On Two Classes of Closely Related Monomial Ideals
aa r X i v : . [ m a t h . A C ] J un On Two Classes of Closely Related Monomial Ideals
Maorong Ge, Jiayuan Lin and Yulan Wang
Abstract
In [7] we obtained a formula for the Hilbert depth of squarefree Veronese ideals ina standard graded polynomial ring by relating it to the Hilbert depth of powers of the irrelevantmaximal ideal. In this paper, we prove that these two Hilbert depth formulas are equivalent toeach other. Our result reveals that there is a strong connection between these two classes ofseemingly unrelated monomial ideals. We conjecture that their Stanley depths are equivalent aswell.
In recent years, two classes of monomial ideals, the squarefree Veronese ideals and thepowers of the irrelevant maximal ideal in a graded polynomial ring, have received specialattentions (e.g. [1], [3]-[8]). The
Stanley depths of these two classes of ideals have beenconjectured in [4], [5] and [8] and partially confirmed in [1], [4], [5], [6] and [8]. Their
Hilbert depths have been obtained in [3] and [7]. At the beginning it seemed that therewere no apparent connections between these two classes of ideals. Most results were derivedsolely on one of them. However, in [7] we computed the Hilbert depth of the squarefreeVeronese ideals and found that it could be reduced to the calculation of the Hilbert depth ofthe powers of the irrelevant maximal ideal. In this paper we further exploit this connectionand show that the two formulas for the Hilbert depths of these two classes of ideals areactually equivalent.Let K be a field and R = K [ x , · · · , x n ] be the polynomial ring in n variables. Let M bea finitely generated graded R -module. Analogous to the concepts of Stanley decomposition and
Stanley depths , W. Bruns et al [2] introduced
Hilbert decomposition and
Hilbert depth .For the reader’s convenience, let us recall the definition of
Hilbert decomposition and
Hilbertdepth from [2].
Definition 1.1.
Let M be a finitely generated graded S -module. A Hilbert decomposition of M is a finite family H = (cid:0) S i , s i (cid:1) such that s i ∈ Z m (where m = 1 in the standard graded case and m = n in the multigradedcase), S i is a graded K -algebra retract of R for each i , and M ∼ = M i S i ( − s i )as a graded K -vector space.The number Hdepth H = min i depth S i ( − s i ) is called the Hilbert depth of H . The Hilbert depth of M is defined to beHdepth M = max { Hdepth H : H is a Hilbert decomposition of M } . R -module M = L k ∈ Z M k , the coarse Hilbert series of M is given by the Laurent series H M ( T ) = P k ∈ Z H ( M, k ) T k , where H ( M, k ) = dim K M k is the Hilbert function of M . We say a Laurent series P k ∈ Z a k T k is non-negative if a k ≥ k . It is easy to see that any Hilbert series is non-negative. In [9], J. Uliczka proved that the Hilbert depth of M is equal to max { r :(1 − T ) r H M ( T ) non-negative } . Using this, W. Bruns et al [3] computed the Hilbert depthof the powers of the irrelevant maximal ideal in R . Using the result of [3], we obtainedin [7] that the coarse Hilbert series of the squarefree Veronese ideal I n,d is H I n,d ( T ) = n − P i = d − (cid:0) id − (cid:1) T d (1 − T ) − n + i − d +1 , and its Hilbert depth is equal to d + (cid:4)(cid:0) nd +1 (cid:1) / (cid:0) nd (cid:1)(cid:5) = d + (cid:4) n − dd +1 (cid:5) = d − l n − ( d − d +1 m . We remark that the numbers in the formulas for their Hilbert depth ofboth classes of ideals are exactly those appeared in the corresponding conjectural formulasfor their Stanley depths. We expect the following to be true. Conjecture 1.2.
Let R = K [ x , · · · , x n ] be the polynomial ring in n variables. Let I n,d be the squarefree Veronese ideal generated by all squarefree monomials of degree d and let m be the irrelevant maximal ideal in R . Then the formula Sdepth( m s ) = (cid:6) ns +1 (cid:7) implies Sdepth( I n,d ) = d − l n − ( d − d +1 m and vice versa, where s is a positive integer. In this paper we prove a Hilbert depth version of Conjecture 1 . Theorem 1.3.
Let I n,d be the squarefree Veronese ideal generated by all squarefree mono-mials of degree d and let m be the irrelevant maximal ideal in R = K [ x , · · · , x n ] . Then theformulas Hdepth( m s ) = (cid:6) ns +1 (cid:7) and Hdepth( I n,d ) = d − l n − ( d − d +1 m are equivalent, that is,they imply each other, where s is a positive integer. In Proposition 1 . . Theorem 1.4.
