Structure of local homomorphisms (Q1322522)

The authors study and define a structure theorem for local homomorphisms. Given a local homomorphism \(\varphi : (R,{\mathfrak m}) \to (S,{\mathfrak n})\) then the canonical morphism \(\varphi'\) from \(R\) to the completion of \(S\) is factorized into a flat homomorphism followed by a surjective one and the flat map has a regular closed fiber. This factorisation plays a role similar to the structure theorem of I. S. Cohen for complete rings. The authors extend definitions like dimension, depth, Cohen-Macaulay, defect, from rings to morphisms, show some properties and promise to give more in forthcoming papers.