Introduzione all’algebra commutativa by M. F. Atiyah, , available at Book Depository with free delivery worldwide. Metodi omologici in algebra commutativa by Gaetana Restuccia, , available at Book Depository with free delivery worldwide. Commutative Algebra is a fundamental branch of Mathematics. following are some research topics that distinguish the Commutative Algebra group of Genova: .
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Commutative algebra is the branch of algebra that studies commutative ringstheir idealsand modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Commutative algebra is the main technical tool in the local study of schemes. The study of rings that are not necessarily commutative is known as noncommutative algebra ; it includes ring theoryrepresentation theoryand the theory of Banach algebras.
Commutative algebra is essentially the study of the rings occurring in algebraic number theory and algebraic geometry. In algebraic number theory, the rings of algebraic integers are Dedekind ringswhich constitute therefore an important class of commutative rings.
Considerations related to modular arithmetic have led to the notion of a valuation ring. The restriction of algebraic field extensions to subrings has led to the notions of integral extensions and integrally closed domains as well as the notion of ramification of an extension of valuation rings. The notion of localization of a ring in particular the localization with respect to a prime idealthe localization consisting in inverting a single element and the total quotient ring is one of the main differences between commutative algebra and the theory of non-commutative rings.
It leads to an important class of commutative rings, the local rings that have only one maximal ideal. The set of the prime ideals of a commutative ring is naturally equipped with a topologythe Zariski topology. All these notions are widely used in algebraic geometry and are the basic technical tools for the definition of scheme theorya generalization of algebraic geometry introduced by Grothendieck.
Many other notions of commutative algebra are counterparts of geometrical notions occurring in algebraic geometry. This is the case of Krull dimensionprimary decompositionregular ringsCohen—Macaulay ringsGorenstein rings and many other notions. The subject, first known as ideal theorybegan with Richard Dedekind ‘s work on idealsitself based on the earlier work of Ernst Kummer and Leopold Kronecker. Later, David Hilbert introduced the term ring to generalize the earlier term number ring.
Hilbert introduced a more abstract approach to replace the more concrete and computationally oriented methods grounded in such things as complex analysis and classical invariant theory. In turn, Hilbert strongly influenced Emmy Noetherwho recast many earlier results in terms of an ascending chain conditionnow known as the Noetherian condition. Another important milestone was the work of Hilbert’s student Emanuel Laskerwho introduced primary ideals and proved the first version of the Lasker—Noether theorem.
Commutative Algebra (Algebra Commutativa) L
The main figure responsible commutatuva the birth of commutative algebra as a mature subject was Wolfgang Krullwho introduced the fundamental notions of localization and completion of a ring, as well as that of regular local rings. He established the concept of the Krull dimension of a ring, first for Noetherian rings before moving on to expand his theory to cover general valuation rings and Krull rings. To this day, Krull’s principal ideal theorem is widely considered the single most important foundational theorem in commutative algebra.
These results paved the way for the introduction of commutative algebra into algebraic geometry, an idea which would revolutionize the latter subject.
Much of the modern development of commutative algebra emphasizes modules. Both ideals of a ring R and R -algebras are special cases of Aglebra -modules, so module theory encompasses both ideal theory and the theory of ring extensions.
Though it was already incipient in Kronecker’s work, the modern approach to commutative algebra using module theory is usually credited to Krull and Noether. In mathematicsmore specifically in the area of modern algebra known as ring theorya Noetherian ringnamed after Emmy Noetheris a ring in which every non-empty set of ideals has a alggebra element.
Equivalently, a ring is Noetherian if it satisfies the ascending chain condition on ideals; that is, given any chain:. For a commutative ring to be Noetherian it suffices that every prime ideal of the ring is finitely generated. The result is due to I. The notion of a Noetherian ring is of fundamental importance in both commutative and noncommutative ring theory, due to the role it plays in simplifying the ideal structure of a ring. For instance, the ring of integers and the polynomial ring over a field are both Noetherian rings, and consequently, such theorems as the Lasker—Noether theoremthe Krull intersection theoremand the Hilbert’s basis theorem hold for them.
Furthermore, if a ring is Noetherian, then it satisfies the descending chain condition on prime ideals. This property suggests a deep theory of dimension for Noetherian rings beginning with the notion of the Krull dimension. If R is a left resp. In Zthe primary ideals are precisely the ideals of the form p e where p is prime and e is a positive integer. Thus, a primary decomposition of n corresponds to representing n as the intersection of finitely many primary ideals.
The Lasker—Noether theoremgiven here, may be seen as a certain fommutativa of the fundamental theorem of arithmetic:. Let R be a commutative Noetherian ring and let I be an ideal of R.
Then I may be written as the intersection of finitely many primary ideals with distinct radicals ; that is:. The localization is a formal way to introduce the “denominators” to a given ring or a module. The archetypal example is the construction of the ring Q of rational numbers from the ring Z commuttativa integers.
A completion is any of several related functors on rings and modules that result in complete topological rings and modules. Completion is similar to localizationand together they are among the most basic tools in analysing commutative rings. Complete commutative rings have simpler structure than the general ones and Hensel’s lemma applies to them.
Metodi omologici in algebra commutativa
The Zariski topology defines a topology on the spectrum of a ring the set of prime ideals. This is defined in analogy with the classical Zariski topology, where closed sets in affine space are those defined by polynomial equations. To see the connection with the classical picture, note that for any set S of polynomials over an algebraically closed fieldit follows from Hilbert’s Nullstellensatz that the points of V S in the old sense are exactly the tuples a 1Thus, V S is “the same as” the maximal ideals containing S.
Grothendieck’s innovation in defining Spec was to replace maximal ideals with all prime ideals; aglebra this formulation it is natural to simply generalize this observation to the definition of a closed set in the spectrum of a ring.
The existence of primes and the unique factorization theorem laid the foundations for concepts such as Noetherian rings and the primary decomposition. Commutative algebra in the form of commutaiva rings and their quotients, used in the definition of algebraic varieties has always been a part of algebraic geometry. However, in the late s, algebraic varieties were subsumed into Alexander Grothendieck ‘s concept of a scheme.
Their local objects are affine schemes or prime spectra, which are locally ringed spaces, which form a category that is antiequivalent dual to the category of commutative unital rings, extending the duality between the category of affine algebraic varieties over a field kand the category of finitely generated reduced k -algebras.
The gluing algebrz along the Zariski topology; one can glue within the category of commutatuva ringed spaces, but also, using the Yoneda embedding, within the more abstract category of presheaves of sets over the category of affine schemes. The Zariski topology in the set-theoretic sense is then replaced by a Zariski topology in the sense of Grothendieck topology.
Nowadays some other examples have become prominent, including the Nisnevich topology. Sheaves can be furthermore generalized to stacks in the sense of Grothendieck, usually with some additional representability conditions, leading to Artin stacks and, even finer, Deligne-Mumford stacksboth often called algebraic stacks.
From Wikipedia, the free encyclopedia. This article is about the branch of algebra that studies commutative rings. For algebras that are commutative, see Commutative algebra structure. Abstract Algebra 3 ed.