Gorenstein rings generated by strongly stable sets of quadratic monomials
GGorenstein rings generated by strongly stable sets ofquadratic monomials
Ralf Fr¨oberg and Lisa Nicklasson ∗ Stockholm University
Abstract
We characterize all Gorenstein rings generated by strongly stable sets of monomialsof degree two. We compute their Hilbert series in several cases, which also providesan answer to a question by Migliore and Nagel [10].
Strongly stable sets of monomials are an important tool in commutative algebra, and pro-vides a link to combinatorics. One reason for studying strongly stable sets of monomialsis the following. When studying graded algebras K [ x , . . . , x n ], a much used techniqueis to make a general change of coordinates and then determine the Gr¨obner basis ofthe transformed ideal. It is well known [6], that the initial ideal of this Gr¨obner basisis Borel fixed. In characteristic zero this is the same as strongly stable. (In positivecharacteristic strongly stable implies Borel fixed only.)In the context of Hilbert schemes, it is known that each component and each inter-section of components contains at least one point corresponding to a scheme defined bya Borel-fixed ideal, and these ideals can be used to understand its local structure, see[12], [9].Strongly stable sets also play an important role in the algebraic theory of shifting,see [8, Chapter 11].In [1] Boij and Conca study subrings K [ f , . . . , f r ] of the polynomial ring K [ x , . . . , x n ], f i of degree d , and are interested in, given n, r, d , how to choose the f i ’s in order to haveminimal Hilbert series. They show that the f i ’s should constitute a strongly stable setof monomials, but it is not clear which strongly stable sets that occur. The secondauthor of this paper made a thorough investigation of the case d = 2 [11]. A subringgenerated by a strongly stable set of quadratic monomials can be realized as a quotientby a polynomial ring and a determinantal ideal. This connection is also studied in [4].A nice feature of these rings is the combinatorial interpretation of their Hilbert series. ∗ (cid:0) : [email protected] a r X i v : . [ m a t h . A C ] A ug ings generated by strongly stable sets of monomials is an interesting topic in itself,and was studied by De Negri in [5]. In this paper we continue the study of subringsgenerated by strongly stable sets of quadratic monomials, asking when such a ring isGorenstein. Our result is a complete characterization of which strongly stable sets indegree two that give Gorenstein rings. We also provide explicit expressions of theirHilbert series in several cases. We find that Gorenstein rings are rather ubiquitous inour situation. Among other things we find lots of more Hilbert functions of Gorensteinideals generated by quadrics than the ones given in [10]. Let K be a field, and let R = K [ x , . . . , x n ] be the standard graded polynomial ring in n variables. Let R d denote the K -space of homogeneous polynomials of degree d in R . For alinearly independent subset W ⊆ R d , let K [ W ] ⊆ R be the subring of R generated by theelements in W . Define the Hilbert function of such an algebra K [ W ] as HF( K [ W ] , i ) =dim K (span W i ), and the Hilbert series of K [ W ] to be (cid:80) ∞ i =0 HF( K [ W ] , i ) t i . Definition 2.1.
A set W of monomials in R d is called strongly stable if m ∈ W and x i | m implies ( x j /x i ) m ∈ W for all j < i .We use the notation st( m , . . . , m s ) for the smallest strongly stable set containingthe monomials m , . . . , m s , and we say that m , . . . , m s are strongly stable generatorsof this set. We are interested in characterizing Gorenstein rings K [ W ] in the case when W is a strongly stable set of quadratic monomials. Strongly stable sets of quadraticmonomials can be illustrated by a shifted Ferrers diagram, as in Figure 1. The box inrow i and column j corresponds to the monomial x i x j . Since x i x j = x j x i we only needto consider boxes on and above the diagonal in the diagram. That the set is stronglystable means precisely that if the x i x j -box is included in the diagram, so is everythingabove and to the left of it. Figure 1: st( x x , x x )In searching for Gorenstein rings we will use the following theorem. Theorem 2.2. [14, Theorem 4.4] If a graded Cohen-Macaulay domain A has Hilbertseries (cid:80) ki =0 h i t i / (1 − t ) d , d = dim A , and the numerator is symmetric, i.e. h i = h k − i ,then A is Gorenstein.