Let I n,d be the squarefree Veronese ideal generated by all squarefree mono-mials of degree d and let m be the irrelevant maximal ideal in R = K [ x , · · · , x n ] . Let ˆ m = m ∩ ˆ R be the irrelevant maximal ideal in ˆ R = K [ x , · · · , x n − d +1 ] and ˆ m d be the idealgenerated by all monomials of degree d in ˆ R . Then Hdepth( ˆ m d ) + d − I n,d ) . Proposition 1.5.
Theorem . implies Theorem . .Proof. Suppose that Hdepth m s = (cid:6) ns +1 (cid:7) is true for any positive integer s and the irrelevantmaximal ideal m in R = K [ x , · · · , x n ]. Then applying this formula to ˆ m = m ∩ ˆ R inˆ R = K [ x , · · · , x n − d +1 ] for s = d gives that Hdepth( ˆ m d ) = (cid:6) n − d +1 d +1 (cid:7) = l n − ( d − d +1 m . ByTheorem 1 .
4, we have Hdepth( I n,d ) = Hdepth( ˆ m d ) + d − l n − ( d − d +1 m + d − I n,d ) = d − l n − ( d − d +1 m is true for any n ≥ d .Applying this formula to the squarefree Veronese ideal I n + s − ,s generated by all squarefree2onomials of degree s in the polynomial ring K [ x , · · · , x n , x n +1 , · · · , x n + s − ], we have thatHdepth( I n + s − ,s ) = s − l n + s − − ( s − s +1 m = s − (cid:6) ns +1 (cid:7) . By Theorem 1 .
4, Hdepth( m s ) + s − I n + s − ,s ) = s − (cid:6) ns +1 (cid:7) . So Hdepth( m s ) = (cid:6) ns +1 (cid:7) . This completes theproof of Proposition 1 . .
4. The proof of Theorem 1 . I n,d (as a Z n -graded module) in R = K [ x , · · · , x n ] and ˆ m d (as a Z n − d +1 -graded module) inˆ R = K [ x , · · · , x n − d +1 ]. Then we show that H I n,d ( T ) = (1 − T ) − ( d − H ˆ m d ( T ). Theorem 1 . M ) =max { r : (1 − T ) r H M ( T ) non-negative } . Denote S the set of subsets of { , · · · , n } with cardinality at least d . We first deduce thefine Hilbert series of the squarefree Veronese ideal I n,d . Proposition 2.1.
The fine Hilbert series of the squarefree Veronese ideal I n,d is given by H I n,d ( T , · · · , T n ) = n Y i =0 (1 − T i ) − X S ∈S T S (1 − T ) S c , where S c is the complement of S in the set { , · · · , n } , T S = Q i ∈ S T i , and (1 − T ) S c = Q j ∈ S c (1 − T j ) .Proof. The squarefree Veronese ideal I n,d is generated by all squarefree monomials of degree d in R = K [ x , · · · , x n ]. A monomial x α · · · x α n n is in I n,d if and only if at least d of the α i ’s are positive. So the fine Hilbert series of the squarefree Veronese ideal I n,d is given by H I n,d ( T , · · · , T n ) = X at least d α i > T α · · · T α n n . By arranging the terms containing the same distinct factors in H I n,d ( T , · · · , T n ) intothe same group, we have that H I n,d ( T , · · · , T n ) = X S ∈S T S ( X α i ≥ Y i ∈ S T α i i ) = X S ∈S Y i ∈ S T i − T i = n Y i =0 (1 − T i ) − X S ∈S T S (1 − T ) S c . T i = T in Proposition 2 .
1, we immediately have that the coarse Hilbert series ofthe squarefree Veronese ideal is H I n,d ( T ) = P S ∈S Q i ∈ S T − T = n P k = d (cid:0) nk (cid:1) T k (1 − T ) k . Recall that we haveobtained H I n,d ( T ) = n − P i = d − (cid:0) id − (cid:1) T d (1 − T ) − n + i − d +1 in [7]. We will show that these twoformulas are equal. We need the following lemma. Lemma 2.2.
For any integer ≤ i ≤ n − d , we have (cid:18) i + d − i (cid:19) = i X l =0 (cid:18) ni − l (cid:19) ( − l (cid:18) n − d − i + ll (cid:19) . Proof.
