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A note on a conjecture regarding the weak Lefschetz property of a special class of Artinian algebras. (English) Zbl 1436.13036

Let \(R = K[x_1,\dots,x_r]\), the polynomial ring over an infinite field \(K\). Let \(I\) be a homogeneous ideal and let \(A = R/I\); we assume that \(A\) is an Artinian \(K\)-algebra with the standard grading. \(A\) is said to have the weak Lefschetz property (WLP) if, for a general linear form \(\ell\), the induced multiplication \(\times \ell : [A]_i \rightarrow [A]_{i+1}\) has maximal rank in each degree. Typically, one either studies the structure of \(A\) that may force or preclude the WLP, or one shows that this property holds for a broad family of algebras. A clean example of this is the result that for any Artinian complete intersection in three variables, WLP holds (Harima, Nagel, J. Watanabe and the reviewer). Many results have also been obtained about monomial ideals, either possessing or failing to possess this property; here even the characteristic of the field plays an important role. Extending work of the reviewer et al. [Trans. Am. Math. Soc. 363, No. 1, 229–257 (2011; Zbl 1210.13019)], in this paper the authors consider monomial ideals of the form \[ I = (x_1^{\alpha_1},\dots,x_r^{\alpha_r}) + (\hbox{all squarefree monomials of degree \(d\)}), \] where \(2 \leq \alpha_i \leq 4\) for \(1 \leq i \leq r\) and \(2 \leq d \leq r\), and they study the WLP for \(R/I\) and how the characteristic plays a role. In their first main result they give a complete and elegant characterization for when the WLP holds, for the case where \(\hbox{char}(K) \neq 0\), all the \(\alpha_i\) are equal to 2 and \(4 \leq d \leq r\). In their second main result, they show that if \(2 \leq \alpha_i \leq 4\) for all \(i\), \(\alpha_i = 4\) for some \(i\), \(r \geq 5\), \(d \geq 5\) and \(\hbox{char}(K) = 2\) then \(A = R/I\) does not satisfy the WLP.

MSC:

13E10 Commutative Artinian rings and modules, finite-dimensional algebras
13C13 Other special types of modules and ideals in commutative rings

Citations:

Zbl 1210.13019
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References:

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