Associated prime ideals of powers of ideals

Primary decompositions play a role in different mathematical areas. For example, in algebraic geometry, decomposing an algebraic set into irreducible components is algebraically encoded in the primary decomposition of the defining ideal I. Moreover, the factor ring of the polynomial ring modulo I n carries information about the derivatives (of order up to n) on the algebraic set. Therefore, not only the decomposition of I but also of its powers are of great interest. Other well-known examples can be found in combinatorial commutative algebra; the associated prime ideal of an edge ideal represent the set of minimal vertex cover of the underlying graph. This speaks to the complexity of the computation of primary decompositions (even in the case of monomial ideals with square-free generators of degree 2) as the computation of minimal vertex covers is NP-hard. Also in this context, associated prime ideal of powers of ideals are desired. The associated prime ideals of powers of edge ideals contain a lot of information about colorings of the graph, that is, the associated prime ideals up to the k-th power of an edge ideals contain to chromatically k + 1-critically subgraphs of the given graph.

This talk is an introduction to the primary decompositions in general and discuss associated primes for powers of certain ideals which show interesting behaviour.