Source: Nagoya Math. J. Volume 207
(2012), 79-93.
For a monomial ideal $I$ of a polynomial ring $S$, a polarization of $I$ is a square-free monomial ideal $J$ of a larger polynomial ring $\widetilde {S}$ such that $S/I$ is a quotient of $\widetilde {S}/J$ by a (linear) regular sequence. We show that a Borel fixed ideal admits a nonstandard polarization. For example, while the usual polarization sends $xy^{2}\in S$ to $x_{1}y_{1}y_{2}\in \widetilde {S}$, ours sends it to $x_{1}y_{2}y_{3}$. Using this idea, we recover/refine the results on square-free operation in the shifting theory of simplicial complexes. The present paper generalizes a result of Nagel and Reiner, although our approach is very different.
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References
[1] A. Aramova, J. Herzog, and T. Hibi, Squarefree lexsegment ideals, Math. Z. 228 (1998), 353–378.
[2] A. Aramova, J. Herzog, and T. Hibi, Shifting operations and graded Betti numbers, J. Algebraic Combin. 12 (2000), 207–222.
[3] A. Björner and M. L. Wachs, Shellable nonpure complexes and posets, II, Trans. Amer. Math. Soc. 349 (1997), 3945–3975.
[4] W. Bruns and J. Herzog, Cohen-Macaulay Rings, rev. ed., Cambridge University Press, Cambridge, 1998.
[5] D. Eisenbud, Commutative Algebra: With a View Toward Algebraic Geometry, Grad. Texts in Math. 150, Springer, New York, 1995.
[6] J. Herzog and D. Popescu, Finite filtrations of modules and shellable multicomplexes, Manuscripta Math. 121 (2006), 385–410.
[7] G. Kalai, “Algebraic shifting” in Computational Commutative Algebra and Combinatorics (Osaka. 1999), Adv. Stud. Pure Math. 33, Math. Soc. Japan, Tokyo, 2002, 121–163.
[8] H. Lohne, Polarizations of powers of the maximal ideal in a polynomial ring, in preparation.
[9] S. Murai, Generic initial ideals and squeezed spheres, Adv. Math. 214 (2007), 701–729.
[10] U. Nagel and V. Reiner, Betti numbers of monomial ideals and shifted skew shapes, Electron. J. Combin. 16 (2009), Research Paper 3.
[11] E. Sbarra, Upper bounds for local cohomology for rings with given Hilbert function, Comm. Algebra 29 (2001), 5383–5409.
[12] R. Stanley, Combinatorics and Commutative Algebra, 2nd ed., Progr. Math. 41, Birkhäuser, Boston 1996.
[13] K. Yanagawa, Sliding functor and polarization functor for multigraded modules, Comm. Algebra, 40 (2012), 1151–1166.
