Construction of Neron Desingularization for Two Dimensional Rings

Let u : A → A′ be a regular morphism of Noetherian rings and B an A-algebra of finite type. Then any A-morphism v : B → A′ factors through a smooth A-algebra C, that is v is a composite A-morphism B → C → A′. This theorem called General Neron Desingularization was first proved by the second author (Popescu, Nagoya Math J 100:97–126, 1985). Later different proofs were given by André (Cinq exposés sur la désingularisation. Handwritten manuscript Ecole Polytechnique Fédérale de Lausanne, 1991), Swan (Neron-Popescu desingularization. In: Kang (ed) Algebra and geometry. International Press, Cambridge, pp 135–192, 1998) and Spivakovsky (J Am Math Soc 294:381–444, 1999). All the proofs are not constructive. In Pfister and Popescu (J Symb Comput 80:570–580, 2017) the authors gave a constructive proof together with an algorithm to compute the Neron Desingularization for 1-dimensional local rings. In this paper we go one step further. We give an algorithmic proof of the General Neron Desingularization theorem for 2-dimensional local rings and morphisms with small singular locus. The main idea of the proof is to reduce the problem to the one-dimensional case. Based on this proof we give an algorithm to compute the desingularization.