Hilbert's Syzygy Theorem for monomial ideals

Year 2022, , 80 - 85, 16.07.2022

Guillermo Alesandronı

https://doi.org/10.24330/ieja.1102307

Abstract

We give a new proof of Hilbert's Syzygy Theorem for monomial ideals. In addition, we prove the following. If $S=k[x_1,\dots ,x_n]$ is a polynomial ring over a field, $M$ is a squarefree monomial ideal in $S$, and each minimal generator of $M$ has degree larger than $i$, then $\text{pd}\left(S/M\right)\le n-i$.

Keywords

Betti number, free resolution, monomial ideal

References

• G. Alesandroni, Monomial ideals with large projective dimension, J. Pure Appl. Algebra, 224(6) (2020), 106257 (13 pp).
• H. Cartan and S. Eilenberg, Homological Algebra, Princeton University Press, Princeton, NJ, 1956.
• D. Eisenbud, Commutative Algebra, Springer-Verlag, New York, 1995.
• V. Gasharov, T. Hibi and I. Peeva, Resolutions of a-stable ideals, J. Algebra, 254(2) (2002), 375-394.
• V. Gasharov, I. Peeva and V. Welker, Coordinate subspace arrangements and monomial ideals, Proceedings of the Computational Commutative Algebra and Combinatorics (Osaka, 1999), Adv. Stud. Pure Math., 33 (2002), 65-74.
• D. Hilbert, Über die Theorie von algebraischen Formen, Math. Ann., 36 (1890), 473-534.
• J. Mermin, Three Simplicial Resolutions, Progress in Commutative Algebra 1, De Gruyter, Berlin, 2012.
• E. Miller and B. Sturmfels, Combinatorial Commutative Algebra, Grad. Texts in Math., vol. 227, Springer-Verlag, New York, 2004.
• I. Peeva, Graded Syzygies, Algebra and Applications, vol. 14, Springer-Verlag, London, 2011.
• F. O. Schreyer, Die Berechnung von Syzygien mit dem verallgemeinerten Weierstra\ss schen Divisionssatz und eine Anwendung auf analytische Cohen-Macaulay Stellenalgebren minimaler Multiplizit\"at, Master's thesis, Universit\"at Hamburg, 1980.