Jutta Rath, University of Klagenfurt
Authors: Clemens Heuberger, Jutta Rath, Roswitha Rissner
2023 AWM Research Symposium
Combinatorial and Homological Methods in Commutative Algebra [Organized by Francesca Gandini and Selvi Kara]

To every ideal $I$ in a ring one can associate a unique set of prime ideals, the so-called associated primes of $I$. In many settings, these primes can be interpreted as structure revealing building blocks of $I$. The associated primes of an ideal in $\mathbb{Z}$ correspond to the prime divisors of the generator of the ideal; the associated primes of the inducing ideal of an algebraic variety correspond to the its irreducible components; for the edge ideal of a finite simple graph, the associated primes are precisely the prime ideals generated by the minimal vertex covers of the graph. Furthermore, also associated primes of powers of ideals have a connection to graph theory: The index after which the sequence of associated primes of powers of the cover ideal of a graph is constant gives an upper bound for the chromatic number of the underlying graph. The phenomenon of occurring changes in the set of associated primes when considering powers of an ideal has been observed in many different settings. In the Noetherian case, it is well known that the sequence of associated primes of powers of an ideal stabilizes. This talk focuses on the stabilization of associated primes of powers of monomial ideals. A technique to develop upper bounds for the power of an ideal after which the sequence is non-increasing is presented.

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