The $s$-multiplicity function of $2 \times 2$-determinantal rings
HTML articles powered by AMS MathViewer
- by Lance Edward Miller and William D. Taylor PDF
- Proc. Amer. Math. Soc. 146 (2018), 2797-2810 Request permission
Abstract:
This article generalizes joint work of the first author and I. Swanson to the $s$-multiplicity recently introduced by the second author. For $k$ a field and $X = [ x_{i,j}]$ an $m \times n$-matrix of variables, we utilize Gröbner bases to give a closed form the length $\lambda ( k[X] / (I_2(X) + \mathfrak {m}^{ \lceil sq \rceil } + \mathfrak {m}^{[q]} ))$, where $s \in {\mathbf Z}[p^{-1}]$, $q$ is a sufficiently large power of $p$, and $\mathfrak {m}$ is the homogeneous maximal ideal of $k[X]$. This shows this length is always eventually a polynomial function of $q$ for all $s$.References
- Kazufumi Eto, Multiplicity and Hilbert-Kunz multiplicity of monoid rings, Tokyo J. Math. 25 (2002), no. 2, 241–245. MR 1948662, DOI 10.3836/tjm/1244208851
- Kazufumi Eto and Ken-ichi Yoshida, Notes on Hilbert-Kunz multiplicity of Rees algebras, Comm. Algebra 31 (2003), no. 12, 5943–5976. MR 2014910, DOI 10.1081/AGB-120024861
- Lance Edward Miller and Irena Swanson, Hilbert-Kunz functions of $2\times 2$ determinantal rings, Illinois J. Math. 57 (2013), no. 1, 251–277. MR 3224570
- P. Monsky, The Hilbert-Kunz function, Math. Ann. 263 (1983), no. 1, 43–49. MR 697329, DOI 10.1007/BF01457082
- Craig Huneke, Moira A. McDermott, and Paul Monsky, Hilbert-Kunz functions for normal rings, Math. Res. Lett. 11 (2004), no. 4, 539–546. MR 2092906, DOI 10.4310/MRL.2004.v11.n4.a11
- Marcus Robinson and Irena Swanson, Explicit Hilbert-Kunz functions of $2\times 2$ determinantal rings, Pacific J. Math. 275 (2015), no. 2, 433–442. MR 3347376, DOI 10.2140/pjm.2015.275.433
- W. Taylor, Interpolating between Hilbert-Samuel and Hilbert-Kunz multiplicity, arXiv:1706.07445.
- K.-i. Watanabe, Hilbert-Kunz multiplicity of toric rings, The Inst. of Natural Sciences, Nihon Univ., Proc. of the Inst. of Natural Sciences 35 (2000), 173–177.
Additional Information
- Lance Edward Miller
- Affiliation: Department of Mathematical Sciences, University of Arkansas, Fayetteville, Arkansas 72701
- MR Author ID: 761821
- Email: lem016@uark.edu
- William D. Taylor
- Affiliation: Department of Mathematical Sciences, University of Arkansas, Fayetteville, Arkansas 72701
- Email: wdtaylor@uark.edu
- Received by editor(s): August 19, 2017
- Received by editor(s) in revised form: October 5, 2017, and October 8, 2017
- Published electronically: February 21, 2018
- Communicated by: Irena Peeva
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 2797-2810
- MSC (2010): Primary 13D40; Secondary 05A15, 05A10
- DOI: https://doi.org/10.1090/proc/13979
- MathSciNet review: 3787344