Thursday, April 15, 2010

Homework #5

This homework is not due until May 11th, and as such I will be adding to it little by little as we cover more material.

1. Let 0 --> M' --> M --> M'' --> 0 be a short exact sequence of R-modules. Prove that one of the three following situations holds:
a) depth M' = depth M <= depth M'' or
b) depth M = depth M'' < depth M' or
c) depth M'' = depth M'-1 < depth M

2. Let M be a f.g. module of dimension n over a Noetherian local ring R. Suppose that for every part of a system of parameters of M (x_1,...,x_r) and for every p in Ass(M/(x_1,...,x_r)M), one has dim (R/p) = n-r. Show that M is Cohen-Macaulay.

3. Let R be a Noetherian local ring, M a f.g. R-module, and (x_1,...,x_r) an M-regular sequence. Show that pdim_{R} M/xM = pdim_R M + n (be careful with the infinite case!)

4. Give an example of a two dimensional ring that is not Cohen-Macaulay. (Note that it must necessarily not be normal)

5. Let R be a ring, and let M and N be R-modules. Suppose that (x_1,...,x_r) is an M-regular sequence in the annihilator of N. Show that Ext^r_R(N,M) \cong Hom_R(N,M/(x_1,...,x_r)M).

New Problems:

6. Let R be a Noetherian ring, M an R-module, N a f.g. R-module, and n > 0 an integer. Suppose that Ext^n_R(R/p,M) = 0 for all p \in Supp(N). Show that Ext^n_R(N,M) = 0.

7. Let (R,m,k) be a Noetherian local ring, \phi : M --> N a homomorphism of f.g. R-modules, and (x) = x_1,...,x_s an N-regular sequence. If \phi \otimes R/(x) is an isomorphism, then \phi is an isomorphism.

8. Let (R,m,k) be a Gorenstein local ring of dimension d, and M a f.g. module of finite projective dimension. Show that Tor_i^R(k,M) \cong Ext^d-i_R(k,M).