Local ring

A commutative ring with a unit that has a unique maximal ideal. If $ A $ is a local ring with maximal ideal $ \mathfrak m $, then the quotient ring $ A / \mathfrak m $ is a field, called the residue field of $ A $.

Examples of local rings. Any field or valuation ring is local. The ring of formal power series $ k [ [ X _ {1} \dots X _ {n} ] ] $ over a field $ k $ or over any local ring is local. On the other hand, the polynomial ring $ k [ X _ {1} \dots X _ {n} ] $ with $ n \geq 1 $ is not local. Let $ X $ be a topological space (or a differentiable manifold, an analytic space or an algebraic variety) and let $ x $ be a point of $ X $. Let $ A $ be the ring of germs at $ x $ of continuous functions (respectively, differentiable, analytic or regular functions); then $ A $ is a local ring whose maximal ideal consists of the germs of functions that vanish at $ x $.

Some general ring-theoretical constructions lead to local rings, the most important of which is localization (cf. Localization in a commutative algebra). Let $ A $ be a commutative ring and let $ \mathfrak p $ be a prime ideal of $ A $. The ring $ A _ {\mathfrak p } $, which consists of fractions of the form $ a / s $, where $ a \in A $, $ s \in A \setminus \mathfrak p $, is local and is called the localization of the ring $ A $ at $ \mathfrak p $. The maximal ideal of $ A _ {\mathfrak p } $ is $ \mathfrak p A _ {\mathfrak p } $, and the residue field of $ A _ {\mathfrak p } $ is identified with the field of fractions of the integral quotient ring $ A / \mathfrak p $. Other constructions that lead to local rings are Henselization (cf. Hensel ring) or completion of a ring with respect to a maximal ideal. Any quotient ring of a local ring is also local.

A property of a ring $ A $( or an $ A $- module $ M $, or an $ A $- algebra $ B $) is called a local property if its validity for $ A $( or $ M $, or $ B $) is equivalent to its validity for the rings $ A _ {\mathfrak p } $( respectively, modules $ M \otimes _ {A} A _ {\mathfrak p } $ or algebras $ B \otimes _ {A} A _ {\mathfrak p } $) for all prime ideals $ \mathfrak p $ of $ A $( see Local property).

The powers $ \mathfrak m ^ {n} $ of the maximal ideal $ \mathfrak m $ of a local ring $ A $ determine a basis of neighbourhoods of zero of the so-called local-ring topology (or $ \mathfrak m $- adic topology). For a Noetherian local ring this topology is separated (Krull's theorem), and any ideal of it is closed.

From now on only Noetherian local rings are considered (cf. also Noetherian ring). A local ring is called a complete local ring if it is complete with respect to the $ \mathfrak m $- adic topology; in this case $ A = \lim\limits _ {\leftarrow n } A / \mathfrak m ^ {n} $. In a complete local ring the $ \mathfrak m $- adic topology is weaker than any other separated topology (Chevalley's theorem). Any complete local ring can be represented as the quotient ring of the ring $ S [ [ X _ {1} \dots X _ {n} ] ] $ of formal power series, where $ S $ is a field (in the case of equal characteristic) or of a complete discrete valuation ring (in the case of different characteristic). This theorem makes it possible to prove that complete local rings have a number of specific properties that are absent in arbitrary Noetherian local rings (see [5]); for example, a complete local ring is an excellent ring.

A finer quantitative investigation of a local ring $ A $ is connected with the application of the concept of the adjoint graded ring $ \mathop{\rm Gr} ( A) = \oplus _ {n \geq 0 } ( \mathfrak m ^ {n} / \mathfrak m ^ {n+} 1 ) $. Let $ H _ {A} ( n) $ be the dimension of the vector space $ \mathfrak m ^ {n} / \mathfrak m ^ {n+} 1 $ over the residue field $ A / \mathfrak m $; as a function of the integer argument $ n $ it is called the Hilbert–Samuel function (or characteristic function) of the local ring $ A $. For large $ n $ this function coincides with a certain polynomial $ \overline{H}\; _ {A} ( n) $ in $ n $, which is called the Hilbert–Samuel polynomial of the local ring $ A $( see also Hilbert polynomial). This fact can be expressed in terms of a Poincaré series: The formal series

