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Commutative algebra

Yoshifumi Tsuchimoto

\fbox{Smoothness and finiteness properties}

DEFINITION 09.1   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 \to N $ and for any surjective $ A$ -module homomorphism $ \pi:M \to N$ , $ f$ ``lifts'' to an $ A$ -module morphism $ \hat f: M\to I$ .

$\displaystyle \begin{CD}
P @>\hat f >>M \\
@\vert @V\pi VV \\
P @>f >> N
\end{CD}$

LEMMA 09.2   An $ A$ -module $ M$ is projective if and only if it is a direct summand of a free $ A$ -module.

PROPOSITION 09.3   Let $ B$ be a 0 -smooth algebra over a ring $ A$ . Then $ \Omega^1_{B/A}$ 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

$\displaystyle 0\to I/I^2 \to \Omega^1_{C}\otimes_C B \to \Omega^1_B \to 0
$

is split exact. So $ \Omega^1_B$ is a direct sum of $ \Omega^1_{C}\otimes_C B$ . On the other hand, $ \Omega^1_{C}$ is free $ C$ -module so that $ \Omega^1_{C}\otimes_C B$ is also a free $ B$ -module.

% latex2html id marker 791
$ \qedsymbol$

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 09.4   Let us put % latex2html id marker 824
$ A=\mathbb{C}[\{\sqrt[2^n]{T}\}_{n=1}^\infty ]
=\mathbb{C}[T,\sqrt{T},\sqrt[2]{T},\sqrt[4]{T},\dots,]$ and put

% latex2html id marker 826
$\displaystyle I=\{f \in A; f(0)=0\}=\sum_{n=1}^\infty \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 09.5   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,\dots,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 $ \varphi$ from a finitely generated polynomial ring $ P=A[X_1,X_2,\dots,X_r]$ to $ B$ such that its kernel is a finitely generated ideal of $ P$ . is a finitely generated ideal of $ P$ .

DEFINITION 09.6   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 09.7   A ring is called Noetherian if its ideals are always finitely generated.

PROPOSITION 09.8  

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.


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2011-07-21