Sequences of linear type
Let A be a commutative ring and I an ideal of A. I is called an ideal of linear type if the canonical surjection from the symmetric algebra of I to the Rees algebra of I is an isomorphism. A sequence \(a_ 1,..,a_ n\) of A is of linear type if each ideal \(I_ i=(a_ 1,...,a_ i)\), \(i=1,...,n\), is an ideal of linear type. It is shown that if \((I_ iI^ t:a_ i)\cap I^ t=I_{i-1}I^{t-1}\) for \(i=1,...,n\) and each integer \(t\geq 1\), then \(I=(a_ 1,...,a_ n)\) is an ideal of linear type. This yields Huneke's result that a d-sequence is a sequence of linear type. A characterization of sequences of linear type in the form of an induction theorem is given and this is used to derive a result on monomial sequences.
scientific article; zbMATH DE number 3957255