Symbolic powers of sums of ideals
aa r X i v : . [ m a t h . A C ] A p r SYMBOLIC POWERS OF SUMS OF IDEALS
HUY T `AI H `A, HOP DANG NGUYEN, NGO VIET TRUNG, AND TRAN NAM TRUNG
Abstract.
Let I and J be nonzero ideals in two Noetherian algebras A and B over a field k . Let I + J denote the ideal generated by I and J in A ⊗ k B . We prove the followingexpansion for the symbolic powers:( I + J ) ( n ) = X i + j = n I ( i ) J ( j ) . If A and B are polynomial rings and if char( k ) = 0 or if I and J are monomial ideals, wegive exact formulas for the depth and the Castelnuovo-Mumford regularity of ( I + J ) ( n ) ,which depend on the interplay between the symbolic powers of I and J . The proof involvesa result of independent interest which states that under the above assumption, the inducedmap Tor Ai ( k, I ( n ) ) → Tor Ai ( k, I ( n − ) is zero for all i ≥ n ≥
0. We also investigate otherproperties and invariants of ( I + J ) ( n ) such as the equality between ordinary and symbolicpowers, the Waldschmidt constant and the Cohen-Macaulayness. Introduction
Let R be a commutative Noetherian ring and let Q be an ideal in R . For an integer n ≥ n -th symbolic power of Q is defined by Q ( n ) := R ∩ (cid:16) \ P ∈ Min( Q ) Q n R P (cid:17) . In other words, Q ( n ) is the intersection of the primary components of Q n associated to theminimal primes of Q .When R is a polynomial ring over an algebraically closed field k of characteristic zeroand Q is a radical ideal, Nagata and Zariski showed that Q ( n ) consists of polynomials in R whose partial derivatives of orders up to n − Q (see e.g. [11]).Therefore, symbolic powers carry more geometric information than ordinary powers of anideal. In general, it is difficult to study properties of symbolic powers.Let A and B be commutative Noetherian algebras over an arbitrary field k . Let I ⊆ A and J ⊆ B be nonzero proper ideals. For simplicity we also use I and J to denote theextensions of I and J in the algebra R := A ⊗ k B . The main aim of this paper is to studythe depth and the Castelnuovo-Mumford regularity (or simply regularity) of the symbolicpowers of the sum I + J in R . Such sums of ideals appear naturally in various contexts: • Fiber product of affine schemes : Let X and let Y be the affine schemes Spec( A/I )and Spec(
B/J ), then the fiber product X × k Y is the affine scheme Spec( R/I + J ); Mathematics Subject Classification.
Primary 13C05, 14B05; Secondary 13D07, 18G15.
Key words and phrases.
Symbolic power, sum of ideals, associated prime, tensor product, binomial ex-pansion, depth, Castelnuovo-Mumford regularity, Tor-vanishing, depth function. Join of simplicial complexes : Let ∆ and Γ be simplicial complexes over disjoint vertexsets with Stanley-Reisner ideals I ∆ and I Γ , then I ∆ + I Γ is the Stanley-Reisner idealof the join complex ∆ ⋆ Γ; • Edge ideal of a graph : Let I ( G ) denote the edge ideal a simple graph G . If G , ..., G n are the connected components of G , then I ( G ) = I ( G ) + · · · + I ( G n ) . Though symbolic powers have been studied extensively (see e.g. [2, 7, 8, 9, 20, 21, 23, 22,25, 26, 28, 33, 34, 35, 37]), symbolic powers of such sums of ideals have not been considered inthis general setting. We shall see from this paper and our sequential work [18] that studyingsums of ideals may indeed provide new insights to many problems on symbolic powers.Several results on the depth and the regularity of the ordinary powers ( I + J ) n have beenrecently established in [19, 29]. These results have had a number of interesting consequences.It is quite natural to ask whether there are similar results on the symbolic powers ( I + J ) ( n ) .The first step is to characterize ( I + J ) ( n ) in terms of I and J . In general, if I and J areprime ideals, I + J needs not be a primary ideal. This indicates that such a characterizationwould be complicated. Surprisingly, we can show that there is a binomial expansion for thesymbolic power ( I + J ) ( n ) : Theorem 3.4. ( I + J ) ( n ) = P i + j = n I ( i ) J ( j ) . This formula was not known even in the simple case when B = k [ x ] is a polynomial ringand J = ( x ). It was known before only for squarefree monomial ideals by Bocci et al [3]. Theproof of Theorem 3.4 is based on a thorough study of associated primes of tensor productsof modules over A and B , which is of independent interest.Theorem 3.4 allows us to study several aspects of ( I + J ) ( n ) . First, we show that when A and B are local rings or domains, ( I + J ) ( n ) = ( I + J ) n if and only if I ( t ) = I t and J ( t ) = J t for all t ≤ n , and that when I and J are homogeneous ideals of polynomial rings, thenˆ α ( I + J ) = min { ˆ α ( I ) , ˆ α ( J ) } , where ˆ α ( I ) denotes the Waldschmidt constant of an ideal, which appears in several areas ofmathematics [6, 14, 20, 38]. This formula for ˆ α ( I + J ) was known before only for squarefreemonomial ideals [3].Our main results on the depth and the regularity of ( I + J ) ( n ) can be summarized asfollows. Theorem 4.6 . Let A and B be polynomial rings over a field k . Let I ⊆ A and J ⊆ B benonzero proper homogeneous ideals. Then (i) depth R (cid:14) ( I + J ) ( n ) ≥ min i ∈ [1 ,n − j ∈ [1 ,n ] (cid:8) depth A/I ( n − i ) + depth B/J ( i ) + 1 , depth A/I ( n − j +1) + depth B/J ( j ) (cid:9) , (ii) reg R (cid:14) ( I + J ) ( n ) ≤ max i ∈ [1 ,n − j ∈ [1 ,n ] (cid:8) reg A/I ( n − i ) + reg B/J ( i ) + 1 , reg A/I ( n − j +1) + reg B/J ( j ) (cid:9) . heorems 5.6 and 5.11 . The inequalities of Theorem 4.6 are equalities if char( k ) = 0 orif I and J are monomial ideals. We expect that equalities hold regardless of the characteristic of the field k .The above results are intricate in the sense that the right-hand sides of the above formulaeinvolve the minimum or maximum value of two sets of different terms, which can be attainedseparately. It is a distinctive feature of polynomial rings because we can show that they donot hold if one of the rings A and B is not a polynomial ring.Using the same approach we further obtain exact formulas for the depth and the regularityof the quotients ( I + J ) ( n ) / ( I + J ) ( n +1) , that are independent of the characteristic of the field k . Theorem 4.7.
