Exercise 1 Let R be a Noetherian local ring. Let M be a f.g. R-module, and take a free resolution of M. Let N be the n-th syzygy. Prove that

depth N ≥ min (n, depth R)

Exercise 2 Let R be a Cohen-Macaulay ring, dim R = d. Let M be a f.g. R-module, and take a free resolution of M. Let N be the d-th syzygy. Prove that

N = 0 or N is a maximal Cohen-Macaulay module.

PROOF: Suppose N ≠ 0. In general,

depth N ≤ dim N ≤ dim R

Now R is Cohen-Macaulay so dim R = depth R and we use ex. 1 to deduce

depth N ≥ min (d, d) = d = dim R

so this yields

depth N = dim N = dim R

and therefore, N is a maximal Cohen-Macaulay module.