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The Weak Lefschetz Property for monomial complete intersection in positive characteristic


Authors: Andrew R. Kustin and Adela Vraciu
Journal: Trans. Amer. Math. Soc. 366 (2014), 4571-4601
MSC (2010): Primary 13D02; Secondary 13A35, 13E10, 13C40
Published electronically: March 12, 2014
MathSciNet review: 3217693
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Abstract: Let $ A=\boldsymbol {k} [x_1,\dots ,x_n]/{(x_1^d,\dots ,x_n^d)}$, where $ \boldsymbol {k}$ is an infinite field. If $ \boldsymbol {k}$ has characteristic zero, then Stanley proved that $ A$ has the Weak Lefschetz Property (WLP). Henceforth, $ \boldsymbol {k}$ has positive characteristic $ p$. If $ n=3$, then Brenner and Kaid have identified all $ d$, as a function of $ p$, for which $ A$ has the WLP. In the present paper, the analogous project is carried out for $ 4\le n$. If $ 4\le n$ and $ p=2$, then $ A$ has the WLP if and only if $ d=1$. If $ n=4$ and $ p$ is odd, then we prove that $ A$ has the WLP if and only if $ d=kq+r$ for integers $ k,q,r$ with $ 1\le k\le \frac {p-1}2$, $ r\in \left \{\frac {q-1}2,\frac {q+1}2\right \}$, and $ q=p^e$ for some non-negative integer $ e$. If $ 5\le n$, then we prove that $ A$ has the WLP if and only if $ \left \lfloor \frac {n(d-1)+3}2\right \rfloor \le p$. We first interpret the WLP for the ring $ {{\boldsymbol {k}}[x_1, \ldots , x_{n}]}/{(x_1^d, \ldots , x_{n}^d)}$ in terms of the degrees of the non-Koszul relations on the elements $ x_1^d, \ldots , x_{n-1}^d, (x_1+ \ldots +x_{n-1})^d$ in the polynomial ring $ \boldsymbol {k}[x_1, \ldots , x_{n-1}]$. We then exhibit a sufficient condition for $ {{\boldsymbol {k}}[x_1, \ldots , x_{n}]}/{(x_1^d, \ldots , x_{n}^d)}$ to have the WLP. This condition is expressed in terms of the non-vanishing in $ \boldsymbol {k}$ of determinants of various Toeplitz matrices of binomial coefficients. Frobenius techniques are used to produce relations of low degree on $ x_1^d$, $ \ldots $, $ x_{n-1}^d$, $ {(x_1+ \ldots +x_{n-1})^d}$. From this we obtain a necessary condition for $ A$ to have the WLP. We prove that the necessary condition is sufficient by showing that the relevant determinants are non-zero in $ \boldsymbol {k}$.


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Additional Information

Andrew R. Kustin
Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
Email: kustin@math.sc.edu

Adela Vraciu
Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
Email: vraciu@math.sc.edu

DOI: https://doi.org/10.1090/S0002-9947-2014-05944-8
Keywords: Artinian rings, complete intersections, Frobenius endomorphism, Weak Lefschetz Property
Received by editor(s): October 12, 2011
Received by editor(s) in revised form: August 20, 2012
Published electronically: March 12, 2014
Additional Notes: Both authors were supported in part by the NSA
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.