Set of Values of Fractional Ideals of Rings of Algebroid Curves

Abstract

The aim of this work is to study sets of values of fractional ideals of rings of algebroid curves and explore more deeply the symmetry that exists among sets of values of dual pairs of ideals when the ring is Gorenstein. We also express the codimension of a fractional ideal in terms of the maximal points of the value set of the ideal. Finally we also show that the Gorensteiness of a ring of an algebroid curve is equivalent to some conditions relating certain codimensions of fractional ideals and of their duals.

Standard Bases for fractional ideals of the local ring of an algebroid curve

“…Notice that v(g 4 ) = v(xg 1 ) and v(h 3 ) = v(x 2 z + f 1 ), so we can discard g 4 and h 3 from G 1 in such way that we obtain, by Proposition 12, a minimal Standard Basis (8,8), (10,10), (∞, 21), (∞, 25), (∞, 32), (∞, 34), (∞, 36), (25, ∞), (27, ∞), (29, ∞), (31, ∞), (32, ∞), (34, ∞), (17,17), (19,19)} is the minimal set of generators of the semiring Γ and its conductor is (31, 31).…”