Normality of Rees algebras of generalized mixed product ideals

Year 2024, , 168 - 185, 09.01.2024

Monica La Barbıera Roya Moghımıpor

https://doi.org/10.24330/ieja.1402961

Abstract

Let $K$ be a field and $K[x_1,x_{2}]$ the polynomial ring in two
variables over $K$ with each $x_i$ of degree $1$. Let $L$ be the
generalized mixed product ideal induced by a monomial ideal
$I\subset K[x_1,x_2]$, where the ideals substituting the monomials
in $I$ are squarefree Veronese ideals. In this paper, we study the
integral closure of $L$, and the normality of $\mathcal{R}(L)$, the
Rees algebra of $L$. Furthermore, we give a geometric description of
the integral closure of $\mathcal{R}(L)$.

Keywords

Integral closure, normality, Rees algebra, generalized mixed product ideal

References

• I. Al-Ayyoub, M. Nasernejad and L. G. Roberts, Normality of cover ideals of graphs and normality under some operations, Results Math., 74(4) (2019), 140 (26 pp).
• S. Bayati and J. Herzog, Expansions of monomial ideals and multigraded modules, Rocky Mountain J. Math., 44(6) (2014), 1781-1804.
• C. A. Escobar, R. H. Villarreal and Y. Yoshino, Torsion freeness and normality of blowup rings and monomial ideals, Commutative Algebra, Lect. Notes Pure Appl. Math., 244 (2006), 69-84.
• J. Herzog, R. Moghimipor and S. Yassemi, Generalized mixed product ideals, Arch. Math. (Basel), 103(1) (2014), 39-51.
• C. Huneke and I. Swanson, Integral Closure of Ideals, Rings and Modules, London Mathematical Society, Lecture Note Series, 336, Cambridge University Press, Cambridge, 2006.
• R. Moghimipor, Algebraic and homological properties of generalized mixed product ideals, Arch. Math. (Basel), 114 (2020), 147-157.
• R. Moghimipor, On the normality of generalized mixed product ideals, Arch. Math. (Basel), 115(2) (2020), 147-157.
• R. Moghimipor, On the Cohen-Macaulayness of bracket powers of generalized mixed product ideals, Acta Math. Vietnam., 47(3) (2022), 709-718.
• R. Moghimipor and A. Tehranian, Linear resolutions of powers of generalized mixed product ideals, Iran. J. Math. Sci. Inform., 14(1) (2019), 127-134.
• G. Restuccia and R. H. Villarreal, On the normality of monomial ideals of mixed products, Comm. Algebra, 29(8) (2001), 3571-3580.
• W. V. Vasconcelos, Computational Methods in Commutative Algebra and Algebraic Geometry, Algorithms Comput. Math., Springer-Verlag, 1998.
• R. H. Villarreal, Monomial Algebras, Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, Inc., New York, 238, 2001.
• R. H. Villarreal, Monomial Algebras, Second Edition, Chapman & Hall/CRC Monographs and Research Notes in Mathematics, 2015.
• D. B. West, Introduction to Graph Theory, Second Edition, Pearson, 2017.