Math 515 homework #14

Due date: 19980313

D. J. Bernstein

19980306

The Artin-Rees lemma. Let I and L be ideals of a Noetherian ring R. The Artin-Rees lemma for ideals states that there is some u ≥ 0 such that I^{n} ∩ L = I^{n−u}(I^{u} ∩ L) for all n ≥ u.

More generally, let I be an ideal of a Noetherian ring R, let M be a finitely generated R-module, and let L be an R-submodule of M. The Artin-Rees lemma for modules states that there is some u ≥ 0 such that I^{n}M ∩ L = I^{n−u}(I^{u}M ∩ L) for all n ≥ u.

Idealization. In the following exercises, R is a ring, M is an R-module, and S is R × M. S is an R-algebra under (0, 0); (1, 0); (r, m) ↦ (−r, m); (r, m), (r′, m′) ↦ (r + r′, m + m′); (r, m), (r′, m′) ↦ (rr′, rm′ + r′m); and r ↦ (r, 0).

Another construction of S is as 2 × 2 matrices of the form ( \begin{matrix}r & m \\ 0 & r \end{matrix} ), with the usual matrix operations.

S is called the idealization of M over R. These exercises show how to translate questions about submodules of M into questions about ideals of S.

Exercise 36. Let I be an ideal of R. Let L be an R-submodule of M containing IM. Prove that I × L is an ideal of S. (In particular, I × M and 0 × L are ideals of S.)

Exercise 37. Let I, I′ be ideals of R. Let L, L′ be R-submodules of M containing IM and I′M respectively. Prove that (I × L)(I′ × L′) = II′ × (IL′ + I′L).

Exercise 38. Let I be an ideal of R. Prove that (I × M)^{n} = I^{n} × I^{n−1}M for n ≥ 1.

Exercise 39. Assume that R is a Noetherian ring, and that M is a finitely generated R-module. Prove that S is a Noetherian ring.

Exercise 40. Assume the Artin-Rees lemma for ideals. Use idealization to prove the Artin-Rees lemma for modules.