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In ( Stanley, 1978 ), Stanley constructs an example of an Artinian Gorenstein (AG) ring A with non-unimodal H -vector ( 1 , 13 , 12 , 13 , 1 ) . Migliore-Zanello show in ( Migliore and Zanello, 2017 ) that for regularity r = 4 , Stanley's example has the smallest possible codimension c for an AG ring with non-unimodal H -vector. The weak Lefschetz property (WLP) has been much studied for AG rings; it is easy to show that an AG ring with non-unimodal H -vector fails to have WLP. In codimension c = 3 it is conjectured that all AG rings have WLP. For c = 4 , Gondim shows in ( Gondim, 2017 ) that WLP always holds for r ≤ 4 and gives a family where WLP fails for any r ≥ 7 , building on Ikeda's example ( Ikeda, 1996 ) of failure for r = 5 . In this note we study the minimal free resolution of A and relation to Lefschetz properties (both weak and strong) and Jordan type for c = 4 and r ≤ 6 . Introduction Let S = K [ x 1 , … , x n ] be a standard graded polynomial ring over a field K , I a nondegenerate (containing no linear form) homogeneous ideal, and A = S / I . The ring A is Gorenstein if it is Cohen-Macaulay and the canonical module ω A is isomorphic to a shift of A : ω A = Ext S c ( A , S ) ( − n ) ≃ A ( r ) where I has codimension c and regularity r . The graded Betti numbers b i , j ( A ) = dim K ⁡ Tor i S ( A , K ) j satisfy a certain symmetry when A is Gorenstein: b i , j = b i , j ( A ) = b c − i , r + n − j ( A ) . As A is Cohen-Macaulay, for a linear system of parameters L , b i , j ( A ) = dim K ⁡ Tor i S / L ( A / L , K ) j , and henceforth we assume that A is an Artin Gorenstein (AG) ring. For an AG ring c = n , so Equation (1) becomes b i , j ( A ) = b n − i , r + n − j ( A ) . Macaulay's famed apolarity theorem (Macaulay, 1927) shows that any AG ring arises as the inverse system of homogeneous polynomial F : there is an apolarity pairing obtained by defining a ring R = K [ y 1 , … , y n ] , and letting S act on R by differentiation: x i ( y j ) = ∂ ∂ ( y i ) ( y j ) = δ i j . If we define I F = ann S ( F ) for a homogeneous polynomial F of degree r , then Macaulay shows that S / I F is Gorenstein of regularity and socle degree both r ; and furthermore that every AG ring arises in this way, with the caveat that in positive characteristic it is necessary to use divided powers. In this note, we investigate Lefschetz properties, which are known (e.g. Boij et al., 2014; Migliore et al., 2011) to depend on characteristic. A main tool we employ is the technique of generic initial ideals, which require an infinite ground field (Green, 1998), so we assume throughout that char ( K ) = 0 . The Hilbert Syzygy Theorem (Eisenbud, 1995) guarantees that any finitely generated Z -graded S -module M has a minimal graded finite free resolution : an exact sequence 0 ⟶ F i ⟶ F i − 1 ⟶ ⋯ ⟶ F 0 ⟶ M ⟶ 0 , where i ≤ n and F i ≃ ⊕ j S ( − j ) b i , j with b i , j ∈ Z ; in particular dim K ⁡ Tor i S ( M , K ) j = b i , j . This data is compactly encoded in the betti table (Eisenbud, 1995): an array whose entry in position ( i , j ) (reading over and down) is b i , i + j . The reason for this odd indexing is that the index of the bottom row of the betti table encodes the regularity of M . Example 1.1 For F = y 1 y 2 y 3 ∈ K [ y 1 , y 2 , y 3 ] , we have I F = 〈 x 1 2 , x 2 2 , x 3 2 〉 and the minimal free resolution is given by the Koszul complex 0 ⟶ S ( − 6 ) ⟶ S ( − 4 ) 3 ⟶ S ( − 2 ) 3 ⟶ S ⟶ S / I F ⟶ 0 , which in betti table notation is written as +--------------+ | 0 1 2 3| |total: 1 3 3 1| | 0: 1 . . .