2e will refer to ( h , . . . , h k ) as the h -vector of K [ W ]. Our algebras can be realizedas algebras defined by a symmetric ladder determinantal ideal. For a proof of this factsee e. g. [4, Theorem 4.2]. Conca, [3], studies the Hilbert series of such algebras. Thekey idea is to take an initial ideal of the ladder determinantal ideal, which then definesa Stanley-Reisner ring of a shellable simplicial complex. For Stanley-Reisner rings ofshellable simplicial complexes there is a combinatorial interpretation of the h -vectorknown as the McMullen-Walkup formula. This also proves that the rings are Cohen-Macaulay. The proof by Conca applies directly also in our case, and the result is thefollowing.Define an NE-path to be a lattice path in the diagram that can only go up or right(north or east). We say that an NE-path is maximal if it is of maximal length, whichimplies that it starts in x i (on the diagonal) for some i , and goes to x x n (the upperright corner). The dimension of K [ W ] is n , and h i is the number of maximal paths with i maximal N-parts. An N-part of a path is a subsequence x a x b − x a − x b − · · · − x a − m x b of N-steps, and it is maximal if it can’t be extended to a longer N-part of the path. Thedescription in [3] is slightly different but equivalent. Instead of maximal N-parts, h i iscounted by the number of corners . A corner, in this context, is the same as the startingpoint of a N-part. Example 2.3. If W is as in Figure 1, h = 1, since x − x x − x x − x x − x x − x x is the only maximal path without N-parts. We have h = 7 corresponding to the paths x − x x − x x − x x − x x − x x , x − x x − x x − x x − x x − x x ,x − x x − x x − x x − x x − x x , x − x x − x x − x x − x x − x x ,x − x x − x x − x x − x x − x x , x − x x − x x − x x − x x − x x , and x − x x − x x − x x − x x − x x . We have h = 5 corresponding to the paths x − x x − x x − x x − x x − x x , x − x x − x x − x x − x x − x x ,x − x x − x x − x x − x x − x x , x − x x − x x − x x − x x − x x , and x − x x − x x − x x − x x − x x . There is no path with more than two maximal N-parts. Thus the Hilbert series is(1 + 7 t + 5 t ) / (1 − t ) . (cid:52) For a diagram of a strongly stable set V , we define a partial ordering by ( a , b ) ≤ ( a , b ) if a ≤ a and b ≤ b . The corners of a NE-path in the diagram constitutes anantichain (i.e. sets of points with no relation between any two) in this partial ordering.Hence h i can be determined as the number of antichains of size i in the diagram of V with the top row deleted. We state this fact as a lemma. Lemma 2.4.
Let W be a strongly stable set of quadratic monomials, and let (cid:80) ki =0 h i t i / (1 − t ) d be the Hilbert series of K [ W ] . Then h i equals the number of antichains of size i ofthe diagram of W with the top row removed. , , , , , , , { (3 , , (2 , } , { (3 , , (2 , } , { (3 , , (2 , } , { (3 , , (2 , } , and { (3 , , (2 , } . For a strongly stable set W ⊂ R we may assume, when searching for Gorenstein rings,that x x n is not the only monomial in the last column. Indeed, it follows from Lemma2.4 that we may add or remove boxes as we like in the first row of the diagram (as longas the diagram still corresponds to a strongly stable set) without affecting the h -vector.The number of boxes in the first row determines the dimension of K [ W ]. As it is naturalto work with lowest possible dimension, we will assume from now on that the first rowhas as few boxes as possible. This means that the last columns has at least two boxes.Or in algebraic terms, we are assuming that x x n ∈ W . We say that K [ W ] has no freevariable.Let V k = st( x k +1 , x k x k +2 , x k − x k +3 , . . . , x x k ). Theorem 3.1.
