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Commutative algebra

Yoshifumi Tsuchimoto

\fbox{10. Minimal free resolution of finitely generated modules over local rings}

PROPOSITION 10.1   Let $ A$ be a local ring with the residue field $ k$ . Let $ M,N$ be $ A$ -modules. We assume $ N$ is a finite module. Then an $ A$ -module homomorphism $ f:M\to N$ is surjective if and only if $ f\otimes 1_k: M\otimes_A k \to N\otimes_A k$ is surjective.


$\displaystyle 0 \leftarrow
M \overset{\epsilon}{\leftarrow}
F_0 \overset{d_1}{\leftarrow}
F_1 \overset{d_2}{\leftarrow}
F_2 \overset{d_3}{\leftarrow}\dots

be a resolution of $ M$ . We say that it is minimal free resolution if the following conditions are satisfied.
  1. each $ F_i$ is a finite free $ A$ -module.
  2. $ \overline{d_i}=0$ .
  3. $ \overline{\epsilon}: L_0 \otimes_A k \to M\otimes k$ is an isomorphism.

LEMMA 10.3   A finite module $ M$ over a Noetherian local ring has a minimal free resolution.

DEFINITION 10.4   Let $ M$ be an $ A$ -module over a ring $ A$ . We define the projective dimension of $ M$ ( $ \operatorname{proj\ dim}{M}$ ) to be the minimal length of the projective resolution of $ M$ .

PROPOSITION 10.5   Let $ A$ be a local ring with the residue field $ k$ . Then:
  1. If we have a minimal resolution $ 0 \leftarrow
M \leftarrow
F_0 \leftarrow
F_1 \leftarrow
F_2 \leftarrow \dots
$ of an $ A$ -module $ M$ , then we have $ \operatorname{Tor}_i^A(M,k)=\operatorname{rank}(F_k)$ .
  2. % latex2html id marker 739
$ \operatorname{proj\ dim}(M)=\sup\{i\vert \operatorname{Tor}^A_i(M,k)\neq 0\} \leq \operatorname{proj\ dim}_A k$ .

DEFINITION 10.6   We define the global dimension of a ring $ A$ by

$\displaystyle \operatorname{gl\ dim}(A)=sup\{\operatorname{proj\ dim}M \vert M \in \operatorname{Ob}(A\operatorname{-modules})\}

LEMMA 10.7   Let $ A$ be a local ring. Let $ k$ be its residue field. Then we have

$\displaystyle \operatorname{gl\ dim}(A)=\operatorname{proj\ dim}(k).

THEOREM 10.8 (Serre)   Let $ A$ be a Noetherian local ring. Then:

$\displaystyle A$ is regular$\displaystyle \iff \operatorname{gl\ dim}A=\dim A \iff \operatorname{gl\ dim}A< \infty.

THEOREM 10.9 (Serre)   A localization of a regular local ring at a prime ideal is also a regular local ring.

DEFINITION 10.10   A Noetherian ring is said to be regular if its localilzation at every prime is a regular local ring.

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