
Mathematics 2013
Finitely Supported *Simple Complete Ideals in a Regular Local RingAbstract: Let I be a finitely supported complete mprimary ideal of a regular local ring (R, m). A theorem of Lipman implies that I has a unique factorization as a *product of special *simple complete ideals with possibly negative exponents for some of the factors. The existence of negative exponents occurs if the dimension of R is at least 3 because of the existence of finitely supported *simple ideals that are not special. We consider properties of special *simple complete ideals such as their Rees valuations and point basis. Let (R, m) be a ddimensional equicharacterstic regular local ring with m = (x_1, ..., x_d)R. We define monomial quadratic transforms of R and consider transforms and inverse transforms of monomial ideals. For a large class of monomial ideals I that includes complete inverse transforms, we prove that the minimal number of generators of I is completely determined by the order of I. We give necessary and sufficient conditions for the complete inverse transform of a *product of monomial ideals to be the *product of the complete inverse transforms of the factors. This yields examples of finitely supported *simple monomial ideals that are not special. We prove that a finitely supported *simple monomial ideal with linearly ordered base points is special *simple.
