COMMUTATIVE ALGEBRA

YOSHIFUMI TSUCHIMOTO

06. Length, Hilbert function, Samuel function

Lemma 6.1 (Artin-Rees). Let I be an ideal of a Noetherian ring A. Let M be an A module with a submodule N. Then there exists an integer c > 0 such that

I^{n}M ∩ N = I^{n−c}(I^{c}M ∩ N)

holds for all n > c.

Theorem 6.2 (Krull). Let A be a ring with an ideal I. Let M be a finite A-module. We set N = ∩_{k=1}^{∞}I^{k}M. Then there exists a ∈ A such that a ≡ 1 mod I and aN = 0.

Theorem 6.3 (the Krull intersection theorem). Let A be a Noetherian ring.

(1) If I is in the Jacobson radical rad A of A, then for any finite A-module M, we have ∩_{n}I^{n}M = 0. Furthermore, for any submodule N of M, we have ∩_{n}I^{n}M ∩ N = 0.

(2) If A is an integral domain and I is a proper ideal of A, then we have ∩_{n}I^{n} = 0.

Proposition 6.4. Let A be a local ring. The following conditions are equivalent:

(1) l(A) < ∞ (which is also equivalent to saying that d(A) = 0 or that δ(A) = 0).

(2) dim(A) = 0.

(3) Any descending chain

I_{0} ⊃ I_{1} ⊃ I_{2} ⊃ …

of ideals of A stops.

Lemma 6.5. Let A be a ring with an ideal I. Let M be an A/Imodule. then we may (of course) consider M as an A-module. The dimensions dim(M), d(M), δ(M) are irrelevant of whether we consider M as an A/I-module or as an A-module.

Lemma 6.6. Let 0 → M′ → M → M′′ → 0 be an exact sequence of finite A-modules over a Noetherian local ring A. Then:

(1) d(M) = max(d(M′), d(M′′)).

(2) For any ideal I of definition of A, The leading coefficient of χ_{M}^{I} − χ_{M′′}^{I} coincides with that of χ_{M′}^{I}.

Theorem 6.7. Let (A, 𝔪) be a d-dimensional regular local ring with the residue field k = A/𝔪. Then

gr_{𝔪}(A) ≅ k[X_{1}, …, X_{d}].