The ring K [ V k ] is Gorenstein with Hilbert series (cid:80) ki =0 (cid:0) ki (cid:1) t i / (1 − t ) k .Proof. The diagram of V k with the top row deleted is a triangle with 1 + 3 + 5 + · · · +2 k − k points. There are (cid:0) ki (cid:1) antichains of length i , by [13, Theorem 1] and[16, Corollary 2.4]. The result also follows from Exercise 3.47(f) in [15]. We thankRichard Stanley for informing us about this. Since (cid:80) ki =0 (cid:0) ki (cid:1) t i is symmetric, the ring isGorenstein.As we always have h = 1, a necessary condition for Gorenstein is h k = 1. Lemma 3.2.
Let W be a strongly stable set of quadratic monomials, and suppose that K [ W ] has no free variable. If K [ W ] is Gorenstein with Hilbert series h ( t ) / (1 − t ) n , deg ( h ( t )) = k , then n = 2 k and V k ⊆ W ⊆ st( x k ) .Proof. An NE-path with k maximal N-parts must start in x i for some i > k , so x k +1 ∈ W . Such a path must also have at least k − n ≥ k . The alternating path P = x k +1 − x k x k +1 − x k x k +2 − · · · − x x k has k maximal N-parts, and every other pathwith k maximal N-parts must lie to the right of P . Thus, if h k = 1, then P is the uniquepath with k maximal N-parts. It follows that V k ⊆ W and n = 2 k , so W ⊆ st( x k )since V = st( x k ) is the largest set with n = 2 k .Let D = { x i x j | i + j = n + 2 } , so that if n = 2 k the set D is the set of stronglystable generators of V k . Theorem 3.3.
Let W be a strongly stable set of quadratic monomials. The ring K [ W ] is Gorenstein if and only if for some k we have W = V k or W = V k ∪ st( m , . . . , m t ) ⊂ K [ x , x , . . . , x k ] where m , . . . , m t are monomials satisfying the following conditions,for any ≤ r, s ≤ t . . m r = x i x j where i ≤ k + 1 < j or i = j > k + 1 .2. st( m r ) ∩ st( m s ) ∩ D = ∅ Before we prove Theorem 3.3 we shall introduce the
Narayana numbers N ( k, i ) = 1 k (cid:18) ki (cid:19)(cid:18) ki − (cid:19) . The Narayana number N ( k, i ) counts the number of Dyck paths of length 2 k with i peaks.In our setting N ( k, i ) is the number of maximal NE-paths in V k with i maximal N-partsstarting in x k +1 . Notice that the Narayana numbers satisfies the relation N ( k, i ) = N ( k, k − i + 1). Lemma 3.4.
The ring K [ V k ∪ st( x j )] , for k + 1 ≤ j ≤ k is Gorenstein. The proof of Lemma 3.4 is illustrated in Figure 2.Figure 2: V k ∪ st( x j ) in the proof of Lemma 3.4 Proof.
