX m for some 0 = r ∈ R and m ∈ N such that, for all sufficiently large powers q of p , rz q X m + nq = rX m · ( zX n ) q ∈ J ( s,q ) = I ( s,q ) S. Therefore, for all large q , rz q ∈ I ( s,q ) , and so z ∈ I { s } , a contradiction. Hence X n / ∈ (cid:0) J { s } : S z (cid:1) for any n ,and so S ⊇ (cid:16) J { s } : S z (cid:17) + ( X ) ⊇ (cid:16) J { s } : S z (cid:17) + ( X ) ⊇ (cid:16) J { s } : S z (cid:17) + ( X ) ⊇ · · · is an infinite chain of descending ideals each of which contain (cid:0) J { s } : S z (cid:1) . Therefore λ (cid:0) S/ (cid:0) J { s } : S z (cid:1)(cid:1) = ∞ ,and so λ (cid:0) J { t } /J { s } (cid:1) = ∞ . Property ( ID s ) : When Weak s -Closure is Equal to s -Closure. As given in Definition 2.3, the weak s -closure is not obviously a closure. We now consider classes of ideals for which the two notions align. Definition 4.2.
Let R be a ring of characteristic p > s ≥ I of R has property ( ID s ) if the weak s -closure is idempotent on I , i.e. I { s } = I cl s . We say the ring R has property( ID s ) if every ideal of R has property ( ID s ).Since weak 1-closure is integral closure, any ring R of positive characteristic has property ( ID ). If,further, ( R, m ) is local with infinite residue field, then R has property ( ID s ) for any s > dim R , since in thiscase weak s -closure is tight closure. Property ( SM s ) : When s -Closure is Characterized by s -Multiplicity. In this section we restrictour attention to ideals of finite colength in local or graded local rings, homogeneous in the graded localcase. Membership in such ideals’ tight or integral closure can be tested using the Hilbert-Kunz or Hilbert-Samuel multiplicity, under certain conditions. The analogous multiplicity function for s -closure is called s -multiplicity. Definition 4.3. [Tay18, Definition 1.3] Let ( R, m ) be a local (resp. graded local) ring of characteristic p > d , I ⊆ R an m -primary (homogeneous) ideal, and s ≥
1. The s -multiplicity of I is e s ( I ) = lim q →∞ λ (cid:0) R/I ( s,q ) (cid:1) q d H s ( d ) , where H s ( d ) = vol { x ∈ [0 , d | | x | ≤ s } .Originally, the s -multiplicity was defined only for local rings, but the graded local case is completelyanalogous.By [Tay18, Theorem 4.6], if x ∈ I { s } , then e s ( I + ( x )) = e s ( I ). When the converse also holds, the s -multiplicity becomes a powerful tool for studying the s -closure. Definition 4.4.
Let ( R, m ) be a (graded) local ring of characteristic p > s ≥ m -primary (homogeneous) ideal I of R has property ( SM s ) if one can test membership in the weak s -closure of I using s -multiplicity, i.e. if e s ( I + ( x )) = e s ( I ) then x ∈ I { s } . We say the ring R has property( SM s ) if every m -primary (homogeneous) ideal of R has property ( SM s ).Property ( SM s ) is stronger than property ( ID s ). Theorem 4.5.
Let ( R, m ) be a (graded) local ring of characteristic p > and I a (homogeneous) m -primaryideal of R . If I has property ( SM s ) , then I has property ( ID s ) .Proof. By [Tay18, Theorem 4.6], if x ∈ I cl s , then e s ( I + ( x )) = e s ( I ), and since I has property ( SM s ), wehave that x ∈ I { s } . Therefore I { s } = I cl s . (cid:3) The following theorem shows that membership in the weak s -closure can be tested modulo minimal primes,which we will use to show that a large class of ideals has ( SM s ). The proof of this result is based closely onthe proof of part (e) of [BH93, Proposition 10.1.2]. In the following proof, the notation x always indicatesthe image of x in the currently considered quotient ring Theorem 4.6.