Let A and B be polynomial rings over a field k . Let I ⊆ A and J ⊆ B benonzero proper homogeneous ideals. Then (i) depth( I + J ) ( n ) / ( I + J ) ( n +1) = min i + j = n { depth I ( i ) /I ( i +1) + depth J ( j ) /J ( j +1) } , (ii) reg( I + J ) ( n ) / ( I + J ) ( n +1) = max i + j = n { reg I ( i ) /I ( i +1) + reg J ( j ) /J ( j +1) } . As a consequence of Theorem 4.7, we show that R/ ( I + J ) ( i ) is Cohen-Macaulay for all i ≤ n if and only if A/I ( i ) and B/J ( i ) are Cohen-Macaulay for all i ≤ n .The above results hold in a more general framework. Given two filtrations of ideals { I n } n ≥ in A and { J n } n ≥ in B , we give bounds for the depth and the regularity of the binomial sum Q n := X i + j = n I i J j . For the filtrations of ordinary or symbolic powers of the ideals I and J , we have Q n = ( I + J ) n or Q n = ( I + J ) ( n ) , respectively. This approach can be also applied to filtrations of integralclosures or of saturations of powers of I and J .We say that a filtration of ideals { I n } n ≥ is Tor-vanishing if the induced map Tor Ai ( k, I n ) → Tor Ai ( k, I n − ) is zero for all i ≥ n ≥
0. We show that the bounds for the depth and theregularity of the binomial sum Q n become equalities if the filtrations { I n } n ≥ and { J n } n ≥ are Tor-vanishing. Theorems 5.6 and 5.11 follow from this result because Tor-vanishing holdsfor filtrations of symbolic powers in these cases: Propositions 5.5 and 5.10 . Let I be a homogeneous ideal in a polynomial ring A over afield k . If char( k ) = 0 or if I is a monomial ideal, the filtration { I ( n ) } n ≥ is Tor-vanishing. The Tor-vanishing of the symbolic powers are of independent interest because they canbe used to investigate homological relationships between I ( n − and I ( n ) for a homogeneousideal I . They can be also considered as a higher order generalization of the inclusion I ( n ) ⊆ m I ( n − , where m is the ideal generated by the variables of R . This inclusion was proved byEisenbud and Mazur [12] under the same assumption of Propositions 5.5 and 5.10. Usingan example of [12] and the polarization trick of McCullough and Peeva [27] we can findhomogeneous ideals whose filtration of symbolic powers is not Tor-vanishing if char( k ) > I + J ) ( n ) . In Section , we present bounds for the depth and the regularity of R/ ( I + J ) ( n ) and the exact formulasfor the depth and the regularity of ( I + J ) ( n − / ( I + J ) ( n ) in terms of those of I and J . InSection 5 we use the technique of Tor-vanishing to study the problem when the obtainedbounds for the depth and the regularity of R/ ( I + J ) ( n ) become exact formulas.We assume that the reader is familiar with basic properties of associated primes, depth andregularity, which we use without references. For other unexplained notions and terminology,we refer the reader to [5, 10]. Acknowledgement.
H.T. H`a is partially supported by the Simons Foundation (grant
Associated primes of tensor products
Let A and B be two Noetherian algebras over a field k such that R := A ⊗ k B is Noetherian.For our investigation on the symbolic powers of sums of ideals, we need to know the associatedprimes of R -modules of the form M ⊗ k N , where M and N are nonzero finitely generatedmodules over A and B .Let Min A ( M ) and Ass A ( M ) denote the sets of minimal associated primes and associatedprimes of M as an A -module, respectively. The aim of this section is to describe Min R ( M ⊗ k N ) and Ass R ( M ⊗ k N ) in terms of those of M and N .We begin with the following observations. Lemma 2.1.
Let P be a prime ideal of R , p = P ∩ A , and q = P ∩ B . Then (i) P ∈ Min R ( M ⊗ k N ) if and only if p ∈ Min A ( M ) , q ∈ Min B ( N ) and P ∈ Min R ( R/ p + q ) ; (ii) P ∈ Ass R ( M ⊗ k N ) if and only if p ∈ Ass A ( M ) , q ∈ Ass B ( N ) and P ∈ Ass R ( R/ p + q ) .Proof. It is clear that ( M ⊗ k N ) P = M p ⊗ A p ( A ⊗ k N ) P . Since the map A → R is flat, themap A p → R P is also flat. Applying [17, Chap. IV, (6.1.2)], we havedim( M ⊗ k N ) P = dim M p + dim k ( p ) ⊗ A p ( A ⊗ k N ) P , where k ( p ) denotes the residue field of A p . Since k ( p ) = ( A/ p ) p , we have (( A/ p ) ⊗ k N ) P = k ( p ) ⊗ A p ( A ⊗ k N ) P . Therefore,dim( M ⊗ k N ) P = dim M p + dim(( A/ p ) ⊗ k N ) P . Since the map B → R is flat, we can also show thatdim(( A/ p ) ⊗ k N ) P = dim N q + dim(( A/ p ) ⊗ k ( B/ q )) P . Note that ( A/ p ) ⊗ k ( B/ q ) = R/ p + q . From the above equalities we get dim( M ⊗ k N ) P = 0if and only if dim M p = dim N q = dim( R/ p + q ) P = 0 . or an arbitrary finite R -module E , we know that P ∈ Min R ( E ) if and only if dim E P = 0.Therefore, P ∈ Min R ( M ⊗ k N ) if and only if p ∈ Min A ( M ), q ∈ Min B ( N ) and P ∈ Min R ( R/ p + q ).Similarly, we can apply [17, Chap. IV, (6.3.1)] to show that depth( M ⊗ k N ) P = 0 if andonly if depth M p = depth N q = depth( R/ p + q ) P = 0 . We also know that P ∈ Ass R ( E ) if and only if depth E P = 0. Therefore, P ∈ Ass R ( M ⊗ k N )if and only if p ∈ Ass A ( M ), q ∈ Ass B ( N ) and P ∈ Ass R ( R/ p + q ). (cid:3) Remark 2.2.
We need the assumption on the Noetherian property of A ⊗ k B for applying[17, Chap. IV, (6.1.2) and (6.3.1)]. In general, A ⊗ k B is not necessarily Noetherian, evenwhen A and B are field extensions of k . For more information on this topic see [36].Notice that p + q is not necessarily a prime or even a primary ideal as illustrated by thefollowing example. Example 2.3.
Let p := ( x + 1) ⊂ A = Q [ x ] and q := ( y + 1) ⊂ B = Q [ y ]. Both p and q are prime ideals. However, p + q = ( x + 1 , y + 1) is not primary in R = Q [ x, y ] because x − y = ( x + y )( x − y ) ∈ p + q . Lemma 2.4.
Let p and q be prime ideals of A and B , respectively. Let P ∈ Ass( R/ p + q ) .Then (i) P ∩ A = p and P ∩ B = q , (ii) P ∈ Min R ( R/ p + q ) .Proof. Note that R/ p + q = ( A/ p ) ⊗ k ( B/ q ). Applying Lemma 2.1 (ii) to ( A/ p ) ⊗ k ( B/ q )we obtain P ∩ A ∈ Ass A ( A/ p ) = { p } and P ∩ B ∈ Ass B ( B/ p ) = { p } , which implies (i).Let k ( p ) and k ( q ) denote the residue fields of A p and B q . Because of (i) we can consider(( A/ p ) ⊗ k ( B/ q )) P as a localization of the algebra k ( p ) ⊗ k k ( q ) at a prime ideal P ′ . Since P is an associated prime of ( A/ p ) ⊗ k ( B/ q ), P ′ is an associated prime of k ( p ) ⊗ k k ( q ). By[30, Theorem 3], ( k ( p ) ⊗ k k ( q )) P ′ is a primary ring, i.e. any of its zero divisors is a nilpotentelement. From this it follows that P ′ is a minimal associated prime of k ( p ) ⊗ k k ( q ). Hence, P must be a minimal associated prime of ( A/ p ) ⊗ k ( B/ q ), which proves (ii). (cid:3) By Lemma 2.4(ii), the ideal p + q is always unmixed though it may be not a primary ideal.Now we can describe the associated and the minimal associated primes of M ⊗ k N in termsof M and N as follows. This description gives more precise information on Ass R ( M ⊗ k N )than [31, Corollary 3.7(1)]. Theorem 2.5.
