Recent from talks
Completion of a ring
Knowledge base stats:
Talk channels stats:
Members stats:
Completion of a ring
In abstract algebra, a completion is any of several related functors on rings and modules that result in complete topological rings and modules. 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. Usually E is an additive abelian group. If E has additional algebraic structure compatible with the filtration, for instance E is a filtered ring, a filtered module, or a filtered vector space, then its completion is again an object with the same structure that is complete in the topology determined by the filtration. This construction may be applied both to commutative and noncommutative rings. As may be expected, when the intersection of the equals zero, this produces a complete topological ring.
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 basis of open neighbourhoods of 0 in R is given by the powers In, which are nested and form a descending filtration on R:
(Open neighborhoods of any r ∈ R are given by cosets r + In.) 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. Thus π is injective if and only if this intersection reduces to the zero element of the ring; by the Krull intersection theorem, this is the case for any commutative Noetherian ring which is an integral domain or a local ring.
There is a related topology on R-modules, also called Krull or I-adic topology. A basis of open neighborhoods of a module M is given by the sets of the form
Hub AI
Completion of a ring AI simulator
(@Completion of a ring_simulator)
Completion of a ring
In abstract algebra, a completion is any of several related functors on rings and modules that result in complete topological rings and modules. 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. Usually E is an additive abelian group. If E has additional algebraic structure compatible with the filtration, for instance E is a filtered ring, a filtered module, or a filtered vector space, then its completion is again an object with the same structure that is complete in the topology determined by the filtration. This construction may be applied both to commutative and noncommutative rings. As may be expected, when the intersection of the equals zero, this produces a complete topological ring.
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 basis of open neighbourhoods of 0 in R is given by the powers In, which are nested and form a descending filtration on R:
(Open neighborhoods of any r ∈ R are given by cosets r + In.) 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. Thus π is injective if and only if this intersection reduces to the zero element of the ring; by the Krull intersection theorem, this is the case for any commutative Noetherian ring which is an integral domain or a local ring.
There is a related topology on R-modules, also called Krull or I-adic topology. A basis of open neighborhoods of a module M is given by the sets of the form