Problems, 215B

Do 15 problems. Due Dec 5.

1. Let A = A_{0} ⊕ A_{1} ⊕ … be a graded commutative ring. Prove that A is Noetherian if and only if A_{0} is Noetherian and the A_{0}-algebra A is finitely generated.

2. Let R be a local ring, f ∈ R[t_{1}, …, t_{n}] a polynomial. Suppose that one of the coefficients of f is invertible in R. Prove that f is a non-zero-divisor in R[t_{1}, …, t_{n}].

3. Let R be a complete Noetherian local ring with maximal ideal P. Suppose there is a subfield K ⊂ R mapping isomorphically onto R/P. Let x_{1}, …, x_{d} be a system of parameters of R. Prove that R is a finitely generated module over the subring K[[x_{1}, …, x_{d}]].

4. Let f and g be functions {0, 1, 2, … } → ℤ such that f(n+1)−f(n) = g(n) for all n ≥ 0. Prove that if g is a polynomial of degree d then f is a polynomial of degree d + 1.

5. Let R be a Noetherian local ring with maximal ideal P and x ∈ P. Prove that

dim R ≥ dim(R/xR) ≥ dim R − 1.

6. Let R be a Noetherian local ring with maximal ideal P and a_{1}, a_{2}, …, a_{k} ∈ P. Prove that {a_{1}, a_{2}, …, a_{k}} is a part of a system of parameters if and only if dim(R/⟨a_{1}, a_{2}, …, a_{k}⟩) = dim R − k.

7. Let R → S be a flat local homomorphism of Noetherian local rings, P the maximal ideal of R. Prove that is R and S/P S are regular then so is S.

8. Let a_{1}, … a_{r} be an R-regular sequence. Prove that for any positive integers k_{1}, … k_{r} the sequence a_{1}^{k_{1}}, …, a_{r}^{k_{r}} is also R-regular.

9. Let a_{1}, …, a_{n} be a n R-regular sequence of a Noetherian local ring R and I = ⟨a_{1}, …, a_{n}⟩. Prove that for any finitely generated R-module M and any i there are canonical isomorphisms

Tor_{i}^{R}(R/I, M) ≃ H_{i}(a, M) ≃ Ext_{R}^{n−i}(R/I, M).

10. Let R = F [t_{1}, t_{2}, t_{3}] with F a field and a_{1} = t_{1}(t_{2} − 1), a_{2} = t_{2}, a_{3} = t_{3}(t_{2} − 1). Prove that a_{1}, a_{2}, a_{3} is an R-regular sequence while a_{1}, a_{3}, a_{2} is not.

11. Show that a Noetherian local ring R is regular if and only if the maximal ideal of R can be generated by an R-regular sequence.

12. Let R be a local C.M. ring and G a finite group of ring automorphisms of R. Prove that if the order of G is invertible in R then the subring R^{G} of G-invariant elements in R is also C.M.

13. Let I be a prime ideal of the ring R = F [t_{1}, t_{2}, t_{3}] with F a field. Prove that the ring R/I is C.M.

14. Prove that the subring F [t_{1}^{4}, t_{1}^{3}t_{2}, t_{1}t_{2}^{3}, t_{2}^{4}] of F [t_{1}, t_{2}] with F a field is not C.M.

15. Let R be the localization of F [t_{1}, t_{2}, t_{3}, t_{4}] (F a field) at the maximal ideal ⟨t_{1}, t_{2}, t_{3}, t_{4}⟩ and I the ideal of R generated by t_{1}t_{2}, t_{3}t_{4} and t_{1}t_{3} + t_{2}t_{4}. Determine dim(R/I) and depth(R/I).

16. Let f : L → R and f′ : L′ → R be two R-linear maps. Let g : L ⊕ L′ → R be the map defined by g(l + l′) = f(l) + f′(l′). Show that the Koszul complex K(g) is isomorphic to K(f) ⊗ K(f′).

17. Prove that if I is a prime ideal of a complete intersection ring R then the localization R_{I} is also a complete intersection ring.

18. Find an example of a Noetherian local ring R and a finitely generated R-module M such that depth M > depth R.

19. Let M be a finitely generated module over a Noetherian local ring R with maximal ideal P. Prove that M is free if and only if Tor_{1}^{R}(R/P, M) = 0.

20. Let R be a Noetherian local ring with maximal ideal P. Show that if the R-module R/P has a finite free resolution of length n then every finitely generated R-module has a finite free resolution of length n.