Commutative algebra

The branch of algebra studying the properties of commutative rings and objects relating to them (ideals, modules, valuations, etc., cf. Ideal; Module; Valuation).

Commutative algebra evolved from problems arising in number theory and algebraic geometry. These problems refer, as a rule, to concrete classes of rings. The fundamental object in number theory is the ring $ \mathbf Z $ of integers, and the fundamental fact of its arithmetic is that, in essence, any integer has a unique factorization as a product of primes. In the 1830's C.F. Gauss, E. Kummer and others discovered a connection between various questions in number theory (for example, on quadratic forms, on Fermat's theorem) and the arithmetic of quadratic and cyclic extensions of the field $ \mathbf Q $ of rational numbers (see [11]). However, the extension of classical arguments to rings of algebraic numbers were hindered by the fact that in these rings factorization into indecomposable factors is not unique (see Algebraic number theory). The conjecture of Kummer was that if one appends to the ordinary numbers certain "ideal" numbers (in the same way as one appends points at infinity in projective geometry), then uniqueness of factorization will be restored (see Ideal number).

Kummer succeeded in constructing such ideal numbers, or, as one now says, divisors (cf. Divisor), only for cyclic fields; however, his results stimulated R. Dedekind and L. Kronecker to extend the theory of divisors to arbitrary rings of algebraic numbers. In Dedekind's theory (1882) the role of the integral elements of a field emerged; but, even more important, it was here that the concepts of an ideal and a prime ideal made their appearance. Thus, the foundations of one-dimensional commutative algebra were laid.

In parallel with this, the formation of multi-dimensional commutative algebra was taking place in algebraic geometry. Algebraic geometry of that time studied the properties of algebraic curves in the plane and also the more general algebraic varieties defined as sets $ M \subset \mathbf C ^ {n} $ of common zeros of given polynomials $ P _ {1} \dots P _ {k} \in \mathbf C [ x _ {1} \dots x _ {n} ] $( the distinction here between affine and projective varieties is not important). The same variety $ M $ can also be defined by other equations, so that the ideal $ \mathfrak A $ of all polynomials that vanish on $ M $ is related to the variety $ M $ in a more invariant manner. This is another way leading to the notion of an ideal. However, until 1890 the algebraic foundations of algebraic geometry were in an embryonic state. The situation changed after the publication of the works of D. Hilbert. In 1893 he proved his "Nullstellensatz" (see Hilbert theorem). Somewhat earlier he established the following facts, which in many respects determined the subsequent direction of development of commutative algebra; the basis theorem (an ideal in $ \mathbf C [ x _ {1} \dots x _ {n} ] $ is generated by a finite number of polynomials), the theorem on syzygies, and the existence of the Hilbert polynomial for a homogeneous ideal in a ring of polynomials.

The experience of working with algebraic varieties of small dimension suggested that in general they consist of a finite number of irreducible subvarieties. This led to the algebraic problem of representing an ideal in the form of an intersection of ideals of simpler structure. This problem was solved by E. Lasker (see [4]), who introduced the notion of a primary ideal, which replaces, for the multi-dimensional case, a power of a prime ideal. He also defined the prime ideal associated to a primary ideal, and proved the existence of a primary decomposition of an arbitrary ideal in a rings of polynomials. The question of the uniqueness of such a decomposition was considered by F. Macaulay (1913). It was established that, although primary decomposition itself is not unique, the set of prime ideals associated with it is uniquely determined, as are the "isolated" primary components. The natural tendency to get rid of the non-uniqueness of a primary decomposition induced B.L. van der Waerden (1931) to introduce an equivalence relation for ideals that is coarser than equality. This led to the theory of divisorial ideals (cf. Divisorial ideal) or divisors, and made it possible to generalize the Kummer–Dedekind theory to a wider class of rings (see Krull ring).

Beginning with Kronecker and Lasker's dimension, defined then as the transcendence degree of the quotient ring corresponding to a prime ideal in a ring of polynomials, the modern combinatorial definition of dimension was subsequently proposed by W. Krull. If all prime ideals associated with an ideal $ \mathfrak A $ have the same dimension, then the ideal $ \mathfrak A $ is called unmixed. In a ring of polynomials, the ideals of the principal series are unmixed or, in modern terminology, a ring of polynomials is a Cohen–Macaulay ring.

Finally, Lasker extended a number of his results to the ring of convergent power series by considering these from an algebraic point of view.

