Fitting ideals without a presentation

In this paper, we investigate alternative construction of Fitting ideals of pushforward modules [Formula: see text] for finite and holomorphic map-germs from an n-dimensional Cohen–Macaulay space [Formula: see text] to [Formula: see text]. For corank 1 map-germs, we generalize a result of Mond and Pellikaan to iteratively calculate Fitting ideals as ideal quotients of lower ones. We also show that for a stable map-germ of any corank, the first Fitting ideal can be calculated as a quotient ideal of the Jacobian of the image and the pushforward of the ramification ideal over the target, which is a modification of classical result due to Piene.