hasWLP(p,L)
hasWLP(R,I)
hasWLP(p,L) check if the monomial complete intersections $A=k[x_1,...,x_n]/(x_1^{a_1}, …, x_n^{a_n})$, where k is a field of characteristic p has the Weak Lefschetz property. For p=2 this method uses Theorem 8.1 in [KMRR,24]. For p>2, it uses the Han-Monsky multiplication obtained with Conjecture 4.1 [KMRR,25].
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It is possible to check the WLP without employing the Han-Monsky multiplication, setting the option UseConjecture to be false.
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hasWLP(R,I) check if the graded Artinian algebra R/I, where R is a standard graded polynomial ring, has the Weak Lefschetz property.
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The options GorensteinAlg and MonomialAlg allow simplifying the computation when the input I is respectively a Gorenstein or a monomial ideal.
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The instances hasWLP(R,I) and hasWLP(R,I, GorensteinAlg=>true) require R to be a standard graded polynomial ring over a sufficiently large field k, e.g., R=QQ[x_1,…,x_n]. If R is a polynomial ring over a finite field, then when hasWLP(R, I) outputs false, it confirms that R/I fails the WLP. However, when hasWLP(R, I) outputs true, we can only conclude that R/I has the WLP over a field extension. The same holds for hasWLP(R,I, GorensteinAlg=>true).
This does not apply to hasWLP(p,L) and hasWLP(R,I, MonomialAlg=>true), which work over any field k.
The object hasWLP is a method function with options.
The source of this document is in /build/reproducible-path/macaulay2-1.25.05+ds/M2/Macaulay2/packages/IncidenceCorrespondenceCohomology.m2:1467:0.