COMMUTATIVE ALGEBRA

YOSHIFUMI TSUCHIMOTO

05. Formal smoothness, unramifiedness.

Definition 5.1. Let A be a ring. Let I be an ideal of A. The Iadic topology on A is a topology defined by introducing for each a ∈ A, {a + I^{n}}_{n=1}^{∞} as the neighbourhood base.

It is easy to see that the I-adic topology is given by a quasi-metric defined by

d(a, b) = inf\{\frac{1}{2^{n}}; a − b ∈ I^{n}\}

Proposition 5.2. Let A be a ring. Let I be an ideal of A. We equip A with the I-adic topology. Then A is Hausdorff if and only if

⋂_{n} I^{n} = 0.

If this is the case, the completion of A is equal to \underleftarrow{lim} A/I^{n}. Thus A is complete Hausdorff if and only if a cannonically defined map

A → \underleftarrow{lim} A/I^{n}

is an isomorphism.

Example 5.3. Let p be a prime number. The ring ℤ of rational integers equipped with the pℤ-adic topology is Hausdorff. Its completion is denoted by ℤ_{p} and is called the ring of p-adic integers.

Definition 5.4. Let A be a ring. Let B be an A-algebra. Let I be an ideal of B. We equip B with the I-adic topology. B is I-smooth over A if for any A-algebra C, any ideal N of C satisfying N^{2} = 0 and any A-algebra homomorphism u : B → C/N which is continuous with respect to the discrete topology of C/N, there exists a lifting v : B → C of u to C, as an A-algebra homormophism.

Definition 5.5. Let A be a ring. Let B be an A-algebra. Let I be an ideal of B. We equip B with the I-adic topology. A-algebra B is Iunramified over A if for any A-algebra C, any ideal N of C satisfying N^{2} = 0 and any A-algebra homomorphism u : B → C/N which is continuous with respect to the discrete topology of C/N, there is at most one lifting v : B → C of u to C, as an A-algebra homormophism.

Definition 5.6. An A-algebra B is I-étale over A if it is both Ismooth and I-unramified.

Note that the conditions I-smooth/unramified/étale become weaker if we take I larger.

In the “strongest” case where I = 0, the continuity of the homomorphism u is automatic (any homomorphism is continuous.) 0smoothness (respectively, 0-unramifiedness, respectively, 0-étale-ness) is also refered to as formal smoothness (respectively, formal unramifiedness, respectively, formal étale-ness).