On the Symmetric Algebra of the First Syzygy of a Graded Maximal Ideal

Abstract

Let Syz₁(𝔪) be the first syzygy of the graded maximal ideal 𝔪 of a polynomial ring K[x₁,…, x_n] over a field K. Using the theory of s-sequences, the dimension and depth of the symmetric algebra Sym(Syz₁(𝔪)) are calculated. As a conclusion, Sym(Syz₁(𝔪)) is not Cohen–Macaulay for any n ≥ 4.

Key Words:

• Symmetric algebra
• Syzygy
• s-sequence

2010 Mathematics Subject Classification:

• 16W50
• 13D02
• 13C15

ACKNOWLEDGMENTS

The article was carried out when the second author was visiting the University of Messina, the author would like to thank INDAM (Istituto Nazionale di Alta Matematica “F. Severi,” Roma, Italy) and the National Natural Science Foundation of China (No. 11471234) for financial supports, and is grateful to the Department of Mathematics and Computer Science of the University of Messina for its hospitality.