Commutative Algebra

Excercises 2

1. Let (I, ≥) be a partially ordered set which is directed. Let A be a ring and let (N_{i}, φ_{i,i′}) be a directed system of A-modules indexed by I. Suppose that M is another A-module. Prove that

lim_{i∈I} M ⊗_{A} N_{i} ≅ M ⊗_{A} ( lim_{i∈I} N_{i}).

Remark. A module M over R is said to be of finite presentation over R if it is isomorphic to the cokernel of a map of finite free modules R^{⊕n} → R^{⊕m}.

2. Prove that any module over any ring is a. the limit of its finitely generated submodules, and b. in some way a limit of finitely presented modules.

3. Let S be a multiplicative subset of the ring A. a. For an A-module M show that S^{−1}M = S^{−1}A ⊗_{A} M. b. Show that S^{−1}A is flat over A.

4. Find an injection M_{1} → M_{2} of A-modules such that M_{1} ⊗ N → M_{2} ⊗ N is not injective in the following cases: a. A = k[x, y] and N = (x, y) ⊂ A. (Here and below k is a field.) b. A = k[x, y] and N = A/(x, y).

5. Give an example of a ring A and a finite A-module M which is a flat but not a projective A-module.

Remark. If M is of finite presentation and flat over A, then M is projective over A. Thus your example will have to involve a ring A which is not Noetherian. I know of an example where A is the ring of C^{∞}-functions on ℝ.

6. Let A = k[x, y]_{(x,y)} be the local ring of the affine plane at the origin. Make any assumption you like about the field k. Suppose that f = x^{3} +x^{2}y^{2} +y^{100} and g = y^{3} −x^{999}. What is the length of A/(f, g) as an A-module? (Possible way to proceed: think about the ideal that f and g generate in quotients of the form A/𝔪_{A}^{n} = k[x, y]/(x, y)^{n} for varying n. Try to find n such that A/(f, g) + 𝔪_{A}^{n} ≅ A/(f, g) + 𝔪_{A}^{n+1} and use NAK.)