Introduction

The questions treated in this Book arose during the development of the theory of algebraic numbers and (later) algebraic geometry (cf. the Historical Note). From the 19th century onwards these two theories began to show remarkable analogies; the attempt to solve the problems they posed led to the isolation of a number of general ideas whose field of application is not limited to rings of algebraic numbers or algebraic functions; and, as always, it is advantageous to consider these in their most general form in order to see their true significance and the repercussions of one study on another. The concepts treated in this Book can be applied in principle to all commutative rings and modules over such rings; it must however be pointed out that substantial results are often obtained only under certain hypotheses of jniteness (which always hold in the classical cases), for example by assuming the modules to be finitely generated or the rings to be Noetherian.

The chief notions central to the first chapters are the following:

I. Localization and globalization. Let us begin for example with a system of Diophantine equations :

(*)

P_i(x_1, \ldots, x_r) = 0 \quad (1 \leq i \leq n)

where the $P_i$ are polynomials with integer coefficients and solutions $(x_i)$ are sought consisting of rational integers. It is possible to start approaching the problem by looking for solutions consisting of rational numbers, which involves looking at the same problem with the coefficientsof the $P_i$ considered as elements of the field of fractions $\mathbf{Q}$ of $\mathbf{Z}$ and the solutions sought with values in $\mathbf{Q}$ . A second step consists of seeing whether, given a prime number $p$ , there exist rational solutions whose denominators are not divisible by $p$ (integer solutions clearly satisfy this condition); this amounts, in this case, to lying in the subring $\mathbf{Z}_{(p)}$ of $\mathbf{Q}$ consisting of the rational numbers of this form, called the local ring of $\mathbf{Z}$ corresponding to the prime number $p$ . Clearly the passage from $\mathbf{Z}$ to $\mathbf{Q}$ and that from $\mathbf{Z}$ to $\mathbf{Z}_{(p)}$ are of the same form: in the two cases, the only denominators allowed do not belong to a certain prime ideal (the ideal $(0)$ and the ideal $(p)$ respectively). The same name "local ring" arises in algebraic geometry, where this notion appears in a more natural way: for example for the ring $\mathbf{C}[X]$ of polynomials in one variable with complex coefficients, the local ring corresponding to the prime ideal $(X - a)$ is the ring of rational fractions "regular" at the point $a$ (that is, without a pole at that point).

Every Diophantine problem and more generally every problem on A-modules (A a commutative ring) can be decomposed into two subsidiary problems: its solution is sought in the local rings A, corresponding to the different prime ideals $\mathfrak{p}$ of A ("localization"), then the question is asked whether it is possible to conclude from the existence for all $\mathfrak{p}$ of a solution to the "localized" problem that a solution exists to the problem posed initially ("passage from the local to the global"). Chapter II is devoted to the study of this double process and it is also seen that "localization" is not related only to prime ideals, but has a wider range.

11. Completion & local rings. A local ring A shares with fields the property of having only one maximal ideal m. This fact is used to transform, to a certain extent, a problem on A-modules into an analogous problem on vector spaces by passing to the quotient ring $A/m$ , as this latter is a field. If we return for example to the Diophantine system (*) this idea is none other than the principle of "reduction modulo K", transforming the equations into congruences mod. $p$ , which occurred naturally beginning with the very first works in the theory of numbers.

This being so, we clearly cannot hope in this way to obtain complete results for the original problem and it was quickly realized that to obtain more precise information it was necessary to consider, not only congruences modulo m, but also "higher" congruences modulo $m^n$ , for arbitrary integers $n > 0$ . It is thus found that, the larger $n$ , the closer in some way the original problem is "approached" (in the case $A = \mathbf{Z}$ for example, the reason is that an integer $\neq 0$ cannot be divisible by all the powers $p^n$ of a given prime number $p$ ; this number will therefore make its presence felt in the reduction mod. $p^n$ provided $n$ is taken large enough). The mathematical translation of this idea consists of considering on A a ring topology (cf. General Topology, Chapter III, § 6) in which the $m^n$ form a fundamental system of neighbourhoods of 0. But when we have for example, solved the system of congruences

\text{P}_i(x_1, \ldots, x_m) \equiv 0 \ (\text{mod. } p^k) \qquad (1 \leq i \leq n)

for every integer $k > 0$ , it still does not follow that the system () has a solution in the local ring $\mathbf{Z}_{(p)}$ ; the above hypothesis can be interpreted as saying that () admits a solution in the completion $\hat{\mathbf{Z}}_{(p)}$ of the topological ring $\mathbf{Z}_{(p)}$ .