The coefficient of T i in (1 + T ) n · (1 + T ) − ( n − d − i +1) is i X l =0 (cid:18) ni − l (cid:19)(cid:18) − ( n − d − i + 1) l (cid:19) = i X l =0 (cid:18) ni − l (cid:19) ( − l (cid:18) n − d − i + ll (cid:19) . Now Lemma 2 . T i on both sides of theidentity (1 + T ) i + d − = (1 + T ) n · (1 + T ) − ( n − d − i +1) . Proposition 2.3. n P k = d (cid:0) nk (cid:1) T k (1 − T ) − k = n − P i = d − (cid:0) id − (cid:1) T d (1 − T ) − n + i − d +1 . Proof.
It is sufficient to show that n X k = d (cid:18) nk (cid:19) T k (1 − T ) n − k = n − X i = d − (cid:18) id − (cid:19) T d (1 − T ) i − d +1 . (2.1)Dividing both sides of Equation (2 .
1) by T d and re-numerating indices, we only needto prove n − d X k =0 (cid:18) nk + d (cid:19) T k (1 − T ) n − k − d = n − d X i =0 (cid:18) i + d − d − (cid:19) (1 − T ) i . (2.2)Expanding T k = [1 − (1 − T )] k and combining like terms on the left-hand side of Equation(2 . n − d P k =0 (cid:0) nk + d (cid:1) T k (1 − T ) n − k − d = n − d P k =0 (cid:0) nk + d (cid:1) k P l =0 ( − l (cid:0) kl (cid:1) (1 − T ) l (1 − T ) n − k − d = n − d P i =0 P k − l = n − d − i (cid:0) nk + d (cid:1) ( − l (cid:0) kl (cid:1) (1 − T ) i = n − d P i =0 i P l =0 (cid:0) nn + l − i (cid:1) ( − l (cid:0) n − d − i + ll (cid:1) (1 − T ) i = n − d P i =0 i P l =0 (cid:0) ni − l (cid:1) ( − l (cid:0) n − d − i + ll (cid:1) (1 − T ) i = n − d P i =0 (cid:0) i + d − i (cid:1) (1 − T ) i = n − d P i =0 (cid:0) i + d − d − (cid:1) (1 − T ) i . The lasttwo equalities follows from Lemma 2 . (cid:0) i + d − i (cid:1) = (cid:0) i + d − d − (cid:1) . This completes the proof ofProposition 2 .
3. 4
Fine and coarse Hilbert series of the powers of theirrelevant maximal ideal
Let m be the irrelevant maximal ideal in R = K [ x , · · · , x n ] and let ˆ m t = m ∩ ˆ R be theirrelevant maximal ideal in ˆ R = K [ x , · · · , x n − t +1 ]. Let ˆ m ts be the ideal generated by allmonomials of degree s in ˆ R and h ˆ m ts i be the ideal generated by ˆ m ts in R . We will deducethe fine and coarse Hilbert series of m s , ˆ m ts and h ˆ m ts i . All of those results are well-known.For the reader’s convenience, we include these formulas with a short proof here. Proposition 3.1.
The fine Hilbert series of m s , ˆ m ts and h ˆ m ts i are H m s ( T , · · · , T n ) = n Q i =0 (1 − T i ) − − s − P k =0 P P i α i = k T α · · · T α n n ,H ˆ m ts ( T , · · · , T n − t +1 ) = n − t +1 Q i =0 (1 − T i ) − − s − P k =0 P P i α i = k T α · · · T α n − t +1 n − t +1 , and H h ˆ m ts i ( T , · · · , T n ) = H ˆ m ts ( T , · · · , T n − t +1 ) · n Q i = n − t +2 (1 − T i ) − .Proof. A monomial x α · · · x α n n is in m s if and only if P i α i ≥ s . So the fine Hilbert series of m s is given by H m s ( T , · · · , T n ) = P P i α i ≥ s T α · · · T α n n = P α i ≥ T α · · · T α n n − s − P k =0 P P i α i = k T α · · · T α n n = n Q i =0 (1 − T i ) − − s − P k =0 P P i α i = k T α · · · T α n n . H ˆ m ts ( T , · · · , T n − t +1 ) follows easily from H m s ( T , · · · , T n ) by replacing n with n − t + 1.Note that a monomial x α · · · x α n n is in h ˆ m ts i if and only if n − t +1 P i =1 α i ≥ s . So H h ˆ m ts i ( T , · · · , T n ) = P n − t +1 P i =1 α i ≥ s T α · · · T α n n = P n − t +1 P i =1 α i ≥ s T α · · · T α n − t +1 n − t +1 · P α i ≥ n Q i = n − t +2 T α i i = H ˆ m ts ( T , · · · , T n − t +1 ) · n Q i = n − t +2 (1 − T i ) − .Let T i = T in Proposition 3 .
1, we immediately have
Corollary 3.2.
