COMMUTATIVE ALGEBRA

YOSHIFUMI TSUCHIMOTO

Smoothness and finiteness properties

Definition 9.1. Let A be a ring. an A-module P is said to be projective if it satisfies the following condition: For any A-module morphism f : P → N and for any surjective A-module homomorphism π : M → N, f “lifts” to an A-module morphism f̂: M → I.

\begin{matrix}P & ⟶^{f̂} & M \\ ∥ & & ^{π}↓ \\ P & ⟶^{f} & N \end{matrix}

Lemma 9.2. An A-module M is projective if and only if it is a direct summand of a free A-module.

Proposition 9.3. Let B be a 0-smooth algebra over a ring A. Then Ω_{B/A}^{1} is projective.

Proof. Let us express the algebra B as a quotient C/I where C is a polynomial algebra and I is an ideal of C. Then by Theorem 08.5, we know that

0 → I/I^{2} → Ω_{C}^{1} ⊗_{C} B → Ω_{B}^{1} → 0

is split exact. So Ω_{B}^{1} is a direct sum of Ω_{C}^{1} ⊗_{C} B. On the other hand, Ω_{C}^{1} is free C-module so that Ω_{C}^{1} ⊗_{C} B is also a free B-module.

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We would like to define “smoothness” as a something good. Especially, we would expect “smooth algebras” to be flat. But that is not always true if we regard “smoothness” as 0-smoothness. The following example is an easy case of [1, example 7.2].

Example 9.4. Let us put A = ℂ[\{ \sqrt[2^{n}]{T} \}_{n=1}^{∞}] = ℂ[T, \sqrt{T} , \sqrt[2]{T} , \sqrt[4]{T} , …,] and put

I = \{f ∈ A; f(0) = 0\} = ∑_{n=1}^{∞} \sqrt[2^{n}]{T} A.

Then we see that I^{2} = I. Thus A/I is 0-smooth over A. where as A/I is not flat over A.

Definition 9.5. Let A be a ring.

(1) An A-algebra B is said to be finitely generated over A if B is generated by a finite set as an A-algebra. In other words, it is finitely generated if there exists a surjective A-algebra homomorphism from a finitely generated polynomial ring A[X_{1}, X_{2}, …, X_{r}] to B.

(2) An A-algebra B is said to be finitely presented over A if there exists a surjective A-algebra homomorphism φ from a finitely generated polynomial ring P = A[X_{1}, X_{2}, …, X_{r}] to B such that its kernel is a finitely generated ideal of P. is a finitely generated ideal of P.

YOSHIFUMI TSUCHIMOTO

Definition 9.6. Let A be a ring. An A-algebra B is said to be smooth over A if it is 0-smooth and finitely presented over A.

We may define unramified/étale algebras in a same manner. Let us recall the definition of Noetherian ring.

Definition 9.7. A ring is called Noetherian if its ideals are always finitely generated.

Proposition 9.8. If A is Noetherian, then:

(1) Any of its quotient ring is Noetherian.

(2) The polynomial ring A[X] is Noetherian. It follows that any finitely generated A-algebra B is also Noetherian. We note also that B is finitely presented over A in this case.

References

[1] M. Maltenfort, A new look at smoothness, Expo. Math. (2002), 59–93.