101. Integral closures of powers of sums of ideals
- Author
-
Banerjee, Arindam and Ha, Huy Tai
- Subjects
Mathematics::Commutative Algebra ,Optimization and Control (math.OC) ,FOS: Mathematics ,13C13, 90C05, 13D07 ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,Mathematics - Optimization and Control - Abstract
Let $k$ be a field, let $A$ and $B$ be polynomial rings over $k$, and let $S = A \otimes_k B$. Let $I$ and $J$ be monomial ideals in $A$ and $B$, respectively. We establish a binomial expansion for rational powers of $I+J \subseteq S$ in terms of those of $I$ and $J$. We give a sufficient condition for this formula to hold for the integral closures of powers of $I+J$. Under this condition, we also provide precise formulas for the depth and the regularity of the integral closures of powers of $I+J$. Our methods are linear algebra in nature, making use of the characterization of rational powers of a monomial ideal via optimal solutions to linear programming problems., 10 pages
- Published
- 2022