Let R be a ring of characteristic p > , I ⊆ R an ideal, and s ≥ . For any x ∈ R , x ∈ I { s } if and only if x ∈ ( IR/ p ) { s } for all p ∈ Min R . UNDAMENTAL RESULTS ON s -CLOSURES 7 Proof. If x ∈ I { s } , then there exists c ∈ R ◦ such that for all q ≫ cx q ∈ I ( s,q ) , and therefore for anyminimal prime p , c · x q = cx q ∈ I ( s,q ) R/ p = ( IR/ p ) ( s,q ) . Hence x ∈ ( IR/ p ) { s } .Let p , p . . . , p m be the minimal primes of R , suppose that for every i , x ∈ ( IR/ p i ) { s } , and choose c i / ∈ p i such that c i x q ∈ ( IR/ p i ) ( s,q ) for all q ≫
0. Therefore c i x q ∈ I ( s,q ) + p i for all i and all q ≫
0. We mayassume that c i ∈ R ◦ ; if not, by prime avoidance we may choose c ′ i such that c ′ i ∈ p j if and only if c i / ∈ p j .Therefore c i + c ′ i ∈ R ◦ , and furthermore, since c ′ i ∈ p i , we have that ( c i + c ′ i ) x q ∈ I ( s,q ) + p i for all q ≫ c i with c i + c ′ i .For each i , let d i ∈ (cid:16)Q j = i p j (cid:17) \ p i , and let d = P i c i d i . We have that d ∈ R ◦ . Now p p · · · p m ⊆ √
0, solet q ′ be large enough that ( p p · · · p m ) [ q ′ ] = 0. For all q ≫
0, and for any i , we have that( c i d i ) q ′ x qq ′ = ( c i x q ) q ′ d q ′ i ∈ (cid:16) I ( s,q ) + p i (cid:17) [ q ′ ] Y j = i p [ q ′ ] i = (cid:18)(cid:16) I ( s,q ) (cid:17) [ q ′ ] + p [ q ′ ] i (cid:19) Y j = i p [ q ′ ] i ⊆ I ( s,qq ′ ) . Therefore, d q ′ x qq ′ ∈ I ( s,qq ′ ) for all q ≫
0, and so x ∈ I { s } . (cid:3) Theorem 4.6 allows us to generalize [Tay18, Theorem 4.6] to the non-domain case.
Theorem 4.7.
Let ( R, m ) be a (graded) local ring of characteristic p > and I a (homogeneous) m -primaryideal. If R is F -finite, complete, and unmixed, then I has property ( SM s ) .Proof. Suppose that x ∈ R such that e s ( I + ( x )) = e s ( I ). By the Associativity Formula for s -multiplicity,[Tay18, Corollary 3.10], we have that X p ∈ Assh R e R/ p s (( I + ( x )) R/ p ) λ R p ( R p ) = e s ( I + ( x )) = e s ( I ) = X p ∈ Assh R e R/ p s ( IR/ p ) λ R p ( R p ) . For each p ∈ Assh R , e R/ p s (( I + ( x )) R/ p ) ≤ e R/ p s ( IR/ p ), and so we have equality for all such p . Since R/ p isan F -finite complete domain for all p ∈ Assh R , x ∈ ( IR/ p ) { s } by [Tay18, Theorem 4.6]. Since R is unmixed,Assh R = Min R , and so by Theorem 4.6, we have that x ∈ I { s } . Therefore, I has property ( SM s ). (cid:3) Property ( LC s ) : When Weak s -Closure is Left-Continuous. Next we consider the condition that an s -closure is the intersection of all larger s -closures. This property is enjoyed by rational powers, which arerelated to s -closures. Definition 4.8.