Around 1900 results were obtained about rings of algebraic numbers and polynomials. Here it is worth calling to mind the constructive direction, in which one looks for explicit algorithms for determining the membership of a polynomial to some ideal. However, the concreteness of the material obscured the general regularities and relations. The theory of $ p $- adic numbers due to K. Hensel served as an impetus for the development of modern commutative algebra. The possibility of applying classical methods to such a non-traditional object has enabled one to realize and to see general ideas applying to arbitrary rings (satisfying some property or other, e.g. finiteness). A new stage had begun — the stage of abstract algebra, the systematic study of the structure of various classes of commutative rings. Such a study already appeared in the works of E. Noether. By combining the finite basis condition for any ideal with the maximum condition, that is, the ascending chain condition for ideals (a ring with this property is called Noetherian), she obtained in its most general form the Lasker–Macaulay theory of primary decompositions, which had previously been especially computational and cumbersome. She also gave an axiomatic description of Dedekind rings. At the same time E. Artin was studying rings with the minimum condition, the so-called Artinian rings; H. Grell introduced the concept of localization of an integral ring, an operation subsequently generalized by C. Chevalley and A.I. Uzkov. Krull had proved a theorem on principal ideals, initiating the dimension theory of Noetherian rings, and also a theorem on the intersection of the powers of an ideal in a Noetherian ring, which is the basis of the study of $ \mathfrak A $- adic topologies. The theory of divisorial ideals (1931) and the theory of valuations generalize the earlier investigations of Hensel and A. Ostrowski. Finally, mention should be made of Noether's normalization theorem, the elucidation of the role of the notion of integral dependence within the framework of the general theory of commutative rings, and also Krull's theorem on the lifting of prime ideals for integral extensions.

The paper [7] by Krull on local rings (cf. Local ring) opened up a new direction. Local rings are rings that have a unique maximal ideal; an example is the ring of germs of analytic functions on a complex manifold. In the abstract case, one passes to local rings through the operations of localization and completion or by Henselization (passage to a Hensel ring). Krull developed the dimension theory of local rings and introduced the concepts of a regular local ring, which corresponds to a geometric notion of a non-singular point of a variety. In the 1940s, local algebra and its algebraic-geometric applications was intensively developed. A local ring is endowed with a natural topology; this allows one to use the operation of completion and to compare the properties of a ring and its completion. It has been established that for the rings of algebraic geometry (geometric rings) passage to the completion preserves a number of important properties. Along with the investigations on the finiteness of the integral closure, this has set the direction of ring theory (see Excellent ring). In 1946 I.S. Cohen gave a description of the structure of complete local rings. Another direction relates to the foundations of intersection theory. By developing the ideas of Krull, P. Samuel introduced the notion of the graded ring associated with a local ring and revived the characteristic functions of Hilbert, which leads to yet another definition of the dimension of a local ring, and defined the multiplicity of an ideal.

The next development of the ideas of commutative algebra is connected with homological methods, the functorial approach and further geometrization. This was furthered by the trend, going back to Dedekind and Noether, towards the linearization of commutative algebra, according to which ideals of a ring are regarded as special cases of modules. The latter are a generalization of vector spaces, and correspond in the geometric representation to the notion of a family of vector spaces. The usual constructions of linear algebra such as the direct sum, a module homomorphism and the tensor product, can be applied to modules. The fruitfulness of such a broader view is already evident from the possibility of applying resolutions (cf. Resolution) and, together with them, homological algebra, which was formulated in the 1950s and which represented a further generalization of the theory of syzygies. This drew attention to modules of special type (see Projective module; Injective module; Flat module). The study of an arbitrary class of rings is closely related to the study of the modules over them (see Homological classification of rings). Real progress had been achieved in local algebra: J.-P. Serre characterized regular rings as rings of finite homological dimension and established the $ \mathop{\rm Tor} $- formula for the multiplicity of intersections. M. Auslander and D.A. Buchsbaum proved that a regular ring is factorial. The study was then begun of the functors $ \mathop{\rm Ext} $ and $ \mathop{\rm Tor} $ in connection with homomorphisms and tensor products of modules. In the same vein one studies Gorenstein (cf. Gorenstein ring) and Cohen–Macaulay rings, which admit a homological description, and Betti numbers of local rings.

A second feature of modern commutative algebra is the functorial approach, which emerged in the study of properties not of a single ring or module but of an entire system of such objects interrelated by morphisms. Change of the base ring, the theory of descent and the study of various functorial constructions are all a reflection of the above-mentioned trend. E.g., the preservation of many properties under localization and completion is connected with flatness of the corresponding extension.

A third feature is geometrization, i.e. viewing elements of a ring as functions on some space (which is natural in algebraic geometry and functional analysis). It was initially suggested to take for such a space the set of maximal ideals of a ring, and subsequently, the set of all prime ideals endowed with the Zariski topology (the prime spectrum of a ring). In particular, the possibility emerged of using the theory of sheaves and their cohomology. Commutative algebra has become an essential part of algebraic geometry that, as a result, is significantly expanding its range of applications. For example, the possibility has emerged of using and interpreting in geometric terms rings with divisors of zero or even rings with nilpotent elements. In its turn, geometric methods have re-animated those directions of commutative algebra that can be called global commutative algebra. These relate to the theory of invariants (cf. Invariants, theory of), algebraic $ K $- theory, cohomological constructions (the Picard group; the Brauer group; etc.), the study of groups of automorphisms and various invariants of rings, the theory of resolution of singularities, etc. At this point however, the boundaries between commutative algebra and algebraic geometry disappear.

References

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