Completion is similar to localization, and together they are among the most basic tools in analysing commutative rings.
Complete commutative rings have a simpler structure than general ones, and Hensel's lemma applies to them.
In algebraic geometry, a completion of a ring of functions R on a space X concentrates on a formal neighborhood of a point of X: heuristically, this is a neighborhood so small that all Taylor series centered at the point are convergent.
An algebraic completion is constructed in a manner analogous to completion of a metric space with Cauchy sequences, and agrees with it in the case when R has a metric given by a non-Archimedean absolute value.
Suppose that E is an abelian group with a descending filtration of subgroups.
One then defines the completion (with respect to the filtration) as the inverse limit: This is again an abelian group.
In commutative algebra, the filtration on a commutative ring R by the powers of a proper ideal I determines the Krull (after Wolfgang Krull) or I-adic topology on R. The case of a maximal ideal
is especially important, for example the distinguished maximal ideal of a valuation ring.
The (I-adic) completion is the inverse limit of the factor rings, pronounced "R I hat".
The kernel of the canonical map π from the ring to its completion is the intersection of the powers of I.
Completions can also be used to analyze the local structure of singularities of a scheme.
have similar looking singularities at the origin when viewing their graphs (both look like a plus sign).
Notice that in the second case, any Zariski neighborhood of the origin is still an irreducible curve.
More explicitly, the power series: Since both rings are given by the intersection of two ideals generated by a homogeneous degree 1 polynomial, we can see algebraically that the singularities "look" the same.
This is because such a scheme is the union of two non-equal linear subspaces of the affine plane.