| | 1: . 3 . .| | 2: . . 3 .| | 3: . . . 1| +--------------+ Lefschetz properties are ubiquitous in algebra, combinatorics, geometry, and topology. In the setting of commutative algebra, we have Definition 1.2 An Artinian Z -graded ring A = S / I has (1) the Weak Lefschetz Property (WLP) if there is an ℓ ∈ S 1 such that for all i , the multiplication map μ ℓ : A i ⟶ A i + 1 has maximum rank; if not, we say that A fails WLP in degree i . (2) the Strong Lefschetz Property (SLP) if there is an ℓ ∈ S 1 such that for all i and k , the multiplication map μ ℓ k : A i ⟶ A i + k has maximum rank; if not we say that A fails SLP in degree i . The set of elements ℓ ∈ S 1 with the property that the multiplication map μ ℓ has maximum rank is a (possibly empty) Zariski open set in S 1 , so existence of the Lefschetz element ℓ in Definition 1.2 is equivalent to requiring that multiplication by a general linear form in S 1 has full rank in every degree. It is also clear from Definition 1.2 that if A has SLP then A has WLP. SLP always holds for r =2, and Proposition 3.15 of (Harima et al., 2013) proves that SLP always holds for c ≤ 2 when char ( K ) = 0 , so we focus on c , r ≥ 3 . The simplest AG rings are complete intersections (CI), and Theorem 2.3 of (Harima et al., 2003) shows that for c = 3 a CI always has WLP. For general c this remains an open question; Boij-Migliore-Miró–Roig-Nagel-Zanello make the following Conjecture 1.3 ( Boij et al., 2014 ) For c = 3 and char ( K ) = 0 an AG ring always has WLP. Despite extensive work, Conjecture 1.3 remains open. It is an easy exercise to show that WLP cannot hold for an AG ring with non-unimodal H -vector. Migliore-Zanello (2017) note in Remark 3.2 that in socle degree 4 the example of Stanley (1978) has the smallest possible codimension, in particular, c = 13 . Hence one might hope that WLP holds for AG rings with small values of c and r , and Theorem 3.1 of (Gondim, 2017) shows that SLP (hence WLP) always holds for c = 4 when r ≤ 4 . We explore the connection between WLP and free resolutions. When c = 4 = r , the possible betti tables of AG rings are determined in (Schenck et al., 2022). We prove that there are three betti tables possible for an AG ring with c = 4 and r = 3 . For c = 4 and r = 5 we make a conjecture concerning the connection of WLP and the minimal free resolution of A . For background on inverse systems and free resolutions, we refer to (Eisenbud, 1995), and for Lefschetz properties and Jordan type we refer to (Harima et al., 2013). Theorem 2.2 of (Schenck et al., 2022) proves that there are 16 possible betti tables for an AG algebra with c = 4 = r . The stratification of the parameter space P 34 of quaternary quartics by betti table is described in §6 of (Kapustka et al., 2021), which notes that for an AG algebra A with c = 4 and r = 3 there are only 3 possible betti tables. In §2 we prove this assertion, which is non-trivial. Theorem 1.4 An AG ring A with c = 4 and r = 3 has betti table in the list below: +-----------------+-----------------+-----------------+ | 0 1 2 3 4| 0 1 2 3 4| 0 1 2 3 4| |total: 1 6 10 6 1|total: 1 7 12 7 1|total: 1 9 16 9 1| | 0: 1 . . . .| 0: 1 . . . .| 0: 1 . . . .| | 1: . 6 5 . .| 1: . 6 6 1 .| 1: . 6 8 3 .| | 2: . . 5 6 .| 2: . 1 6 6 .| 2: . 3 8 6 .| | 3: . . . . 1| 3: . . . . 1| 3: . . . . 