Adic topology

A linear topology of a ring $ A $ in which the fundamental system of neighbourhoods of zero consists of the powers $ \mathfrak A ^ {n} $ of some two-sided ideal $ \mathfrak A $. The topology is then said to be $ \mathfrak A $- adic, and the ideal $ \mathfrak A $ is said to be the defining ideal of the topology. The closure of any set $ F \subset A $ in the $ \mathfrak A $- adic topology is equal to $ \cap _ {n \geq 0 } ( F + \mathfrak A ^ {n} ) $; in particular, the topology is separable if, and only if, $ \cap _ {n \geq 0 } \mathfrak A ^ {n} = (0) $. The separable completion $ \widehat{A} $ of the ring $ A $ in an $ \mathfrak A $- adic topology is isomorphic to the projective limit $ \lim\limits _ \leftarrow ( A / \mathfrak A ^ {n} ) $.

The $ \mathfrak A $- adic topology of an $ A $- module $ M $ is defined in a similar manner: its fundamental system of neighbourhoods of zero is given by the submodules $ \mathfrak A ^ {n} M $; in the $ \mathfrak A $- adic topology $ M $ becomes a topological $ A $- module.

Let $ A $ be a commutative ring with identity with an $ \mathfrak A $- adic topology and let $ \widehat{A} $ be its completion; if $ \mathfrak A $ is an ideal of finite type, the topology in $ \widehat{A} $ is $ \widehat{\mathfrak A} $- adic, and $ {\widehat{\mathfrak A} } {} ^ {n} = \mathfrak A ^ {n} \widehat{A} $. If $ \mathfrak A $ is a maximal ideal, then $ \widehat{A} $ is a local ring with maximal ideal $ \widehat{\mathfrak A} $. A local ring topology is an adic topology defined by its maximal ideal (an $ \mathfrak m $- adic topology).

A fundamental tool in the study of adic topologies of rings is the Artin–Rees lemma: Let $ A $ be a commutative Noetherian ring, let $ \mathfrak A $ be an ideal in $ A $, let $ E $ be an $ A $- module of finite type, and let $ F $ be a submodule of $ E $. Then there exists a $ k $ such that, for any $ n \geq 0 $, the following equality is valid:

$$ \mathfrak A ^ {n} ( \mathfrak A ^ {k} E \cap F ) = \ \mathfrak A ^ {k + n } E \cap F . $$

The topological interpretation of the Artin–Rees lemma shows that the $ \mathfrak A $- adic topology of $ F $ is induced by the $ \mathfrak A $- adic topology of $ E $. It follows that the completion $ \widehat{A} $ of a ring $ A $ in the $ \mathfrak A $- adic topology is a flat $ A $- module (cf. Flat module), that the completion $ \widehat{E} $ of the $ A $- module $ E $ of finite type is identical with $ E \otimes _ {A} \widehat{A} $, and that Krull's theorem holds: The $ \mathfrak A $- adic topology of a Noetherian ring is separable if and only if the set $ 1 + \mathfrak A $ contains no zero divisors. In particular, the topology is separable if $ \mathfrak A $ 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)