COMMUTATIVE ALGEBRA

YOSHIFUMI TSUCHIMOTO

Smoothness

Let us recall the universality of polynomial algebras.

Proposition 8.1. Let A be a commutative ring. Let B be an Aalgebra. That means, we assume that there is given a specific homomorphism (called the structure homomorphism) ι : A → B. Then for any family {b_{λ}}_{λ∈Λ} of elements of B, there exists a unique ring homomorphism

φ : A[{X_{λ}}_{λ∈Λ}] → B

such that φ|_{A} = ι and φ(X_{λ}) = b_{λ} for all λ ∈ Λ.

As a corollary, we see:

Proposition 8.2. Let A be a commutative ring. Then any polynomial algebra A[{X_{λ}}_{λ∈Λ} is 0-smooth over A.

Lemma 8.3. Let A be a ring. Let B be an A-algebra. Let I be a finitely generated ideal of B. Let us denote by B̂ = \underleftarrow{lim}(B/I^{n}) (respectively, Î) the completion of B (respectively, I) with respect to the I-adic topology. Then B is I-smooth over A if and only if B̂ is Î-smooth over A.

Proof. B/I^{n} ≅ B̂/Î^{n} for any n.

□

Corollary 8.4. Let A be a ring. Then A[[X_{1}, X_{2}, …, X_{n}]] is Ismooth over A for I = ∑_{i=1}^{n} X_{i}A[[X_{1}, X_{2}, …, X_{n}]].

Note. In general, A[[X]] is not 0-smooth over A. See [1] and the literatures cited there.

Theorem 8.5. Let A be a ring. Let B be an A-algebra with an ideal I. If B/I is 0-smooth over A, then the sequence (which appears in Lemma 04.2)

0 → I/I^{2} → Ω_{B/A}^{1} ⊗ (B/I) → Ω_{B/I}^{1} → 0

is split exact.

The following theorem says that the converse is true if the ring B is 0-smooth.

Theorem 8.6. Let A be a ring. Let B be an A-algebra with an ideal I. Assume B is 0-smooth over A. If the exact sequence

0 → I/I^{2} → Ω_{B/A}^{1} ⊗ (B/I) → Ω_{B/I}^{1} → 0

is split exact, then B/I is 0-smooth over A.

References

[1] H. Matsumura, Commutative ring theory, Cambridge university press, 1986.