Part I The Generalization of Noetherian Methods

1

PART

I

THE GENERALIZATION OF NOETHERIAN METHODS

This section is concerned exclusively with Commutative Algebra and is completely independent of Analytic Geometry.

Its object

is to define the notion of Weierstrass Class and to investigate the properties of rings belonging to such a class.

The motiva-

tion for such a study comes from geometry, which, due to the variaus division theorems, yields numerous examples of such classes.

I n the second part of this work this notion will be

used principally to prove the factorial character of the local rings

,

O(E)

and to prove the Nullstellensatz.

I n fact it is

quite possible to give direct geometric proofs of these theorems as is indicated in Appendix 1 .

However the scope of this first

part is wider than these two results since it initiates the local algebraic study of analytic spaces in arbitrary dimension. It allows us thus to adapt to the infinite dimensional case

many methods used up till now only in the finite dimensional theory (as for example primary decompositions). The principal obstacle to the generalization to arbitrary dimension of finite dimensional algebraic methods arises from the fact that the rings we encounter are no longer Noetherian. The idea used to circumvent this obstacle, perfected by P. GABRIEL in

[GB]

,

is that of the localization of categories of modules.

This allows us to define the notion of being Noetherian relative to a theory of torsion and to develop, within this framework, a theory of primary decomposition. Chapter 2

.

This is the subject of

Such a technique has been introduced by L. CLABORN

and R. FOSSUM in

[CF] ,

However, in that article, the torsion

theory is defined in terms of the height of ideals.

For our

purposes the notion of height is not suitable (in particular

PART

2

for theorem 2 . 1 9 Nullstellensatz).

I

which is subsequently used to prove the We have thus preferred to replace the notion

o f height by the notion of grade.

However the properties of the

grade which are necessary for our developments are classical

only in the case of Noetherian rings. consecrate Chapter 1

Thus we have had to

to the establishment of these properties

without having recourse to Noetherian hypotheses. Finally, in Chapter 3

we obtain a generalization to the non-

Noetherian case of the notion of a Cohen-Macaulay ring.

By

isolating the notion of Weierstrass class we show that the Division Theorems of geometry furnish us with many examples of such rings.