1| +-----------------+-----------------+-----------------+ The classification in (Schenck et al., 2022) uses the theorems of Macaulay and Gotzmann as the main tools. In contrast, to prove Theorem 1.4, we need to analyze certain Groebner strata of the Hilbert scheme. To do this, we use Schreyer's algorithm (Schreyer, 1980) for computing syzygies, described in §15.5 of (Eisenbud, 1995). The key point is showing that certain maps in the Schreyer resolution must have full rank. This implies that an AG algebra whose betti table has top row 6 7 2 must have degree two component I 2 which is not saturated, and an argument with Ext modules then shows the betti table below cannot occur for an AG algebra. +-----------------+ | 0 1 2 3 4 | |total: 1 8 14 8 1| | 0: 1 . . . .| (3) | 1: . 6 7 2 .| | 2: . 2 7 6 .| | 3: . . . . 1| +-----------------+ If we stay in the case of r = 3 but increase c from 4 to 5, then it is easy to find examples where WLP fails–this occurs (and is easy to show) when I 2 consists of the 2 × 2 minors of a 2 × 5 matrix. As noted, for c = 4 = r WLP always holds, and for c = 4 and r = 5 Ikeda (1996) describes an AG algebra A with H ( A ) = ( 1 , 4 , 10 , 10 , 4 , 1 ) which fails to have WLP. Our computations indicate that the following is true: Conjecture 1.5 For an AG ring with c = 4 and r = 5 there are 36 possible betti tables (see §3.2). SLP (and hence WLP) holds for any AG ring of this type with betti table not appearing in Theorem 1.6 below. For an AG ring with c = 4 and r = 6 , WLP always holds. The next theorem provides some evidence for Conjecture 1.5: Theorem 1.6 For an AG algebra A with c = 4 and r = 5 having betti table in the list below, WLP is not determined by the betti table. +-----------------+-------------------+-------------------+-------------------+ | 0 1 2 3 4| 0 1 2 3 4| 0 1 2 3 4| 0 1 2 3 4| |total: 1 9 16 9 1|total: 1 11 20 11 1|total: 1 13 24 13 1|total: 1 16 30 16 1| | 0: 1 . . . .| 0: 1 . . . .| 0: 1 . . . .| 0: 1 . . . .| | 1: . 3 2 . .| 1: . 2 1 . .| 1: . 1 . . .| 1: . . . . .| | 2: . 3 6 3 .| 2: . 5 9 4 .| 2: . 7 12 5 .| 2: . 10 15 6 .| | 3: . 3 6 3 .| 3: . 4 9 5 .| 3: . 5 12 7 .| 3: . 6 15 10 .| | 4: . . 2 3 .| 4: . . 1 2 .| 4: . . . 1 .| 4: . . . . .| | 5: . . . . 1| 5: . . . . 1| 5: . . . . 1| 5: . . . . 1| +-----------------+-------------------+-------------------+-------------------+ For each betti table above, in §3 we give an example where WLP holds, and an example where WLP fails. In §3.3 we show that SLP can fail for c = 4 and r = 6 ; WLP is unknown in this case. For the theorems appearing in (Kapustka et al., 2021) and (Schenck et al., 2022), and the results and conjecture in §1.3 above, evidence was provided by computing in Macaulay2 the inverse system of all polynomials containing up to four monomial terms, but with all coefficients either zero or one. These computations were made first over Z / p for small primes p , and subsequently over Q . We found it surprising that in the cases we considered, all possible betti tables could be generated by polynomials with a small number of monomial terms and simple coefficients, to some extent this is probably due to the strong constraints imposed by the theorems of Macaulay and Gotzmann, combined with the Gorenstein condition. Of course, for codimension 4 and regularity 5, the fact that the list of betti tables is complete is the content of Conjecture 1.5. We expect that as codimension and regularity become large, inverse systems of more complicated polynomials will come into play. As noted earlier, WLP depends on characteristic of the ground field, so in positive characteristic we expect more exotic behavior. The computations in §3.1 on Jordan type were