Determinant is a most useful tool in every branch of mathematics, especially in linear mathematics. What kind of quantities do take a similar universal important role as determinant, in advanced branches of mathematics? In the present articles, showing the usefulness of semi-invariants in the classical invariant theory, we shall give a partial answer of the above question.

In this paper we are concerned with subvarieties which realize Chern classes of holomorphic vector bundles. The existence of these subvarieties is known in some cases (for instance, see A. Grotheridieck [2] for projective algebraic varieties and M. Cornalba and P. Griffiths [1] for Stein manifolds). In the present paper we realize Chern classes by subvarieties with singularities of a certain type. Our main theorem is as follows (see Def. 1.1.3 for the definition of quasilinear subvarieties).

S. C. Kleene developed the theory of recursive functionals of finite types in Kleene [5]. He proved that a set A of natural numbers is recursive in E if and only if A is hyperarithmetical, where E is the type 2 object defined by

Let A be a Noetherian local ring of dimension d and with maximal ideal m. Then A is called Buchsbaum if every system of parameters is a weak sequence. This is equivalent to the condition that, for every parameter ideal q, the difference is an invariant I(A) of A not depending on the choice of q. (See Section 2 for the detail.) The concept of Buchsbaum rings was introduced by Stückrad and Vogel [8], and the theory of Buchsbaum singularities is now developing (cf. [6], [7], [9], [10], and [12]).

The purpose of this paper is to present the results of E. Artin and Furtwängler, with which they proved the principal ideal theorem, as a structure theorem of the idele group of an algebraic number field. Such treatment may be helpful to clarify the Arithmetic nature these results possess.

For a positive integer N, put