The proof is by induction over j . As x k +1 ∈ V k the base case j = k + 1 isTheorem 3.1. We now want to compute how many maximal NE-paths with i maximal5-parts are added when we extend V k ∪ st( x j − ) to V k ∪ st( x j ). The paths in V k ∪ st( x j ) that go outside of V k ∪ st( x j − ) are precisely that paths including the N-step x k − j +3 x j − x k − j +2 x j .Let us first consider the NE-paths to x k − j +2 x j where the last step is an N-step.The NE-paths from x (cid:96) , for some (cid:96) , to x k − j +2 x j are all NE-paths of length 2( j − k − i maximal N-parts not counting the last N-part. To do this we choose 2 i + 1 numbersbetween 1 and 2( j − k − (cid:18) j − k − i + 1 (cid:19) . Next we consider the NE-paths from x k − j +2 x j to x x k +1 . Such a path must startwith an N-step, which then continues the N-part in the end of the path to x k − j +2 x j . Thenumber of paths from x k − j +2 x j to x x k +1 with i maximal N-parts is N (2 k − j + 1 , i ).We can now conclude that the number of maximal NE-paths with i = i + i maximalN-parts and including the step x k − j +3 x j − x k − j +2 x j is (cid:88) (cid:18) j − k − i + 1 (cid:19) N (2 k − j + 1 , i )where the sum is over all i , i such that i + i = i , 0 ≤ i ≤ j − k −
2, and 1 ≤ i ≤ k − j + 1. We get the same result if we replace i by j − k − − i and i by 2 k − j + 2 − i since (cid:18) j − k − j − k − − i ) + 1 (cid:19) = (cid:18) j − k − i + 1 (cid:19) and N (2 k − j + 1 , k − j + 2 − i ) = N (2 k − j + 1 , i ) . In that case we are counting the NE-paths with( j − k − − i ) + (2 k − j + 2 − i ) = k − i maximal N-steps, so we can conclude that the h -vector is symmetric.We are now ready to prove that the conditions in Theorem 3.3 are sufficient. Theproof is illustrated in Figure 3. Proof of the “if ”-part of Theorem 3.3.
We shall now prove that a strongly stable set W = V k ∪ st( m , . . . , m t )with m , . . . , m t as stated in Theorem 3.3 defines a Gorenstein ring. Notice that at mostone of m , . . . , m t can be a square, as st( x i ) ⊂ st( x j ) if i < j . If t = 1 and m = x j theresult is proved in Lemma 3.4. We assume from now on that t > t = 1 but m is6igure 3: The fixed steps in the “if”-part of Theorem 3.3.not a square. We proceed by induction over t . We may order m , . . . , m t so that m t sitsabove and to the right of m , . . . , m t − in the diagram.Suppose the strongly stable set W (cid:48) = V k ∪ st( m , . . . , m t − ) defines a Gorensteinring. If t = 1 this should be interpreted as W (cid:48) = V k , which indeed is Gorenstein. Wewant to prove that when we extend W (cid:48) to W (cid:48) ∪ st( m t ) = W the number of new pathswith i N-parts and the number of new paths with k − i N-parts are the same.Let us fix an E-step x k +1 − b x k +1+ b − x k +1 − b x k +2+ b and an N-step x k +2 − a x k +1+ a − x k +1 − a x k +1+ a inside st( m t ), such that a > b . The idea is to consider paths that staysinside W (cid:48) before the fixed E-step and after the fixed N-step. Every path in W that goesoutside W (cid:48) can be described in this way for a unique pair a, b . Hence is it enough toprove the statement for a fixed a and b .Let us first consider paths to x k +1 − b x k +1+ b in W (cid:48) with i maximal N-parts. Letus call the number of such paths β ( i ). This is the same as the number of paths with i maximal N-parts in V b ∪ st( m , . . . , m t − ) which defines a Gorenstein ring, by theinduction hypothesis. It follows that β ( i ) = β ( b − i ).Next we consider paths from x k +1 − b x k +2+ b to x k +2 − a x k +1+ a with i maximal N-parts. If the path ends with an N-step we will not count the last N-part, as it will becounted in the next step of the proof. These paths stay inside a square with side a − b .To do this we choose i points that represent the endpoints of the maximal N-parts. We7o not choose points from the rightmost column or the bottom row in the square. Thereare (cid:0) a − b − i (cid:1) ways to choose these points.Last we consider paths