On decomposing monomial algebras with the Lefschetz properties

Case of misdiagnosed melioidosis from Hue city, Vietnam

“…The case of artinian k-algebras defined by monomial ideals, while being rather accessible, is far from simple and the literature concerning their Lefschetz properties is quite extensive; see, for instance, [2,3,11,12,23,22,27,24] and the references therein. In this work, we focus on a special class of artinian algebras defined by quadratic monomials, given as follows.…”

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“…Indeed, there are at least 2 elements in the kernel of the multiplication by x 1 + x 2 + x 3 + y 1 + y 2 map from degree 8 to degree 9 of C, and they are given explicitly by f 1 g 1 and f 2 g 1 , where f 1 = x 4 1 , f 2 = x 3 1 x 2 and g 1 = (y 4 1 − y 3 1 y 2 + y 2 1 y 2 2 − y 1 y 3 2 + y 4 2 ), where the latter is in the kernel of the multiplication by y 1 + y 2 map from degree 4 to degree 5 on B. A computation in Macaulay2 shows that these 2 elements in fact span the kernel; the Hilbert series of C/(x 1 + x 2 + x 3 + y 1 + y 2 ) equals 1 + 4T + 10T 2 + 20T 3 + 35T 4 + 47T 5 + 54T 6 + 52T 7 + 38T 8 + 13T 9 , where the coefficient 13 of T 9 satisfies 13 = 272 − 261 + 2.…”

The weak Lefschetz property of equigenerated monomial ideals

“…Let n = 8, d = 4, µ = 13. We have µ ∈ Σ 8,4 = [α(8, 4), β(8, 4)] =[9, 322], so by Theorem 1.1, there exists an ideal I minimally generated by 13 elements so that S/I fails the WLP. Following the proof of Theorem 1.1 we construct such an ideal.…”