Let M and N be nonzero modules over A and B , respectively. Then (i) Min R ( M ⊗ k N ) = [ p ∈ Min A ( M ) q ∈ Min B ( N ) Min R ( R/ p + q ) . (ii) Ass R ( M ⊗ k N ) = [ p ∈ Ass A ( M ) q ∈ Ass B ( N ) Min R ( R/ p + q ) . roof. By Lemma 2.1, we haveMin R ( M ⊗ k N ) = [ p ∈ Min A ( M ) q ∈ Min B ( N ) { P ∈ Min R ( R/ p + q ) (cid:12)(cid:12) P ∩ A = p , P ∩ B = q } . Ass R ( M ⊗ k N ) = [ p ∈ Ass A ( M ) q ∈ Ass B ( N ) { P ∈ Ass R ( R/ p + q ) (cid:12)(cid:12) P ∩ A = p , P ∩ B = q } . By Lemma 2.4, we have P ∩ A = p , P ∩ B = q for all P ∈ Ass R ( R/ p + q ) andAss R ( R/ p + q ) = Min R ( R/ p + q ) . Hence, we can rewrite the above formulas as in the statement of the theorem. (cid:3)
The following immediate consequence of Theorem 2.5 is a generalization of a classicalresult of Seidenberg on unmixed polynomial ideals under base field extensions in [32].
Corollary 2.6.
Ass R ( M ⊗ k N ) = Min R ( M ⊗ k N ) if and only if Ass A ( M ) = Min A ( M ) and Ass B ( N ) = Min B ( N ) . One may ask when is the sum p + q of two prime ideals p ⊂ A and q ⊂ B a prime idealin R ? This question has the following simple answer. Lemma 2.7.
Let k ( p ) and k ( q ) denote the fields of fractions of A/ p and B/ q . Then p + q is a prime ideal if and only if k ( p ) ⊗ k k ( q ) is a domain.Proof. Let p + q be a prime ideal. Then ( A/ p ) ⊗ k ( B/ q ) = R/ p + q is a domain. Since k ( p ) ⊗ k k ( q ) is a localization of ( A/ p ) ⊗ k ( B/ q ), it must be a domain, too. The converse istrue since we have an injection ( A/ p ) ⊗ k ( B/ q ) → k ( p ) ⊗ k k ( q ). (cid:3) By [39, Corollary 1, p. 198], the tensor product of two field extensions of k is a domain if k is algebraically closed. In this case, Theorem 2.5 can be reformulated as follows. Corollary 2.8.
Let k be an algebraically closed field. Then (i) Ass R ( M ⊗ k N ) = { p + q | p ∈ Ass A ( M ) and q ∈ Ass B ( N ) } , (ii) Min R ( M ⊗ k N ) = { p + q | p ∈ Min A ( M ) and q ∈ Min B ( N ) } . Binomial expansion of symbolic powers
Let A and B be two commutative Noetherian algebras over a field k such that R = A ⊗ k B is also a Noetherian ring. Let I and J be nonzero proper ideals of A and B , respectively.We will use the same symbols I, J for the extensions of
I, J in R . The aim of this section isto prove that the symbolic power ( I + J ) ( n ) has a binomial expansion.We shall need the following observations. Lemma 3.1. I ∩ J = IJ .Proof. Let V and W be two sets of elements of A and B . We denote by V ⊗ W the set of theelements f ⊗ g , f ∈ V and g ∈ W . Choose bases V and W of the k -vector spaces I and J andextend them to bases V ∗ and W ∗ of A and B , respectively. Then V ⊗ W ∗ and V ∗ ⊗ W arebases of the k -vector spaces I ⊗ k B and A ⊗ k J , respectively. Since V ⊗ W ∗ and V ∗ ⊗ W are ubsets of V ∗ ⊗ W ∗ , which is a basis of A ⊗ k B , the vector space I ∩ J = ( I ⊗ k B ) ∩ ( A ⊗ k J )is generated by the set ( V ⊗ W ∗ ) ∩ ( V ∗ ⊗ W ) = V ⊗ W. Since IJ = I ⊗ k J is also generatedby V ⊗ W , we conclude that I ∩ J = IJ . (cid:3) Lemma 3.1 was known before for polynomial ideals [24, Lemma 1.1].
Lemma 3.2.
Let I ′ and J ′ be subideals of I and J , respectively. Then ( I/I ′ ) ⊗ k ( J/J ′ ) ∼ = IJ/ ( IJ ′ + I ′ J ) . Proof.
We have(
I/I ′ ) ⊗ k ( J/J ′ ) ∼ = (cid:0) ( I ⊗ k J ) / ( I ⊗ k J ′ ) (cid:1) / (cid:0) ( I ′ ⊗ k J ) / ( I ′ ⊗ k J ′ ) (cid:1) . ∼ = ( IJ/IJ ′ ) / ( I ′ J/I ′ J ′ ) . By Lemma 3.1, I ′ J ′ = I ′ ∩ J ′ = I ′ J ∩ J ′ = I ′ J ∩ IJ ′ . From this it follows that(
I/I ′ ) ⊗ k ( J/J ′ ) ∼ = ( IJ/IJ ′ ) (cid:14)(cid:0) ( IJ ′ + I ′ J ) /IJ ′ (cid:1) ∼ = IJ (cid:14) ( IJ ′ + IJ ′ ) . (cid:3) In the following, we will consider binomial sums of filtrations of ideals, which is defined asfollows.For simplicity, we call a sequence of ideals { Q n } n ≥ in R a filtration if it satisfies thefollowing conditions:(1) Q = R ,(2) Q is a nonzero proper ideal,(3) Q n ⊇ Q n +1 for all n ≥ { Q n } n ≥ , the symbolic powers { Q ( n ) } n ≥ ,and the integral closures of powers { Q n } n ≥ , where Q is a nonzero proper ideal.Let { I i } i ≥ and { J j } j ≥ be two filtrations of ideals in A and B , respectively. For each n ≥
0, we define Q n := X i + j = n I i J j . We call Q n the n -th binomial sum of the filtrations { I i } i ≥ and { J j } j ≥ .Our next result shows that quotients of successive binomial sums have a nice decomposi-tion. Proposition 3.3.
For any integer n ≥ , there is an isomorphism Q n /Q n +1 ∼ = n M i =0 ( I i /I i +1 ) ⊗ k ( J n − i /J n − i +1 ) . Proof.
First, we will show that Q n /Q n +1 ∼ = n M i =0 (cid:0) ( I i J n − i + Q n +1 ) /Q n +1 (cid:1) . or that it suffices to show that for 0 ≤ i ≤ n ,( I i J n − i + Q n +1 ) ∩ (cid:0) X j = i I j J n − j + Q n +1 (cid:1) ⊆ Q n +1 or, equivalently, I i J n − i ∩ (cid:0) X j = i I j J n − j + Q n +1 (cid:1) ⊆ Q n +1 . We have X j = i I j J n − j + Q n +1 = X j = i I j J n − j + X ≤ j ≤ n +1 I j J n − j +1 ⊆ J n − i +1 + I i +1 because J n − j ⊆ J n − i +1 for j < i and I j ⊆ I i +1 for j ≥ i + 1. On the other hand, everyelement of J n − i +1 + I i +1 in I i J n − i is a sum of two elements, one in J n − i +1 ∩ ( I i +1 + I i J n − i )and the other in I i +1 ∩ ( J n − i +1 + I i J n − i ). Therefore, I i J n − i ∩ (cid:0) X ≤ j ≤ n,j = i I j J n − j + Q n +1 (cid:1) ⊆ J n − i +1 ∩ ( I i +1 + I i J n − i ) + I i +1 ∩ ( J n − i +1 + I i J n − i ) ⊆ J n − i +1 ∩ I i + I i +1 ∩ J n − i = I i +1 J n − i + I i J n − i +1 ⊆ Q n +1 , where the equality holds thanks to Lemma 3.1.The above inclusions also show that I i J n − i ∩ Q n +1 = I i +1 J n − i + I i J n − i +1 . Hence,( I i J n − i + Q n +1 ) /Q n +1 ∼ = I i J n − i / (cid:0) Q n +1 ∩ I i J n − i (cid:1) ∼ = I i J n − i / (cid:0) I i +1 J n − i + I i J n − i +1 (cid:1) ∼ = ( I i /I i +1 ) ⊗ k ( J n − i /J n − i +1 ) , where the last isomorphism follows from Lemma 3.2. Therefore, Q n /Q n +1 = n M i =0 (cid:0) ( I i J n − i + Q n +1 ) /Q n +1 (cid:1) ∼ = n M i =0 ( I i /I i +1 ) ⊗ k ( J n − i /J n − i +1 ) . (cid:3) We are now ready to prove the main result of this section.
