Commutative algebra

Yoshifumi Tsuchimoto

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.

DEFINITION 10.2   Let

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

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. $\operatorname{proj\ dim}(M)=\sup\{i\vert \operatorname{Tor}^A_i(M,k)\neq 0\} \leq \operatorname{proj\ dim}_A k$ .

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

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

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

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

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

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

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.