Roswitha Rissner: The stability index of monomial ideals

The stability index of monomial ideals

Roswitha Rissner, University of Klagenfurt

Abstract: In her 1921 work, Emmy Noether established that every ideal in a Noetherian commutative ring admits an irredundant primary decomposition, that is, it can be written as an intersection of primary ideals. Although such decompositions are not unique, the collection of radicals of the primary components—the set of associated primes of the ideal—is uniquely determined. These associated primes may be viewed as the “building blocks” of the ideal or of the mathematical object it encodes: familiar examples include the prime factors of an integer and the irreducible components of an algebraic variety.

We differentiate between two types of associated primes, the minimal and the embedded ones. While the minimal primes are stable under exponentiation, the embedded primes behave erratically. Despite this volatility, a theorem of Brodmann (1979) guarantees eventual stabilization and the power at which the stabilization occurs is called the stability index.

This talk surveys known results on the stability index, with an emphasis on monomial ideals, and reports on ongoing joint work with Jutta Rath. Monomial ideals play an important role in combinatorial commutative algebra. For example, the minimal vertex covers of a (simple, undirected) graph define a monomial ideal $I$---the cover ideal---whose associated primes are in one-to-one correspondence to the edges of the graph. The associated primes of powers of the cover ideal are intrinsically connected to coloring properties of the graph.