Theorem 3.4. ( I + J ) ( n ) = X i + j = n I ( i ) J ( j ) . Proof.
Consider the symbolic filtrations { I ( i ) } i ≥ and { J ( j ) } j ≥ . In this case, we have the n -th binomial sum Q n = X i + j = n I ( i ) J ( j ) . We shall first prove the inclusion ( I + J ) ( n ) ⊆ Q n . For that we need to investigate theassociated primes of R/Q n . It follows from the short exact sequence0 → Q t − /Q t → R/Q t → R/Q t − → , hat Ass R ( R/Q t ) ⊆ Ass R ( Q t − /Q t ) ∪ Ass R ( R/Q t − ) . Hence,Ass R ( R/Q n ) ⊆ n [ t =1 Ass R ( Q t − /Q t ) . By Proposition 3.3, we haveAss R ( Q t − /Q t ) = [ i + j = t − Ass R (cid:0) I ( i ) /I ( i +1) ⊗ k J ( j ) /J ( j +1) (cid:1) . By Theorem 2.5(ii), we haveAss R (cid:0) I ( i ) /I ( i +1) ⊗ k J ( j ) /J ( j +1) (cid:1) = [ p ∈ Ass A ( I ( i ) /I ( i +1) ) q ∈ Ass B ( J ( j ) /J ( j +1) ) Min R ( R/ p + q ) . Since I ( i ) /I ( i +1) and J ( j ) /J ( j +1) are ideals of A/I ( i +1) and B/J ( j +1) , we haveAss A ( I ( i ) /I ( i +1) ) ⊆ Ass A ( A/I ( i +1) ) = Min A ( A/I ( i +1) ) = Min A ( A/I ) , Ass B ( J ( j ) /J ( j +1) ⊆ Ass B ( B/J ( j +1) ) = Min B ( B/J ( j +1) ) = Min B ( B/J ) . Since
R/I + J = ( A/I ) ⊗ k ( B/J ), it follows from Theorem 2.5(i) thatMin R ( R/I + J ) = [ p ∈ Min A ( A/I ) q ∈ Min B ( B/J ) Min R ( R/ p + q ) . Therefore, Ass R (cid:0) I ( i ) /I ( i +1) ⊗ k J ( j ) /J ( j +1) (cid:1) ⊆ Min R ( R/I + J ) . So we get Ass R ( R/Q n ) ⊆ Min R ( R/I + J ) = Min R ( R/ ( I + J ) n ) . This shows that every associated prime of Q n is a minimal associated prime of ( I + J ) n .Since Q n ⊇ P i + j = n I i J j = ( I + J ) n , it follows from the definition of symbolic powers thatevery primary component of Q n contains a primary component of ( I + J ) ( n ) . Therefore, Q n ⊇ ( I + J ) ( n ) .Now, we shall prove the converse inclusion ( I + J ) ( n ) ⊇ Q n . Let P be an arbitraryminimal associated prime of ( I + J ) n . Then P is a minimal associated prime of R/I + J =( A/I ) ⊗ k ( B/J ). Set p = P ∩ A and q = P ∩ B . By Lemma 2.1(i), p and q are minimalassociated primes of I and J . Therefore, ( I ( i ) ) p = ( I i ) p and ( J ( j ) ) q = ( J j ) q for all i, j ≥ I ( i ) ) p ⊗ k ( J ( j ) ) q = ( I i ) p ⊗ k ( J j ) q . Since ( I ( i ) ⊗ J ( j ) ) P and ( I i ⊗ J j ) P are localizations of ( I ( i ) ) p ⊗ k ( J ( j ) ) q and ( I i ) p ⊗ k ( J j ) q ata prime ideal of A p ⊗ B q , we get( I ( i ) J ( j ) ) P = ( I ( i ) ⊗ k J ( j ) ) P = ( I i ⊗ J j ) P = ( I i J j ) P for all i, j ≥
0. Thus,( Q n ) P = X i + j = n ( I ( i ) J ( j ) ) P = X i + j = n ( I i J j ) P = ( I + J ) nP . his shows that every primary component of ( I + J ) ( n ) contains a primary component of Q n . Hence, ( I + J ) ( n ) ⊇ Q n . (cid:3) Theorem 3.4 extends a result on squarefree monomial ideals of Bocci et al [3, Theorem7.8] to arbitrary ideals. It will play a key role in our investigation on invariants of ( I + J ) ( n ) in the next sections.Moreover, Theorem 3.4 yields the following criterion for the equality of symbolic andordinary powers of I + J . Corollary 3.5.
Assume that I t = I t +1 and J t = J t +1 for all t ≤ n − . Then ( I + J ) ( n ) =( I + J ) n if and only if I ( t ) = I t and J ( t ) = J t for all t ≤ n .Proof. Assume that I ( t ) = I t and J ( t ) = J t for all t ≤ n . By Theorem 3.4, we have( I + J ) ( n ) = X i + j = n I ( i ) J ( j ) = X i + j = n I i J j = ( I + J ) n . Conversely, assume that ( I + J ) ( n ) = ( I + J ) n . Since ( I + J ) n − / ( I + J ) n ⊆ R/ ( I + J ) n ,we have Ass R (( I + J ) n − / ( I + J ) n ) ⊆ Ass R ( R/ ( I + J ) n ) = Min R ( R/ ( I + J ) n )= Min R ( R/ ( I + J )) = Min R (cid:0) ( A/I ) ⊗ k ( B/J ) (cid:1) . By Proposition 3.3, we have( I + J ) n − / ( I + J ) n = M i + j = n − ( I i /I i +1 ) ⊗ k ( J j /J j +1 ) . Hence, Ass R (cid:0) ( I + J ) n − / ( I + J ) n (cid:1) = [ i + j = n − Ass R (cid:0) ( I i /I i +1 ) ⊗ k ( J j /J j +1 ) (cid:1) . Therefore, Ass R (cid:0) ( I i /I i +1 ) ⊗ k ( J j /J j +1 ) (cid:1) ⊆ Min R (cid:0) ( A/I ) ⊗ k ( B/J ) (cid:1) . Since I i = I i +1 and J j = J j +1 , we can apply Theorem 2.5 to get Ass A ( I i /I i +1 ) ⊆ Min A ( A/I )for i ≤ n −
1. Similarly as in the proof of Theorem 3.4, we haveAss A ( A/I t ) ⊆ t − [ i =0 Ass A ( I i /I i +1 ) ⊆ Min A ( A/I ) . This implies that Ass A ( A/I t ) = Min A ( A/I ). Hence, I ( t ) = I t for all t ≤ n . By the sameway, we can also show that J ( t ) = J t for all t ≤ n . (cid:3) It is easy to see that the assumption of Corollary 3.5 is satisfied if A and B are local ringsor domains. The following example shows that Corollary 3.5 does not hold if we remove theassumption that I t = I t +1 and J t = J t +1 for all t ≤ n − Example 3.6.
