Factorization of complete ideals
Zariski developed a theory of complete ideals for two-dimensional regular local rings. He proved that in such rings there is unique factorization of complete ideals [\textit{O. Zariski}, Am. J. Math. 60, 151-204 (1938; Zbl 0018.20101); \textit{O. Zariski} and \textit{P. Samuel}, ``Commutative algebra'', Vol. 2 (Princeton 1960; Zbl 0121.278); appendix 5]. In general the product of two complete ideals is not complete, but there is the classical *-product of complete ideals. Let (R,m) be a complete normal local domain of dimension two, and m(R) the semigroup of m-primary complete ideals with the *-product. If R/m is algebraically closed, and R is a U.F.D. then m(R) has unique factorisation [\textit{J. Lipman}, Publ. Math., Inst. Hautes Étud. Sci. 36, 195-279 (1969; Zbl 0181.489)]; the converse is also true [the author, ``On unique and almost unique factorisation of complete ideals'' (preprint)]. In this note the author adresses the question of how the theory of Zariski generalizes to regular local rings of higher dimensions and proves the following theorem: Let (R,m) be any regular local ring of dimension \(t\geq 3\) which is a localisation of a finitely generated k-algebra, for an algebraically closed field k. Then R does not have unique factorisation of complete m-primary ideals.
scientific article; zbMATH DE number 4075233
scientific article