The prime spectra of subalgebras of affine algebras and their localizations (Q1118654)

The paper under review studies the subaffine k-algebras (with k a field), i.e. the k-subalgebras of affine k-algebras. Specifically, one wants to determine to what extent the properties of Spec(A) are shared by Spec(R), where R is a k-subalgebra of the affine k-algebra A. The first result considers the situation where R is a k-subalgebra of \(A_ M\) (where M is a maximal ideal of the affine k-algebra A), and states that \(tr.\deg_ k(R)=\dim (R)\). Next one considers only subaffine domains R and one proves that certain Zariski open sets of Spec(R) do share properties of the prime spectra of affine domains. On the other hand, one presents examples to show that Spec(R) does not share such properties globally. In particular, one exhibits a Noetherian subpolynomial domain D satisfying the first chain condition and having the property that D/P is affine for every \(P\in Spec(D)\), but which is not itself affine over k.