Let R be a ring of characteristic p > s >
1. We say an ideal I of R has property( LC s ) if the weak s -closure is left-continuous on R , i.e. I { s } = T t , I ⊆ R is an ideal, and ≤ t < s , then (cid:0) I { t } (cid:1) { s } = I { t } .Proof. Let J = I { t } . We have that J ⊆ J { s } , since this holds for all ideals. Now let x ∈ J { s } , let c ∈ R ◦ such that cx q ∈ J ( s,q ) for all q ≫
0, and let d ∈ R ◦ such that dJ [ q ] ⊆ I ( t,q ) for all q ≫
0. Finally, let q ′ besuch that q ′ ( s − t ) ≥ µ ( J ). For q ≫
0, we have that cd ⌈ sq ′ − µ ( J ) ⌉ x qq ′ ∈ d ⌈ sq ′ − µ ( J ) ⌉ J ( s,qq ′ ) = d ⌈ sq ′ − µ ( J ) ⌉ J ⌈ sqq ′ ⌉ + d ⌈ sq ′ − µ ( J ) ⌉ J [ qq ′ ] Now ⌈ sq ′ − µ ( J ) ⌉ ≥ q ′ t ≥
1, so d ⌈ sq ′ − µ ( J ) ⌉ J [ qq ′ ] ⊆ I [ qq ′ ] ⊆ I ( t,qq ′ ) . Also, for q ≫ d ⌈ sq ′ − µ ( J ) ⌉ J ⌈ sqq ′ ⌉ ⊆ d ⌈ sq ′ − µ ( J ) ⌉ (cid:16) J [ q ] (cid:17) ⌈ sq ′ − µ ( J ) ⌉ = (cid:16) dJ [ q ] (cid:17) ⌈ sq ′ − µ ( J ) ⌉ ⊆ (cid:16) I ( t,q ) (cid:17) ⌈ sq ′ − µ ( J ) ⌉ ⊆ ( I q ) ⌈ sq ′ − µ ( J ) ⌉ . Now q ⌈ sq ′ − µ ( I ) ⌉ ≥ q ( sq ′ − µ ( I )) ≥ q ( tq ′ ), and so d ⌈ sq ′ − µ ( J ) ⌉ J ⌈ sqq ′ ⌉ ⊆ ( I q ) ⌈ sq ′ − µ ( J ) ⌉ ⊆ I ⌈ tqq ′ ⌉ ⊆ I ( t,qq ′ ) . Hence, we have that for all q ≫ cd ⌈ sq ′ − µ ( J ) ⌉ x qq ′ ∈ I ( t,qq ′ ) , so x ∈ I { t } = J . Thus J { s } ⊆ J . (cid:3) Theorem 4.10. If R is a ring of characteristic p > , s > , and I an ideal of R , then I cl s ⊆ T t
Proof.
Let J ⊆ T t
Let R be a ring of characteristic p > and s ≥ . There exists ǫ > such that I { t } = I { s + ǫ } for any t ∈ ( s, s + ǫ ] .Proof. Consider the chain of ideals I { s +1 } ⊆ I { s +1 / } ⊆ I { s +1 / } ⊆ · · · . Since R is noetherian, there exists m ∈ N such that for all n ≥ m , I { s +1 /n } = I { s +1 /m } . Hence, for any t ∈ ( s, s +1 /m ], there exists some n such that s +1 /n < t , and so I { s +1 /m } ⊆ I { t } ⊆ I { s +1 /n } = I { s +1 /m } . (cid:3) Theorem 4.11 inspires a definition of jumping numbers for s -closure similar to that for test ideals. Definition 4.12.
Let R be a ring of characteristic p > I an ideal of R . We say that a real number s ≥ s -jumping number for I if, for all t > s , I { s } = I { t } .Theorem 4.11 implies that jumping numbers cannot accumulate above a given s ≥
1. That is, for any s there is an ǫ > s -jumping numbers in ( s, s + ǫ ). However, we do not have atheorem disproving the existence of such accumulations below s . Therefore, we define a property based onthat condition, which we show holds for some well-understood classes of ideals. Definition 4.13.
Let R be a ring of characteristic p > s > I of R has property ( LS s ) if the weak s -closure of I is left-stable, i.e. there exists ǫ > I { t } = I { s } forall s − ǫ < t < s . We say the ring R has property ( LS s ) if every ideal of R has property ( LS s ).Left-stability is a very strong condition, implying left-continuity and therefore idempotence. Theorem 4.14. If R is a ring of characteristic p > , s > , I is an ideal of R , and I has ( LS s ) , then I has ( LC s ) .Proof. Since I has ( LS s ), there exists u < s such that I { u } = I { s } . This implies that T t
Theorem 4.15. If R is a ring of characteristic p > and I is an ideal of R such that I { s } = I ∗ + I s for all s ∈ Q , then I has ( LS s ) for all s > .Proof. By Theorem 2.7, there exists e ∈ N such that I α = I ⌈ αe ⌉ /e for all α ∈ Q > . Now let t ≥ ⌈ se ⌉− e < t < s . Finally, let α, β ∈ Q such that ⌈ se ⌉ − e < α ≤ t < s ≤ β ≤ ⌈ se ⌉ e . We have that I { t } ⊆ I { α } = I ∗ + I α = I ∗ + I β = I { β } ⊆ I { s } . Therefore I { t } = I { s } . Hence I has ( LS s ). (cid:3) Corollary 4.16.