from x k +1 − a x k +1+ a to x x k with i maximal N-parts. Thenumber of such paths is N ( k − a, i ). We can now conclude that the number of maximalNE-paths with i N-parts and a fixed E-step x k +1 − b x k +1+ b − x k +1 − b x k +2+ b and N-step x k +2 − a x k +1+ a − x k +1 − a x k +1+ a is (cid:88) β ( i ) (cid:18) a − b − i (cid:19) N ( k − a, i )where the sum is over all i , i , i such that i + i + i = 1, 0 ≤ i ≤ b , 0 ≤ i ≤ a − b − ≤ i ≤ k − a . Notice that we obtain the same result if we replace i by b − i , i by a − b − − i , and i by k − a + 1 − i . In that case we are counting the paths with( b − i ) + ( a − b − − i ) + ( k − a + 1 − i ) = k − i maximal N-parts. As this holds for all possible choices of a and b , we have now provedthat the number of new paths with i N-parts and the number of new paths with k − i N-parts are the same, when we extend W (cid:48) to W .Before we prove the ”only if” part, we give an example. Suppose k = 5 andthat st( m , . . . , m t ) ∩ D = { x x , x x , x x } , so { m , . . . , m t } is either { x x , x x } or { x x } . We will look for a condition for h = h k − = h . The paths with fourmaximal N-parts which are not paths in V all starts with x − x x − x x and allends with x x − x x − x x − x x . There are three possibilities between x x and x x , namely x x − x x − x x − x x − x x , x x − x x − x x − x x − x x , or x x − x x − x x − x x − x x . Thus h increases with three when we extend V by st( x x , x x ) or st( x x ). The increase in h is either or two or three for the twopossibilities. Proof of the “only if ”-part of Theorem 3.3.
Let st( m , . . . , m t ) ∩ D consist of disjointparts D m , where D m = { x i x j ; i + j = n + 2 , a m ≤ i ≤ b m } , and suppose D m = V k ∪ st( m i , . . . , m i s ) ∩ D . The increase in h from V k to V k ∪ st( m i , . . . , m i s ) is thenumber of points in st( m i , . . . , m i s ) \ V k . This number is ≤ (cid:0) b m − a m (cid:1) , with equality ifand only if s = 1. The increase in h k − is always (cid:0) b m − a m (cid:1) . A path of maximal length2 k − k − k − V k and does not contributeto the increase of h k − . Thus we have to choose two points in st( m i , . . . , m i s ) ∩ D , onewhen we enter st( m i , . . . , m i s ) \ V k , and one when we leave st( m i , . . . , m i s ) \ V k . Thiscan be done in (cid:0) b m − a m (cid:1) ways. An increase in h or h k − can only involve one D m so thetotal increase is the sum for each m . This finishes the proof. Corollary 3.5.
For the rings R we consider, generated by a strongly stable set ofquadratic monomials, we have that R is Gorenstein if and only if h i = h k − i for i = 0 and i = 1 . orollary 3.6. If k [ W ] is Gorenstein and has no free variable, then dim( k [ W ]) is even. Example 3.7.
In [10] the authors construct Gorenstein algebras with h -vectors (1 , s + t, st + 2 , s + t,
1) for each s, t ≥ h -vectors (1 , s + t, st + t + 1 , st + t + 1 , s + t, s ≥ t ≥
3. In Question 2.12 they ask if these are the only h -vectors forArtinian Gorenstein rings of socle degrees 4 and 5. We have calculated the h -vectors forour rings of dimension 2 k with k ≤ h -vector and socle degree k . Infact, all our examples with k = 4 or k = 5 give counterexamples to their question. (cid:52) We have seen that if K [ W ] is Gorenstein with no free variable and has Hilbert series h ( t ) / (1 − t ) dim K [ W ] , where deg( h ( t )) = k , then dim K [ W ] = 2 k and W lies between V k and st( x k ). Both extreme sets give Gorenstein rings by Theorem 3.3. In fact, K [st( x k )]is the second Veronese subring of K [ x , . . . , x k ], and it was proved to be Gorenstein in[7, Theorem 3.2.1]. The Hilbert series of K [st( x dn )] was computed for d = 2 in [3] andfor general d in [2]. For d = 2 one can easily verify that the h -vector is symmetric if andonly if n is even. Theorem 3.8. K [st( x n )] has Hilbert series (cid:80) (cid:98) n/ (cid:99) i =1 (cid:0) n i (cid:1) t i / (1 − t ) n . It is Gorenstein ifand only if n is even. Corollary 3.9.
