Adic topology
A linear topology of a ring in which the fundamental system of neighbourhoods of zero consists of the powers
of some two-sided ideal
. The topology is then said to be
-adic, and the ideal
is said to be the defining ideal of the topology. The closure of any set
in the
-adic topology is equal to
; in particular, the topology is separable if, and only if,
. The separable completion
of the ring
in an
-adic topology is isomorphic to the projective limit
.
The -adic topology of an
-module
is defined in a similar manner: its fundamental system of neighbourhoods of zero is given by the submodules
; in the
-adic topology
becomes a topological
-module.
Let be a commutative ring with identity with an
-adic topology and let
be its completion; if
is an ideal of finite type, the topology in
is
-adic, and
. If
is a maximal ideal, then
is a local ring with maximal ideal
. A local ring topology is an adic topology defined by its maximal ideal (an
-adic topology).
A fundamental tool in the study of adic topologies of rings is the Artin–Rees lemma: Let be a commutative Noetherian ring, let
be an ideal in
, let
be an
-module of finite type, and let
be a submodule of
. Then there exists a
such that, for any
, the following equality is valid:
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The topological interpretation of the Artin–Rees lemma shows that the -adic topology of
is induced by the
-adic topology of
. It follows that the completion
of a ring
in the
-adic topology is a flat
-module (cf. Flat module), that the completion
of the
-module
of finite type is identical with
, and that Krull's theorem holds: The
-adic topology of a Noetherian ring is separable if and only if the set
contains no zero divisors. In particular, the topology is separable if
is contained in the (Jacobson) radical of the ring.
References
[1] | O. Zariski, P. Samuel, "Commutative algebra" , 2 , Springer (1975) |
[2] | N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French) |
Adic topology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Adic_topology&oldid=16092