The following classes of ideals have property ( LS s ) for all s > . All rings mentioned havecharacteristic p > . (1) Monomial ideals in polynomial rings over a field. (2)
Monomial ideals in affine semigroup rings over a field.
UNDAMENTAL RESULTS ON s -CLOSURES 9 (3) Homogeneous ideals of positive height in graded rings in which each graded piece has length 1 overthe zeroth piece. (4)
Powers of R + , where R is an N -graded ring, generated in degree over R , and R + is the idealgenerated by all homogeneous elements of degree 1. (5) Principal ideals.Proof.
Items (1) and (2) follow from part (3) when we take the monomial N n -grading. Thus, let R = L g ∈ G R g be a graded ring such that for all g ∈ G , R g has length 1 over R . Let I ⊆ R be a homogeneousideal, and fix g ∈ G . For any q , [ I [ q ] ] g is an R -submodule of R g , and therefore [ I [ q ] ] g = R g or [ I [ q ] ] g = 0.For any rational s >
1, in the first case we have that [ I ⌈ sq ⌉ ] g ⊆ [ I [ q ] ] g and in the second we have that[ I [ q ] ] g ⊆ [ I ⌈ sq ⌉ ] g . Thus, by Theorem 3.4, I { s } = I ∗ + I s . Hence by Theorem 4.15, I has property ( LS s ) forall s . This proves (3).For item (4), let R be an N -graded ring generated in degree 1 as an R -algebra. Let x , . . . , x t be a setof degree 1 generators for R as an R -algebra, and let R + = ( x , . . . , x t ). Finally, let n ∈ N and I = ( R + ) n .Fix g ∈ N , q >
0, and s ∈ Q > . If g ≥ n ⌈ sq ⌉ , then any x ∈ R g can be written as x = rx a · · · x a t t , where r ∈ R and P i a i ≥ n ⌈ sq ⌉ . Therefore, x ∈ ( R + ) n ⌈ sq ⌉ = I ⌈ sq ⌉ . Hence, if g ≥ n ⌈ sq ⌉ , [ I ⌈ sq ⌉ ] g = R g . On theother hand, any homogeneous element of I ⌈ sq ⌉ must have degree at least n ⌈ sq ⌉ since I is generated in degree n . Therefore, if g < n ⌈ sq ⌉ , then [ I ⌈ sq ⌉ ] g = 0. Hence for any g , [ I ⌈ sq ⌉ ] g ⊆ [ I [ q ] ] g or [ I [ q ] ] g ⊆ [ I ⌈ sq ⌉ ] g , and byTheorem 3.4, I { s } = I ∗ + I s . Hence by Theorem 4.15, I has property ( LS s ) for all s .Item (5) follows from the fact that for a principal ideal I , I ∗ = I , and so for all s , I = I ∗ ⊆ I { s } ⊆ I ,hence I { s } = I . (cid:3) Relationships Between the Properties.
The various implications between the properties that we havedefined can be summarized in the following figure.Left-Stable Left-Continuous Idempotent( R, m ) F -finite, unmixed,complete, I m -primary Testable using s -MultiplicityTheorem 4.14 Theorem 4.10Theorem 4.7 ( R, m ) local, I m -primaryTheorem 4.55. A Brianc¸on-Skoda Theorem for s -Closure The classical Brian¸con-Skoda Theorem describes containments between the integral closures of powers ofan ideal and the powers themselves. In particular, when I is an ideal in a regular ring, we have that for all n ∈ N , I n + µ ( I ) − ⊆ I n . The statement is generalized by Hochster and Huneke in [HH90, Theorem 5.4], whoshow that even in singular rings we have I n + µ ( I ) − ⊆ ( I n ) ∗ for all n ∈ N . This, combined with their proofthat in regular rings all ideals are tightly closed, gives a new proof of the Brian¸con-Skoda Theorem. In thissection we develop a generalization of the Brian¸con-Skoda Theorem in positive characteristic. Theorem 5.1.