Let A = k [ x ] / ( x − x ) and I = xA . Then I = I . Let B = k [ y, z, t ] and J = ( y , y z, yz , z , y z t ). Then J (1) = ( y, z ) = J and J (2) = J = ( y, z ) . However,( I + J ) (2) = ( I + J ) = ( x, ( y, z ) ) R . emark 3.7. Corollary 3.5 shows that if ( I + J ) ( n ) = ( I + J ) n then ( I + J ) ( t ) = ( I + J ) t forall t ≤ n −
1. However, for an arbitrary ideal Q in a polynomial ring, Q ( n ) = Q n does notimply Q ( t ) = Q t for all t ≤ n −
1. The ideal J in the above example is such a case.We end this section by giving another interesting application of Theorem 3.4. Recall thatfor a nonzero proper homogeneous ideal I , α ( I ) := min { d | I d = 0 } is the smallest degree ofa nonzero element in I , and the Waldschmidt constant of I is defined byˆ α ( I ) := lim m →∞ α ( I ( m ) ) m . This limit exists and was first investigated by Waldschmidt in complex analysis [38]. Sincethen, it has appeared in different areas of mathematics, e.g., in number theory, algebraicgeometry and commutative algebra [6, 14, 20]. The following consequence of Theorem 3.4extends a result on the Waldschmidt constant of squarefree monomial ideals [3, Corollary7.10] to arbitrary homogeneous ideals.
Corollary 3.8.
Let A and B be standard graded polynomial rings over k . Let I ⊂ A and J ⊂ B be nonzero proper homogeneous ideals. Then ˆ α ( I + J ) = min { ˆ α ( I ) , ˆ α ( J ) } . Proof.
The proof goes in the same line of arguments as that of [3, Corollary 7.10] replacing[3, Theorem 7.8] by our more general statement in Theorem 3.4. (cid:3) Depth and regularity of binomial sums
Throughout this section, let A = k [ X ] and B = k [ Y ] be polynomial rings over an arbitraryfield k in different sets of variables. Then R := A ⊗ k B = k [ X, Y ]. If I ⊂ A and J ⊂ B arehomogeneous ideal, then their extensions in R are also homogeneous. As before, we shallalso denote these ideals by I and J .We shall need the following results of Hoa and Tam in [24]. Lemma 4.1. [24, Lemmas 2.2 and 3.2]
Let I ⊆ A and J ⊆ B be nonzero proper homogeneousideals. Then (i) depth R/IJ = depth
A/I + depth
B/J + 1 . (ii) reg R/IJ = reg
A/I + reg
B/J + 1 . Let { I i } i ≥ and { J j } j ≥ be filtrations of homogeneous ideals in A and B , respectively.Recall that the ideal Q n := X i + j = n I i J j is called the n -th binomial sum of these filtrations. The aim of this section is to give boundsfor the depth and the regularity of R/Q n in terms of those of I i and J j . Theorem 4.2.
For all n ≥ , we have (i) depth R/Q n ≥ min i ∈ [1 ,n − j ∈ [1 ,n ] { depth A/I n − i + depth B/J i + 1 , depth A/I n − j +1 + depth B/J j } , ii) reg R/Q n ≤ max i ∈ [1 ,n − j ∈ [1 ,n ] { reg A/I n − i + reg B/J i + 1 , reg A/I n − j +1 + reg B/J j } . Proof.
We shall only prove the bound for depth. The bound for regularity can be obtainedin the same fashion.For t = 0 , . . . , n , set P n,t := I n J + I n − J + · · · + I n − t J t . Then P n,t = P n,t − + I n − t J t for 1 ≤ t ≤ n . Since P n,t − ⊆ I n − t +1 , we have P n,t − ∩ I n − t J t ⊆ I n − t +1 ∩ J t = I n − t +1 J t by Lemma 3.1. On the other hand, I n − t +1 J t ⊆ I n − t +1 J t − ⊆ P n,t − and I n − t +1 J t ⊆ I n − t J t .This implies that P n,t − ∩ I n − t J t = I n − t +1 J t . Hence, there is a short exact sequence0 −→ R/I n − t +1 J t −→ ( R/P n,t − ) ⊕ ( R/I n − t J t ) −→ R/P n,t −→ . Therefore, we havedepth
R/P n,t ≥ min { depth R/P n,t − , depth R/I n − t J t , depth R/I n − t +1 J t − } . We will use these bounds to successively estimate depth
R/Q n as follows.For t = n , we have P n,n = Q n . Since I J n = J n , we have depth R/I J n = dim A +depth B/J n . Applying Lemma 4.1 to the product I J n , we getdepth R/Q n ≥ min { depth R/P n,n − , dim A + depth B/J n , depth A/I + depth B/J n } . For t = n − , . . . ,
2, by applying Lemma 4.1 to the products I n − t J t and I n − t +1 J t , we getdepth R/P n,t ≥ min { depth R/P n,t − , depth A/I n − t + depth B/J t +1 , depth A/I n − t +1 +depth B/J t } . For t = 1, we have depth R/P n, = depth A/I n +dim B because P n, = I n J = I n . ApplyingLemma 4.1 to the product I n J now yieldsdepth R/P n, ≥ min { depth A/I n +dim B, depth A/I n − +depth B/J +1 , depth A/I n +depth B/J } . Putting all these estimates for depth
R/P n,t together, we obtaindepth
R/Q n ≥ min { dim A + depth B/J n , depth A/I n + dim B, min i ∈ [1 ,n − j ∈ [1 ,n ] { depth A/I n − i + depth B/J i + 1 , depth A/I n − j +1 + depth B/J j } } . Since I = (0), we havedim A + depth B/J n > depth A/I + depth B/J n . Since J = (0), we havedepth A/I n + dim B > depth
A/I n + depth B/J . he right-hand sides of the above inequalities are depth A/I n − j +1 + depth B/J j for j = n, A + depth B/J n and depth A/I n + dim B from theestimate for depth R/Q n to obtain thatdepth R/Q n ≥ min i ∈ [1 ,n − j ∈ [1 ,n ] { depth A/I n − i + depth B/J i + 1 , depth A/I n − j +1 + depth B/J j } . (cid:3) Remark 4.3.
Let I ⊂ A and J ⊂ B be nonzero homogeneous ideals. If I i = I i and J j = J j for all i, j ≥
0, we have Q n = ( I + J ) n . In this case, Theorem 4.2 recovers a main result ofour previous work on depth and regularity of ordinary powers [19, Theorem 2.4]. As pointedout in [19, Propositions 2.6 and 2.7], both terms on the right-hand side of these bounds areessential (i.e., are attainable). Hence, this is also the case for the two terms on the right-handside of the bounds of Theorem 4.2.If we consider the quotients Q n /Q n +1 instead of the quotient rings R/Q n , we can computetheir depth and regularity explicitly in terms of those of quotients of successive I i ’s and J j ’s. Theorem 4.4.
For all n ≥ , we have (i) depth Q n /Q n +1 = min i + j = n { depth I i /I i +1 + depth J j /J j +1 } , (ii) reg Q n /Q n +1 = max i + j = n { reg I i /I i +1 + reg J j /J j +1 } .Proof. Proposition 3.3 gives Q n /Q n +1 = M i + j = n ( I i /I i +1 ) ⊗ k ( J j /J j +1 ) . The desired conclusion now follows from [19, Lemma 2.5], which expresses the depth andthe regularity of a tensor product over k in terms of those of the components. (cid:3) As a consequence of Theorem 4.4, we obtain bounds for the depth and the regularity of
R/Q n in terms of those of quotients of successive I i ’s and J j ’s. Corollary 4.5.
For all n ≥ , we have (i) depth R/Q n ≥ min i + j ≤ n − { depth I i /I i +1 + depth J j /J j +1 } , (ii) reg R/Q n ≤ max i + j ≤ n − { reg I i /I i +1 + reg J j /J j +1 } . Proof.
Using the short exact sequences0 → Q t /Q t +1 → R/Q t +1 → R/Q t → t ≤ n − R/Q n ≥ min t ≤ n − depth Q t /Q t +1 , reg R/Q n ≤ max t ≤ n − reg Q t /Q t +1 . Hence, the assertions follow from Theorem 4.4. (cid:3) f { I i } i ≥ and { J j } j ≥ are the filtrations of symbolic powers of two nonzero homogeneousideals I ⊂ A and J ⊂ B , then Q n = ( I + J ) ( n ) by Theorem 3.4. For the sake of applications,we reformulate Theorem 4.2 and Theorem 4.4 in this case. Theorem 4.6.