In the triangular area consisting of the integer points in { ( x, y ); y ≥ , ≤ x ≤ n − , y ≤ x } with the partial order ( a, b ) ≤ ( c, d ) if a ≤ c and b ≤ d , thereare (cid:0) n i (cid:1) antichains of size i , ≤ i ≤ n/ .Proof. Consider the picture for st( x n ) with the top row deleted. If we move this onestep to the north and one step to the west we get the same number of antichains of eachsize. The Hilbert series of K [st( x n )] is (cid:80) ≤ i ≤ n/ (cid:0) n i (cid:1) t i / (1 − t ) n , [2]. Then use Lemma2.4.We will finish by computing the Hilbert series of two more classes of Gorenstein ringsfrom Theorem 3.3. Theorem 3.10.
The ring K [st( x x k , x k − )] is Gorenstein with Hilbert series k (cid:88) i =0 (cid:18)(cid:18) k − i (cid:19) + (cid:18) k − i − (cid:19)(cid:19) t i (1 − t ) k . Proof.
All maximal NE-paths in the diagram of st( x x k , x k − ) end with a step E orsteps EN. The paths ending with an E can be seen as paths in the diagram of st( x k − )followed by an E. The paths ending with EN can be seen as paths in the diagram ofst( x k − ) followed by EN. The h -vector now follows from Theorem 3.8.The strongly stable sets in the next theorem are precisely those obtained from V k by adding one monomial. 9 heorem 3.11. The ring K [ V k ∪ st( x a x k +3 − a )] , with ≤ a ≤ k + 1 , is Gorensteinwith Hilbert series k (cid:88) i =0 (cid:18) ki (cid:19) t i + 1 a − k − a +1 (cid:88) j =0 (cid:18) k − a + 1 j (cid:19) t j (cid:32) a − (cid:88) m =0 (cid:18) a − m (cid:19)(cid:18) a − m − (cid:19) t m (cid:33) (1 − t ) k . Proof.
Let W = V k ∪ st( x a x k +3 − a ). As mentioned in the paragraph before the theorem W = V k ∪ { x a x k +3 − a } . A maximal NE-path in W that goes outside of V k mustcontain the EN steps x a x k +2+ a − x a x k +3 − a − x a − x k +3 − a . Hence an NE-path thatgoes through x a x k +3 − a consists of a path P from some x b to x a x k +2+ a and a path Q from x a − x k +3 − a to x x k . If P has j maximal N-parts and Q has m maximal N-partsthen the whole path has j + m maximal N-parts. The path P can be considered as apath in V k − a +1) . The number of choices for Q with m maximal N-parts is given bythe Narayana number N ( a − , m ). Applying Theorem 3.1 we get the numerator of theHilbert series of K [ V k ∪ st( x a x k +3 − a )] as k (cid:88) i =0 (cid:18) ki (cid:19) t i + k − a +1 (cid:88) j =0 (cid:18) k − a + 1 j (cid:19) t j (cid:32) a − (cid:88) m =0 N ( a − , m ) t m (cid:33) . eferences [1] M. Boij and A. Conca, On the Fr¨oberg-Macaulay conjectures for algebras , Rend. Istit.Mat. Univ. Trieste, The Veronese construction for formal power series andgraded algebras , Adv. Appl. Math., (4) 545-556 (2009)[3] A. Conca, Symmetric ladders , Nagoya Math. J., , 35–56 (1994)[4] A. Corso, U. Nagel, S. Petrovi´c, and C. Yen,
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