Let R be a ring of characteristic p > , ≤ t < s , and I an ideal of R . If r ≥ ( µ ( I ) − s − t ) t ( s − ,then for all n ∈ N , ( I n + r ) { t } ⊆ ( I n ) { s } .Proof. We consider two cases. First, suppose that n < µ ( I ) − s − . This implies that r ≥ ( µ ( I ) − s − t ) t ( s − > n ( s − t ) t .If q is large enough that n ( s − t ) t + ntq ≤ r , then( n + r ) ⌈ tq ⌉ ≥ ntq + rtq ≥ ntq + n ( s − t ) q + n = nsq + n ≥ n ⌈ sq ⌉ . Therefore, for all q ≫ (cid:0) I n + r (cid:1) ( t,q ) = I ( n + r ) ⌈ tq ⌉ + (cid:0) I n + r (cid:1) [ q ] ⊆ I n ⌈ sq ⌉ + ( I n ) [ q ] = ( I n ) ( s,q ) . Therefore ( I n + r ) { t } ⊆ ( I n ) { s } .Now suppose that n ≥ µ ( I ) − s − . In this case we have that( n + r ) t = n + n ( t −
1) + rt ≥ n + ( µ ( I ) − t − s − µ ( I ) − s − t ) s − n + µ ( I ) − q , (cid:0) I n + r (cid:1) ( t,q ) = I ( n + r ) ⌈ tq ⌉ + (cid:0) I n + r (cid:1) [ q ] ⊆ I ( n + µ ( I ) − q + ( I n ) [ q ] ⊆ ( I n ) [ q ] . Therefore, ( I n + r ) { t } ⊆ ( I n ) ∗ ⊆ ( I n ) { s } . (cid:3) Theorem 5.1 recovers the classical Brian¸con-Skoda Theorem by taking t = 1 and r = µ ( I ) −
1. Inparticular, we note that Theroem 5.1 does not give us a stronger version of the theorem in the case that oneof our closures is integral closure.We record two immediate corollaries, one of the statement of Theorem 5.1 and one of its proof.
Corollary 5.2.
Suppose ( R, m ) is a local ring with dimension d , characteristic p > , and infinite residuefield, let I ⊆ R be an ideal with reduction number w , and let ≤ t < s . If r ≥ ( d − s − t ) t ( s − , then for all n ∈ N , ( I n + r + w ) { t } ⊆ ( I n ) { s } .Proof. Since R has infinite residue field, I has a minimal reduction J with reduction number w and generatedby d elements. Therefore, (cid:0) I n + r + w (cid:1) { s } ⊆ (cid:0) J n + r (cid:1) { t } ⊆ ( J n ) { s } ⊆ ( I n ) { s } . (cid:3) Corollary 5.3.
Let R be a ring of characteristic p > , ≤ s , and I an ideal of R . If n ∈ N and r ≥ s ( µ ( I ) − − n ( s − , then ( I n + r ) { s } ⊆ ( I n ) ∗ . In particular, if n ≥ µ ( I ) − s − , then ( I n ) { s } = ( I n ) ∗ . One way of interpreting Corollary 5.3 is that asymptotically, as we take powers of an ideal, each s -closurewith s > s , we must take ever higherpowers of I to realize this collapse; i.e., there is in general no uniform power beyond which every s -closurefor s > Example 5.4.
Let I = ( x , y ) ⊆ k [ x, y ], where k is a field of characteristic p >
0. By Theorem 3.4, forany rational s , ( I n ) { s } = ( I n ) ∗ + ( I n ) s = I n + I ns . Now I ns is generated by all monomials with degree atleast 3 ns . Thus, if 1 < s < n , we have that deg( x n − y ) = 3 n + 1 = 3 n (cid:0) n (cid:1) ≥ ns . Therefore x n − y ∈ ( I n ) { s } , but x n − y / ∈ I n = ( I n ) ∗ . References [BH93] Winfried Bruns and J¨urgen Herzog.
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London MathematicalSociety Lecture Note Series . Cambridge University Press, Cambridge, 2006.[Tay18] William D. Taylor. Interpolating between Hilbert-Samuel and Hilbert-Kunz multiplicity.
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