Tensor-multinomial sums of ideals: Primary decompositions and persistence of associated primes

Abstract

Given a Noetherian tensor product of two Noetherian algebras over a field and proper ideals I I and J J in the two algebras, we determine the associated primes of each power of I + J I+J in terms of the associated primes of lower powers of I I and of J J . We record two applications. First, in case the field is algebraically closed, we construct primary decompositions for powers of I + J I+J from primary decompositions for powers of I I and J J . Separately, we attack the persistence problem for associated primes of powers of an ideal in case one of I I or J J is a non-zero normal ideal.