# Local ring

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

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)