# On the regularity and defect sequence of monomial and binomial ideals

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## Abstract

When *S* is a polynomial ring or more generally a standard graded algebra over a field *K*, with homogeneous maximal ideal m, it is known that for an ideal *I* of *S*, the regularity of powers of *I* becomes eventually a linear function, i.e., reg(*I*^{m}) = *dm* + *e* for *m* ≫ 0 and some integers *d, e*. This motivates writing reg(*I*^{m}) = *dm*+*e*_{m} for every *m* ≥ 0. The sequence *e*_{m}, called the defect sequence of the ideal *I*, is the subject of much research and its nature is still widely unexplored. We know that *e*_{m} is eventually constant. In this article, after proving various results about the regularity of monomial ideals and their powers, we give several bounds and restrictions on *e*_{m} and its first differences when *I* is a primary monomial ideal. Our theorems extend the previous results about m-primary ideals in the monomial case. We also use our results to obtatin information about the regularity of powers of a monomial ideal using its primary decomposition. Finally, we study another interesting phenomenon related to the defect sequence, namely that of regularity jump, where we give an infinite family of ideals with regularity jumps at the second power.

## Keywords

Castelnuovo-Mumford regularity powers of ideal defect sequence## MSC

13D02 13P10## Preview

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## References

- [1]
*D. Berlekamp*: Regularity defect stabilization of powers of an ideal. Math. Res. Lett.**19**(2012), 109–119.MathSciNetCrossRefGoogle Scholar - [2]
*K. Borna*: On linear resolution of powers of an ideal. Osaka J. Math.**46**(2009), 1047–1058.MathSciNetzbMATHGoogle Scholar - [3]
*M. P. Brodmann, R. Y. Sharp*:*Local Cohomology: An Algebraic Introduction with Geometric Applications*. Cambridge Studies in Advanced Mathematics 60, Cambridge University Press, Cambridge, 1998.CrossRefGoogle Scholar - [4]
*M. Chardin*:*Some results and questions on Castelnuovo-Mumford regularity*. Syzygies and Hilbert Functions (I. Peeva, ed.). Lecture Notes in Pure and Applied Mathematics 254, Chapman & Hall/CRC, Boca Raton, 2007, pp. 1–40.zbMATHGoogle Scholar - [5]
*A. Conca*:*Regularity jumps for powers of ideals. Commutative Algebra: Geometric*, Homological, Combinatorial and Computational Aspects (A. Corso et al., eds.). Lecture Notes in Pure and Applied Mathematics 244, Chapman & Hall/CRC, Boca Raton, 2006, pp. 21–32.zbMATHGoogle Scholar - [6]
*S. D. Cutkosky, J. Herzog, N. V. Trung*: Asymptotic behaviour of the Castelnuovo-Mumford regularity. Compos. Math.**118**(1999), 243–261.MathSciNetCrossRefGoogle Scholar - [7]
*H. T. Hà, N. V. Trung, T. N. Trung*: Depth and regularity of powers of sums of ideals. Math. Z.**282**(2016), 819–838.MathSciNetCrossRefGoogle Scholar - [8]
*V. Kodiyalam*: Asymptotic behaviour of Castelnuovo-Mumford regularity. Proc. Am. Math. Soc.**128**(2000), 407–411.MathSciNetCrossRefGoogle Scholar