The original problem, thus weakened, is finally transformed into the analogous problem for local rings of the type $A/m^n$ , which are also nearer to fields than general rings, since they have a nilpotent radical; in classical algebraic geometry this corresponds to a "differential" study of the problem in the neighbourhood of a given point.

Chapter III deals in a general way with these applications of topological notions to the theory of local rings. In Chapter VI a special aspect of this is studied, adapted on the one hand to more detailed studies of algebraic geometry, and above all to the arithmetic of algebraic number fields, where the local rings encountered (such as $\mathbf{Z}_{(p)}$ ) belong to a particularly simple class, that of "valuation rings", where divisibility is a total ordering (cf. Algebra, Chapter VI, § 1) of the set of principal ideals.

The study of the passage from a ring A to a local ring A, or to a completion $\hat{A}$ brings to light a feature common to these two operations, the property of flatness of the A-modules A, and $\hat{A}$ , which allows amongst other things the use of tensor products of such A-modules with arbitrary A-modules somewhat similar to that of tensor products of vector spaces, that is, without all the precautions surrounding their use in the general case. The properties associated with this notion, which are also applicable to modules over non-commutative rings, are the object of study in Chapter I.

III. Integers and decomposition of ideals. The study of divisibility in algebraic number fields necessitated from the start the introduction of the notion of integer in such a field K, generalizing the notion of rational integer in the field $\mathbf{Q}$ . The general theory of this notion of "algebraic integer", linked, as will be seen, to very strict conditions of finiteness, is developed in Chapter V; it can be applied to all commutative rings and is of great interest not only in arithmetic, but in algebraic geometry and even in the modern theory of "analytic spaces" over the field $\mathbf{C}$ .

One of the major obstacles to the extension of classical arithmetic to rings of algebraic integers has long been that the classical decomposition of a rational integer into prime factors does not extend in general to these rings. The creation of the theory of ideals was necessary to surmount this difficulty: the unique decompositionsoughtisthenestablishedforideals,thenotionofprimeidealbeing substituted for that of prime number. Moreover this result can be considered as a typical case where the "passage from the local to the global" is performed satisfactorily: the knowledge, for $x \in K$ , of the values at $x$ of all the "valuations" on K determines $x$ up to multiplication by an invertible integer.

In less simple rings than rings of algebraic integers (and even for example in ring: of polynomials in several indeterminates) this result is no longer valid. However it is possible to associate in a canonical way with every ideal a well-determined set of prime ideals : in algebraic geometry, if we consider for example in $K^n$ (K any commutative field) a subvariety defined by a system of polynomial equations $P_\alpha = 0$ , the irreducible components of this subvariety correspond bijectively with the minimal elements of the set of prime ideals thus associated with the ideal generated by the $P_\alpha$ . It is moreover possible (if we restrict ourselves to Noetherian rings) to give for every ideal a "decomposition" less precise than a decomposition as a product of prime ideals: here the product is in fact replaced by the intersection and the powers of prime ideals by "primary" ideals connected with the prime ideals associated with the ideal in question (but which are not direct generalizations of powers of prime ideals). The introduction of prime ideals associated with an ideal and the study of their properties is the subject of Chapter IV; here also the existence and certain uniqueness properties of the "primary decompositions" to which we have just alluded are proved; but it seems at present that these decompositions usually only play an accessory role in applications, the essential notion being that of prime ideal associated with an ideal.

In Chapter VII we examine in more detail rings whose properties most nearly approach those of rings of algebraic integers as far as decomposition as a product of prime ideals is concerned; amongst other things it is possible to introduce into these rings the notion of "divisor", which is the geometric aspect of this decomposition and plays an important role in algebraic geometry.

Finally Chapters VIII et seq. will deal with notions of more interest in algebraic geometry than in arithmetic (where they become trivial) and notably the concept of dimension.

With these notions we come to the frontier of algebraic geometry proper, a frontier which is ever moving and difficult to trace. For, if commutative algebra is an essential tool for the development of algebraic geometry in all its generality, conversely (as has already been seen above) the language of geometry proves very convenient for expressing the theorems of commutative algebra and suggesting a certain intuition naturally enough absent from abstract algebra; with the tendency to enlarge more and more the limits of algebraic geometry, algebraic and geometric language tend more than ever to merge.