performed using Macaulay2 scripts of Mats Boij, which were written to support work currently in progress. Section snippets Codimension four and regularity three We quickly review the theorems of Macaulay and Gotzmann (§7.2 of Schenck, 2003): for a graded algebra S / I with Hilbert function h i , write h i = ( a i i ) + ( a i − 1 i − 1 ) + ⋯ and h i 〈 i 〉 = ( a i + 1 i + 1 ) + ( a i − 1 + 1 i ) + ⋯ , where a i > a i − 1 > ⋯ . Then we have Theorem 2.1 Macaulay In the setting above, h i + 1 ≤ h i 〈 i 〉 . Theorem 2.2 Gotzmann If I is generated in a single degree t and equality holds in Macaulay's formula in the first degree t, then h t + j = ( a t + j t + j ) + ( a t − 1 + j − 1 t + j − 1 ) + ⋯ . Suppose that S = K [ x 1 , x 2 , x 3 , x 4 ] ( c = 4 ) and r = 3 . Since the regularity of A is equal to the socle degree and A Codimension four and regularity five We begin by proving Theorem 1.6: Proof For each betti table, we give examples of ideals satisfying the theorem. +-----------------+ | 0 1 2 3 4| |total: 1 9 16 9 1| | 0: 1 . . . .| | 1: . 3 2 . .| | 2: . 3 6 3 .| | 3: . 3 6 3 .| | 4: . . 2 3 .| | 5: . . . . 1| +-----------------+ • WLP holds: 〈 x 3 x 4 , x 2 x 4 , x 3 2 , x 4 3 , x 2 2 x 3 − x 1 x 4 2 , x 2 3 , x 1 3 x 3 , x 1 3 x 2 , x 1 4 〉 . • WLP fails: 〈 x 4 2 , x 3 x 4 , x 3 2 , x 2 2 x 4 , x 2 2 x 3 − x 1 2 x 4 , x 1 2 x 3 , x 2 4 , x 1 2 x 2 2 , x 1 4 〉 . +--------------------+ | 0 1 2 3 4 | |total: 1 11 20 11 1 | | 0: 1 . . . . | | 1: . 2 1 . . | | 2: . 5 9 4 . | | 3: . 4 9 5 . | | 4: . . 1 2 . | | 5: . . . . 1 | +--------------------+ • WLP holds: 〈 x 4 2 , x 3 x 4 , x 3 3 , x 2 Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements All of the computations in this paper were performed using the QuaternaryQuartics (Kapustka et al., 2021) package of Macaulay2, by Grayson and Stillman, which is available at: http://www.math.uiuc.edu/Macaulay2/ . Our collaboration began at the 2019 CIRM workshop “Lefschetz properties in algebra, geometry, and combinatorics”, organized by A. Dimca, R. Miró-Roig, and J. Vallès, and we thank them and CIRM for a great workshop. We thank Nasrin Altafi, Tony Iarrobino, Juan Migliore, and two References (20) N. Abdallah et al. Lefschetz properties of some codimension three Artinian Gorenstein algebras J. Algebra (2023) M. Boij et al. On the weak Lefschetz property for Artinian Gorenstein algebras of codimension three J. Algebra (2014) R. Gondim On higher Hessians and the Lefschetz properties J. Algebra (2017) T. Harima et al. The weak and strong Lefschetz properties for Artinian K -algebras J. Algebra (2003) A. Iarrobino Ancestor ideals of vector spaces of forms, and level algebras J. Algebra (2004) R. Stanley Hilbert functions of graded algebras Adv. Math. (1978) N. Abdallah A note Artin Gorenstein algebras with Hilbert function ( 1 , 4 , k , k , 4 , 1 ) D. Eisenbud Commutative Algebra with a View Towards Algebraic Geometry (1995) M. Green Generic initial ideals Prog. Math. (1998) T. Harima et al. The Lefschetz Properties (2013) There are more references available in the full text version of this article. Cited by (0) Recommended articles (6) Research article Consecutive patterns in Coxeter groups Journal of Algebra, Volume 634, 2023, pp. 650-666 Show abstract For an arbitrary Coxeter group element σ and a connected subset J of the Dynkin diagram, the parabolic decomposition σ = σ J σ J defines σ J as a consecutive pattern of σ , generalizing the notion of consecutive patterns in permutations. 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