For all n ≥ , we have (i) depth R (cid:14) ( I + J ) ( n ) ≥ min i ∈ [1 ,n − j ∈ [1 ,n ] (cid:8) depth A/I ( n − i ) + depth B/J ( i ) + 1 , depth A/I ( n − j +1) + depth B/J ( j ) (cid:9) , (ii) reg R (cid:14) ( I + J ) ( n ) ≤ max i ∈ [1 ,n − j ∈ [1 ,n ] (cid:8) reg A/I ( n − i ) + reg B/J ( i ) + 1 , reg A/I ( n − j +1) + reg B/J ( j ) (cid:9) . We shall see in the next section that the inequalities of Theorem 4.6 are in fact equalitiesif char( k ) = 0 or if I and J are monomial ideals. Theorem 4.7.
For all n ≥ , we have (i) depth( I + J ) ( n ) (cid:14) ( I + J ) ( n +1) = min i + j = n { depth I ( i ) /I ( i +1) + depth J ( j ) /J ( j +1) } , (ii) reg( I + J ) ( n ) (cid:14) ( I + J ) ( n +1) = max i + j = n { reg I ( i ) /I ( i +1) + reg J ( j ) /J ( j +1) } . Using Theorem 4.7 we obtain the following criterion for the Cohen-Macaulayness of R/ ( I + J ) ( n ) . Recall that a finite graded R -module M is Cohen-Macaulay if depth M = dim M . Corollary 4.8.
The following conditions are equivalent: (i) ( I + J ) ( n − (cid:14) ( I + J ) ( n ) is Cohen-Macaulay, (ii) R/ ( I + J ) ( i ) is Cohen-Macaulay for all i ≤ n , (iii) A/I ( i ) and B/J ( i ) are Cohen-Macaulay for all i ≤ n , (iv) I ( i ) /I ( i +1) and J ( i ) /J ( i +1) are Cohen-Macaulay for all i ≤ n − .Proof. It is clear thatdim( I + J ) ( n − / ( I + J ) ( n ) ≤ dim R/ ( I + J ) ( n ) = dim R/ ( I + J ) . For any prime P ∈ Min R ( R/ ( I + J )), we have(( I + J ) ( n − / ( I + J ) ( n ) ) P = (( I + J ) n − / ( I + J ) n ) P = 0if and only if (( I + J ) n − ) P = 0 by Nakayama’s lemma. But this could not happen because R is a domain and I + J = 0. Thus,dim R/ ( I + J ) ≤ dim( I + J ) ( n − / ( I + J ) ( n ) . From this it follows thatdim( I + J ) ( n − / ( I + J ) ( n ) = dim R/ ( I + J ) = dim A/I + dim
B/J.
Similarly, we also have dim I ( i − /I ( i ) = dim A/I and dim J ( j − /I ( j ) = dim B/J for all i, j ≥ I + J ) ( n − / ( I + J ) ( n ) = dim I i /I ( i +1) + dim J ( i ) /J ( i +1) or all n, i, j ≥
1. Using Theorem 4.7(i) we can show thatdepth( I + J ) ( n − / ( I + J ) ( n ) = dim( I + J ) ( n − / ( I + J ) ( n ) if and only if depth I i /I ( i +1) = dim I i /I ( i +1) and depth J ( i ) /J ( i +1) = dim J ( i ) /J ( i +1) for all i ≤ n −
1. Thus, ( I + J ) ( n − / ( I + J ) ( n ) is Cohen-Macaulay if and only if I ( i ) /I ( i +1) and J ( i ) /J ( i +1) are Cohen-Macaulay for all i ≤ n −
1. From this it follows that (i) ⇔ (iv). Inparticular, (i) implies that ( I + J ) ( i ) / ( I + J ) ( i +1) is Cohen-Macaulay for all i ≤ n − R -module M is Cohen-Macaulay if and only if H t m ( M ) = 0 for t < dim M , where H t m ( M ) denotes the t -th local cohomology module of M with respect tothe maximal homogeneous ideal m of R . Using this characterization and the short exactsequence 0 → ( I + J ) ( i ) / ( I + J ) ( i +1) → R/ ( I + J ) ( i +1) → R/ ( I + J ) ( i ) → , we deduce that R/ ( I + J ) ( i ) is Cohen-Macaulay for all i ≤ n if and only if ( I + J ) ( i ) / ( I + J ) ( i +1) is Cohen-Macaulay for all i ≤ n −
1. This proves the implication (i) ⇔ (ii).Similarly, A/I ( i ) and B/J ( i ) are Cohen-Macaulay for all i ≤ n if and only if I ( i ) /I ( i +1) and J ( i ) /J ( i +1) are Cohen-Macaulay for all i ≤ n −
1. This proves the equivalence (iii) ⇔ (iv). (cid:3) The implication (i) ⇒ (ii) is remarkable in the sense that the Cohen-Macaulay property of( I + J ) ( n − (cid:14) ( I + J ) ( n ) alone implies that of R/ ( I + J ) ( t ) for all t ≤ n . The following exampleshows that this implication does not hold if we replace I + J by an arbitrary homogeneousideal. Example 4.9.
Take A = k [ x, y, z, t ] and I = ( x , x y, xy , y , x y ( xz + yt ) , x y ). Then I is a ( x, y )-primary ideal, dim A/I = 2 and I (1) = I . It is clear that z is a regularelement of A/I . Since the socle of A/ ( I, z ) contains the residue class of x y , we havedepth A/I = 1. Therefore,
A/I (1) = A/I is not Cohen-Macaulay. On the other hand, wehave I = ( x, y ) ∩ (cid:0) I + ( z, t ) (cid:1) . Hence, I (2) = ( x, y ) . Now, we can see that z, t is a regularsequence of I (1) /I (2) . Therefore, I (1) /I (2) is Cohen-Macaulay.We end this section with the following formulas for the case where one of the ideals I and J is generated by linear forms. Though this case appears to be simple, these formulas hadnot been known before. The proof is based on the binomial expansion of ( I + J ) ( n ) . Proposition 4.10.
Assume that J is generated by linear forms. Then (i) depth R/ ( I + J ) ( n ) = min i ≤ n { depth A/I ( i ) + dim B/J } , (ii) reg R/ ( I + J ) ( n ) = max i ≤ n { reg A/I ( i ) − i } + n. Proof.