The coarse Hilbert series of m s , ˆ m ts and h ˆ m ts i are H m s ( T ) = (1 − T ) − n − s − P k =0 (cid:0) n + k − k (cid:1) T k , H ˆ m t s ( T ) = (1 − T ) − n + t − − s − P k =0 (cid:0) n − t + kk (cid:1) T k and H h ˆ m t s i ( T ) = (1 − T ) − n − (1 − T ) − t +1 s − P k =0 (cid:0) n − t + kk (cid:1) T k Now we are ready to prove Theorem 1 . Proof of Theorem . In order to prove Theorem 1 .
4, we need the following lemma.
Lemma 4.1.
For any non-negative integer k , we have (cid:18) n + kk + d (cid:19) = n − X i = d − (cid:18) id − (cid:19)(cid:18) n − i + k − k (cid:19) (4.1) Proof.
The coefficient of T n − d in the expansion of (1 − T ) − ( d + k +1) is equal to (cid:0) d + k +1+ n − d − n − d (cid:1) = (cid:0) n + kn − d (cid:1) = (cid:0) n + kd + k (cid:1) . For any 0 ≤ i ≤ n − d , the coefficient of T i in (1 − T ) − d and the coefficientof T n − d − i in (1 − T ) − ( k +1) are (cid:0) d + i − i (cid:1) = (cid:0) i + d − d − (cid:1) and (cid:0) k +1+ n − d − i − n − d − i (cid:1) = (cid:0) k + n − d − ik (cid:1) respectively.So the coefficient of T n − d in the expansion of (1 − T ) − ( d + k +1) = (1 − T ) − d · (1 − T ) − ( k +1) is equal to n − d P i =0 (cid:0) i + d − d − (cid:1)(cid:0) k + n − d − ik (cid:1) = n − P i = d − (cid:0) id − (cid:1)(cid:0) n − i + k − k (cid:1) . Now Lemma 4 . T n − d in the expansion of both sides of (1 − T ) − ( d + k +1) =(1 − T ) − d · (1 − T ) − ( k +1) .By J. Uliczka [9], the Hilbert depth of M is equal to max { r : (1 − T ) r H M ( T ) non-negative } .So to prove Theorem 1 .
4, it is sufficient to show that H I n,d ( T ) = (1 − T ) − ( d − H ˆ m d ( T ). ByCorollary 3 . H ˆ m d ( T ) = H ˆ m d d ( T ) = (1 − T ) − n + d − − d − P k =0 (cid:0) n − d + kk (cid:1) T k . Hence we only need toshow that n X k = d (cid:18) nk (cid:19) T k (1 − T ) − k = (1 − T ) − n − (1 − T ) − ( d − d − X k =0 (cid:18) n − d + kk (cid:19) T k (4.2)Multiply both sides of Equation (4 .
2) by (1 − T ) n and using Equation (2 . n − X i = d − (cid:18) id − (cid:19) T d (1 − T ) i − d +1 = 1 − (1 − T ) n − d +1 d − X k =0 (cid:18) n − d + kk (cid:19) T k (4.3)Dividing both sides of Equation (4 .
3) by (1 − T ) n − d +1 and expanding (1 − T ) i − n and(1 − T ) − ( n − d +1) respectively, we have n − X i = d − (cid:18) id − (cid:19) T d ∞ X k =0 (cid:18) n − i + k − k (cid:19) T k = ∞ X k = d (cid:18) n − d + kk (cid:19) T k (4.4)Dividing Equation (4 .
4) by T d , we have ∞ X k =0 n − X i = d − (cid:18) id − (cid:19)(cid:18) n − i + k − k (cid:19) T k = ∞ X k = d (cid:18) n − d + kk (cid:19) T k − d = ∞ X k =0 (cid:18) n + kk + d (cid:19) T k (4.5)6quation (4 .
5) holds true if and only if (cid:0) n + kk + d (cid:1) = n − P i = d − (cid:0) id − (cid:1)(cid:0) n − i + k − k (cid:1) . The latter onefollows from Lemma 4 .
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DEPARTMENT OF MATHEMATICS, ANHUI UNIVERSITY, HEFEI, ANHUI, 230039, CHINA
E-mail address: [email protected]
DEPARTMENT OF MATHEMATICS, SUNY CANTON, 34 CORNELL DRIVE, CANTON,NY 13617, USA
E-mail address: [email protected]
DEPARTMENT OF MATHEMATICS, SUNY CANTON, 34 CORNELL DRIVE, CANTON,NY 13617, USADEPARTMENT OF MATHEMATICS, ANHUI ECONOMIC MANAGEMENT INSTITUTE, HEFEI,ANHUI, 230059, CHINA