$$ P _ {A} ( t) = \sum _ {n \geq 0 } H _ {A} ( n) \cdot t ^ {n} $$

is a rational function of the form $ f ( t) ( 1 - t ) ^ {-} d( A) $, where $ f ( t) \in \mathbf Z [ t] $ is a polynomial and $ d ( A) - 1 $ is the degree of $ \overline{H}\; _ {A} $. The integer $ d ( A) $ is the (Krull) dimension $ \mathop{\rm dim} A $ of the ring $ A $ and is one of the most important invariants of a ring. Moreover, $ d ( A) $ is equal to the least number of elements $ a _ {1} \dots a _ {d} \in A $ for which the quotient ring $ A / ( a _ {1} \dots a _ {d} ) $ is Artinian (cf. Artinian ring). If these elements can be chosen in such a way that they generate the maximal ideal $ \mathfrak m $, then $ A $ is called a regular local ring. The regularity of $ A $ is equivalent to the fact that $ \mathop{\rm dim} ( \mathfrak m / \mathfrak m ^ {2} ) = \mathop{\rm dim} A $. For a $ d $- dimensional regular ring $ A $,

$$ H _ {A} ( n) = \ \left ( \begin{array}{c} n + d - 1 \\ d - 1 \end{array} \right ) $$

and $ P _ {A} ( t) = ( 1 - t ) ^ {-} d $. Geometrically, regularity means that the corresponding point of the (analytic or algebraic) variety is non-singular.

Besides the characteristic function $ H _ {A} $ and the dimension and multiplicity connected with it, a local ring has various invariants of a homological kind. The main one of these is the depth $ \mathop{\rm depth} A $( see Depth of a module); the condition $ \mathop{\rm depth} A = \mathop{\rm dim} A $ distinguishes among local rings the so-called Cohen–Macaulay rings (cf. Cohen–Macaulay ring). It is not known (1989) whether there is a module $ M $ with $ \mathop{\rm depth} M = \mathop{\rm dim} A $ for an arbitrary or a complete local ring $ A $. Other homological invariants are the so-called Betti numbers $ b _ {i} ( A) $ of a local ring $ A $, that is, the dimensions of the $ k $- spaces $ \mathop{\rm Tor} _ {i} ^ {A} ( k , k ) $, where $ k $ is the residue field of $ A $. The question of the rationality of the Poincaré series $ \sum _ {n \geq 0 } b _ {n} ( A) t ^ {n} $ is open, although for many classes of rings an affirmative answer is known. There are also invariants of an algebraic-geometrical nature; for their definition one uses resolution of the singularity corresponding to the local ring.

A similar theory has been constructed for semi-local rings; that is, rings that have finitely many maximal ideals. The role of a maximal ideal for them is played by the Jacobson radical.

References

[1]	W. Krull, "Dimensionstheorie in Stellenringen" J. Reine Angew. Math. , 179 (1939) pp. 204–226
[2]	C. Chevalley, "On the theory of local rings" Ann. of Math. (2) , 44 (1943) pp. 690–708
[3]	I.S. Cohen, "On the structure and ideal theory of complete local rings" Trans. Amer. Math. Soc. , 59 (1946) pp. 54–106
[4]	P. Samuel, "Algèbre locale" , Gauthier-Villars (1953)
[5]	M. Nagata, "Local rings" , Interscience (1962)
[6]	O. Zariski, P. Samuel, "Commutative algebra" , 2 , Springer (1975)
[7]	J.-P. Serre, "Algèbre locale. Multiplicités" , Lect. notes in math. , 11 , Springer (1965)
[8]	N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)
[9]	M.F. Atiyah, I.G. Macdonald, "Introduction to commutative algebra" , Addison-Wesley (1969)

Comments

For the notion of Krull dimension see Dimension of an associative ring.

A counter-example to the question of the rationality of the Poincaré series was given by D. Anick [a1].

References

[a1]	D. Anick, "Construction d'espaces de lacets et d'anneaux locaux à séries de Poincaré–Betti non rationelles" C.R. Acad. Soc. Paris , 290 (1980) pp. 1729–1732 (English abstract)