Assume that B = k [ y , . . . , y s ]. Without restriction we may assume that J =( y , . . . , y t ), t ≤ s . Then dim B/J = s − t . Set B ′ = k [ y , . . . , y t ], J ′ = ( y , . . . , y t ) B ′ ,and R ′ = A ⊗ k B ′ . It is clear thatdepth R/ ( I + J ) ( n ) = depth R ′ / ( I + J ′ ) ( n ) + s − t, reg R/ ( I + J ) ( n ) = reg R ′ / ( I + J ′ ) ( n ) . Therefore, we only need to prove the case t = s . f t = s = 1, we set y = y . Then B = k [ y ] and J = ( y ). By Theorem 3.4, we have( I, y ) ( n ) = P ni =0 I ( i ) y n − i . If we write R = ⊕ i ≥ Ay i , then R/ ( I, y ) ( n ) = ⊕ i ≤ n ( A/I ( i ) ) y n − i .From this it follows thatdepth R/ ( I, y ) ( n ) = min i ≤ n depth A/I ( i ) , reg R/ ( I, y ) ( n ) = max i ≤ n { reg A/I ( i ) − i } + n. If t = s >
1, we set A ′ = A [ y , . . . , y s − ] and I ′ = ( I, y , . . . , y s − ) A ′ . Using induction wemay assume that depth A ′ / ( I ′ ) ( n ) = min i ≤ n depth A/I ( i ) , reg A ′ / ( I ′ ) ( n ) = max i ≤ n { reg A/I ( i ) − i } + n. Since I + J = ( I ′ , y s ), we havedepth R/ ( I + J ) ( n ) = min i ≤ n depth A ′ / ( I ′ ) ( i ) = min i ≤ n depth A/I ( i ) , reg R/ ( I + J ) ( n ) = max i ≤ n { reg A ′ / ( I ′ ) ( i ) − i } + n = max i ≤ n { reg A/I ( i ) − i } + n. (cid:3) Splitting conditions
The aim of this section is to show that the inequalities of Theorem 4.6 are equalities ifchar( k ) = 0 or if I and J are monomial ideals. Our main tool is the following notion whichallows us to compute the depth and the regularity of a sum of ideals in terms of those of thesummands and their intersection.Let R be a polynomial ring over a field k . Let P, I, J be nonzero homogeneous ideals of R such that P = I + J . The sum P = I + J is called a Betti splitting if the Betti numbersof
P, I, J, I ∩ J satisfy the relation β i,j ( P ) = β i,j ( I ) + β i,j ( J ) + β i − ,j ( I ∩ J )for all i ≥ j ∈ Z .This notion was introduced by Francisco, H`a and Van Tuyl [15]. It generalizes the notionof splittable monomial ideals of Eliahou and Kervaire [13]. The following result explains whyit is useful to have a Betti splitting. Lemma 5.1. [15, Corollary 2.2]
Let P = I + J be a Betti splitting. Then (i) depth R/P = min { depth R/I, depth
R/J, depth
R/I ∩ J − } , (ii) reg R/P = max { reg R/I, reg
R/J, reg
R/I ∩ J − } . A Betti splitting can be characterized by the following property. We say that a homomor-phism φ : M → N of graded R -modules is Tor-vanishing if Tor Ri ( k, φ ) = 0 for all i ≥
0. Thisnotion was due to Nguyen and Vu [29].
Lemma 5.2. [15, Proposition 2.1]
The following conditions are equivalent: (i)
The decomposition P = I + J is a Betti splitting. ii) The inclusion maps I ∩ J → I and I ∩ J → J are Tor-vanishing. We say that a filtration of ideals { P n } n ≥ in R is a Tor-vanishing filtration if for all n ≥ P n → P n − is Tor-vanishing, i.e. Tor Ri ( k, P n ) → Tor Ri ( k, P n − ) is the zeromap for all i ≥ Theorem 5.3.
Let { I i } i ≥ and { J j } j ≥ be Tor-vanishing filtrations in A and B . Let Q n = P i + j = n I i J j . Then (i) depth R/Q n =min i ∈ [1 ,n − j ∈ [1 ,n ] { depth A/I n − i + depth B/J i + 1 , depth A/I n − j +1 + depth B/J j } , (ii) reg R/Q n =max i ∈ [1 ,n − j ∈ [1 ,n ] { reg A/I n − i + reg B/J i + 1 , reg A/I n − j +1 + reg B/J j } . Proof.
We shall only prove the formula for depth. The formula for regularity can be provedsimilarly.For t = 0 , . . . , n , set P n,t := I n J + I n − J + · · · + I n − t J t . Then P n, = I n , P n,n = Q n , and P n,t = P n,t − + I n − t J t for t ≥
1. We will compute depth
R/Q n by successively computing depth R/P n,t . To do that we will first show that P n,t = P n,t − + I n − t J t is a Betti splitting for 1 ≤ t ≤ n .We have seen in the proof of Theorem 4.2 that P n,t − ∩ I n − t J t = I n − t +1 J t . Note that I n − t +1 J t = I n − t +1 ⊗ k J t and I n − t J t = I n − t ⊗ k J t . By the hypothesis, the inclusionmap I n − t +1 → I n − t is Tor-vanishing. Since the tensor product with J t is an exact functor,the inclusion map I n − t +1 J t → I n − t J t is also Tor-vanishing. Similarly, the inclusion map I n − t +1 J t → I n − t +1 J t − is Tor-vanishing. Since I n − t +1 J t ⊆ I n − t +1 J t − ⊆ P n,t − , the inclusionmap I n − t +1 J t → P n,t − is also Tor-vanishing. By Lemma 5.2, it follows that P n,t = P n,t − + I n − t J t is a Betti splitting.Now we can apply Lemma 5.1 to obtaindepth R/P n,t = min { depth R/P n,t − , depth R/I n − t J t , depth R/I n − t +1 J t − } . Proceeding as in the proof of Theorem 4.2 we will get the desired conclusion. (cid:3)
Tor-vanishing filtrations can be found by the following result of Ahangari Maleki [1] in thecase char( k ) = 0. Let ∂ ( Q ) denote the the ideal generated by the partial derivatives of thegenerators of an ideal Q . Using the chain rule for taking partial derivatives, it is not hardto see that ∂ ( Q ) does not depend on the chosen generators of Q . Lemma 5.4. [1, Propostion 3.5]
Assume that char( k ) = 0 . Let Q ⊆ Q ′ be homogeneousideals such that ∂ ( Q ) ⊆ Q ′ . Then the inclusion map Q → Q ′ is Tor-vanishing. roposition 5.5. Let I be a nonzero proper homogeneous ideal in A . If char( k ) = 0 then ∂ ( I ( n ) ) ⊆ I ( n − for all n ≥ . Therefore, the filtration { I ( n ) } n ≥ is Tor-vanishing.Proof. Let Ass ∗ ( I ) := S ∞ n =1 Ass A ( I n ). This is a finite set due to a classical result of Brodmann[4]. Set L = \ P ∈ Ass ∗ ( I ) \ Min( I ) P, where L = A if Ass ∗ ( I ) = Min( I ). It follows from the definition of symbolic powers that I ( m ) = ∪ t ≥ I m : L t for all m ≥ t ≥ I ( n ) L t ⊆ I n . For any f ∈ I ( n ) and g ∈ L t +1 , we have f g ∈ I n . Let ∂ x denote the partial derivative with respect to an arbitrary variable x of A .We have ∂ x ( f ) g + f ∂ x ( g ) = ∂ x ( f g ) ∈ ∂ ( I n ) ⊆ I n − . Since ∂ x ( g ) ∈ ∂ ( L t +1 ) ⊆ L t , we have f ∂ x ( g ) ∈ f L t ⊆ I n . Therefore, ∂ x ( f ) g = ∂ x ( f g ) − f ∂ x ( g ) ∈ I n − . Hence, ∂ x ( f ) ∈ I n − : L t +1 ⊆ I ( n − . So wecan conclude that ∂ ( I ( n ) ) ⊆ I ( n − . (cid:3) Now, we can show that the bounds in Theorem 4.6 are actually the exact values of thedepth and the regularity of R/ ( I + J ) ( n ) if char( k ) = 0. Theorem 5.6.
Let I and J be nonzero proper homogeneous ideals in A and B . Assume that char( k ) = 0 . Then for all n ≥ , we have (i) depth R/ ( I + J ) ( n ) =min i ∈ [1 ,n − j ∈ [1 ,n ] (cid:8) depth A/I ( n − i ) + depth B/J ( i ) + 1 , depth A/I ( n − j +1) + depth B/J ( j ) (cid:9) , (ii) reg R/ ( I + J ) ( n ) =max i ∈ [1 ,n − j ∈ [1 ,n ] (cid:8) reg A/I ( n − i ) + reg B/J ( i ) + 1 , reg A/I ( n − j +1) + reg B/J ( j ) (cid:9) . Proof.
