FUNDAMENTAL PROPERTIES OF GERMS OF ANALYTIC MAPPINGS OF ANALYTIC SETS AND RELATED TOPICS SHUZO IZUMI Abstract. This is a sketch of the theory of germs of analytic mappings of analytic spaces and related theories. There are two major streams in singular- ity theory. One consists of the research of commutative algebra and algebraic geometry. They treat mainly rings and spaces rather than mappings. Another consists of theories of singularities of smooth mappings of manifolds. This is originated by topologists. In this note we are mainly concerned with local properties of singularities of general analytic mappings of singular spaces. This theory has a flavour of analysis and we may say that the theme of this note belongs to differential analysis or analytic singularity theory. We work mostly in the framework of the complex analytic geometry for the sake of simplicity. We point out some relations of this theory to the problems in other areas of mathematics also and show a list of related literature (of course, not complete). This note has overlapping parts with the postscript of the Japanese book [YFI]. 1. Introduction Let ξ be a point of the complex affine space C n , C{x − ξ} the algebra of convergent power series centred at ξ and a its ideal. The residue class algebra A := C{x − ξ}/a is called analytic algebra. We assume that a is a radical, i.e. A is reduced. Let X ξ be the germ at ξ of the analytic subset X of a neighbourhood of ξ defined by the representatives of a finite number of generators of a. Then (by the coherency theorem of Oka-Cartan and the Nullstellensatz of R¨ uckert) A is a C-algebra of germs at ξ of C-valued analytic functions (holomorphic functions) on X. The algebra A is often denoted by O X,ξ . All the germs of functions vanishing at ξ form the unique maximal ideal m A of A. Since C{x − ξ} is Noetherian, A is a Noetherian local algebra. We call ν (f ) := sup{p : f ∈ m p A } the (algebraic) order of f ∈ A. We can define a distance function on A by ρ(f,g) := exp[−ν (f − g)] which is invariant with respect to the addition and continuous with respect to the multiplication. This define a ring topology on A. Let us call it the Krull topology. We can define the completion ˆ A of A using Cauchy sequences. Let X and Y be analytic subsets of an open subset of affine spaces irreducible at ξ and η respectively. This means that the local algebras A := O X, ξ and B := O Y,η are integral domains. Let Φ : Y η −→ X ξ be a germ of an analytic map ˜ Φ: Y −→ X. This induces a homomorphism ϕ : A −→ B by pullback: ϕ(f ) := f ◦ Φ. This is a unitary C-algebra homomorphism and ϕ(m A ) ⊂ m B . Hence, ϕ 1