By Proposition 5.5, the filtrations { I ( i ) } i ≥ and { J ( j ) } j ≥ are Tor-vanishing. There-fore, the conclusion follows from Theorem 3.4 and Theorem 5.3. (cid:3) The following example shows that both equalities of Theorem 5.6 may fail if one of thebase rings is not regular, even if char( k ) = 0. Example 5.7.
By abuse of notations, we will denote the residue class of an element f by f itself. Let S = Q [ a, b, c, d ]. Put A = S/ ( a , a b, ab , b , a c − b d ) and I = ( a − b ) A .Computations with Macaulay2 [16] show that I (1) = I, I (2) = I = ( a b ) A . Hence, thereare isomorphisms of S -modules A/I (1) ∼ = S/ ( a − b , a , a b, a c − b d ) ,A/I (2) ∼ = S/ ( a , a b, a b , ab , b , a c − b d ) . his implies that depth( A/I (1) ) = depth(
A/I (2) ) = 1 , reg( A/I (1) ) = reg(
A/I (2) ) = 3 . Let B = Q [ x, y, z, t ] and J = ( x , x y, xy , y , x y ( xz − yt ) , x y ) ⊆ B . Computationswith Macaulay2 give J (1) = J = ( x , x y, xy , y , x y ( xz − yt ) , x y ), J (2) = ( x, y ) , anddepth( B/J (1) ) = 1 , depth( B/J (2) ) = 2 , reg( B/J (1) ) = 10 , reg( B/J (2) ) = 9 . Let R = A ⊗ k B and Q = I + J ⊆ R . Then R = Q [ a, b, c, d, x, y, z, t ] / ( a , a b, ab , b , a c − b d ) .Q = ( a − b , x , x y, xy , y , x y ( xz − yt ) , x y ) R. By Theorem 3.4, we have Q (2) = I (2) + I (1) J (1) + J (2) = ( a b , ( a − b )( x , x y, xy , y , x y ( xz − yt ) , x y , ( x, y ) ) R. It can be checked thatdepth(
R/Q (2) ) = 3 > A/I (2) ) + depth(
B/J (1) ) , reg( R/Q (2) ) = 13 <
14 = reg(
A/I (1) ) + reg(
B/J (1) ) + 1 . Hence, both equalities of Theorem 5.6 fail in this example.The Tor-vanishing of the symbolic powers in Proposition 5.5 can be considered as a higherorder generalization of the inclusion I ( n ) ⊆ m I ( n − , which was proved by Eisenbud andMazur for char( k ) = 0 [12, Proposition 13] and for I being a monomial ideal [12, Proposition9].Eisenbud and Mazur [12] conjectured that I (2) ⊆ m I (1) if I is a prime ideal in a power se-ries ring over a field of characteristic zero (see [20] for similar conjectures on higher symbolicpowers). They also showed that this conjecture has a negative answer in positive characteris-tic. Using their result we can construct the following example, which shows that Proposition5.5 does not hold in positive characteristic. Example 5.8.
Let R = ( Z / x, y, z, t ] and S = ( Z / u ]. Consider the kernel L of the map R → S given by x u , y u , z u , t u . Clearly, L is a prime ideal. Hence, L (1) = L . Computations with Macaulay2 [16] show that L is minimally generated by the following polynomials: g = x + y , g = yz + xt, g = x y + z , xz + t , x z + yt, xy + zt. The map Tor R ( k, L (2) ) → Tor R ( k, L ) is not zero. In fact, for f = x y − y − xz + t , we have x f = g g + g ∈ L , which shows that f ∈ L (2) . On the other hand, we have f / ∈ m R L ,where m R = ( x, y, z, t ).While the ideal L is not homogeneous in the standard grading, it is weighted homogeneousby setting deg x = 4 , deg y = 6 , deg z = 7 , deg t = 9 . Now we use the polarization trick of . McCullough and I. Peeva [27] to transform L into a homogeneous ideal in the standardgrading. Let A = ( Z / x , x , x , x , y , y , y , y ]. Consider the homogeneous map φ : R → A given by x x y , y x y , z x y , t x y . Let I be the extension of L to A . Then one can check with Macaulay2 that I is a primeideal of A , and more importantly, I is homogeneous in the standard grading of A . Let m A = ( x , . . . , x , y , . . . , y ). Then the relation x f = g g + g shows that φ ( f ) ∈ I (2) \ m A I .Hence, the map I (2) → I (1) is not Tor-vanishing. Using similar arguments and [12, Example,p. 200], we even have similar examples in any positive characteristic.We could not use Example 5.8 to construct any counterexample to the conclusion ofTheorem 5.6 in positive characteristics. We expect that Theorem 5.6 holds regardless of thecharacteristic of k .Our next main result shows that this is the case for monomial ideals. We shall first collectalternatives for Lemma 5.4 and Proposition 5.5 in this case, and then present the result inTheorem 5.11.For a monomial ideal Q we denote by ∂ ∗ ( Q ) the ideal generated by elements of the form f /x , where f is a minimal monomial generator of Q and x is a variable dividing f . Lemma 5.9. [29, Proposition 4.4 and Lemma 4.2]
Let Q ⊆ Q ′ be monomial ideals such that ∂ ∗ ( Q ) ⊆ Q ′ . Then the inclusion map Q → Q ′ is Tor-vanishing. Proposition 5.10.
Let I be a nonzero proper monomial ideal in A . Then ∂ ∗ ( I ( n ) ) ⊆ I ( n − for all n ≥ . Therefore, the filtration { I ( n ) } n ≥ is Tor-vanishing.Proof. Let Q , . . . , Q s be the primary components of I associated to the minimal primes of I . Note that Q , . . . , Q s are monomial ideals. By [22, Lemma 3.1], we have I ( n ) = Q n ∩ · · · ∩ Q ns . It is clear that ∂ ∗ ( Q n ) = ∂ ∗ ( Q ) Q n − ⊆ Q n − for any monomial ideal Q . Therefore, ∂ ∗ ( I ( n ) ) ⊆ ∂ ∗ ( Q n ) ∩ · · · ∩ ∂ ∗ ( Q ns ) ⊆ Q n − ∩ · · · ∩ Q n − s = I ( n − . By Lemma 5.9, this implies that { I ( n ) } n ≥ is Tor-vanishing. (cid:3) Theorem 5.11.
Let I and J be nonzero proper monomial ideals in A and B . Then theequalities of Theorem 5.6 hold regardless of the characteristic of k .Proof. By Proposition 5.10, both the filtrations { I ( i ) } i ≥ and { J ( j ) } j ≥ are Tor-vanishing.Therefore, the conclusion follows from Theorem 3.4 and Theorem 5.3. (cid:3) Our results on Tor-vanishing have some interesting consequences on the relationship be-tween the depth and the regularity of
A/I ( n − , A/I ( n ) , and I ( n − /I ( n ) . Proposition 5.12.
Let I be a nonzero proper homogeneous ideal in A . Assume that char( k ) =0 or I is a monomial ideal. Then for any n ≥ , (i) depth I ( n − /I ( n ) = min { depth A/I ( n − + 1 , depth A/I ( n ) } , (ii) reg I ( n − /I ( n ) = max { reg I ( n − , reg I ( n ) − } . roof. By Proposition 5.5 and Proposition 5.10, the inclusion map I ( n ) → I ( n − is Tor-vanishing, i.e. Tor Ai ( k, I ( n ) ) → Tor Ai ( k, I ( n − ) is the zero map for all i . From the exactsequence 0 → I ( n ) → I ( n − → I ( n − /I ( n ) → , it follows that the long exact sequence of Tor splits into short exact sequences0 → Tor Ai +1 ( k, I ( n − ) → Tor Ai +1 ( k, I ( n − /I ( n ) ) → Tor Ai ( k, I ( n ) ) → . Using the characterization of the depth and the regularity by Tor we can easily deduce theconclusion. (cid:3)
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