Questions: Homological Dimension

Question 1

What is the global dimension of a field k?

• A. 0
• B. 1
• C. Infinite
• D. Undefined

Answer: (A)

Over a field, every module is free (hence projective), so every module has projective dimension 0. The global dimension is the supremum of all projective dimensions, which is 0. More generally, a ring has global dimension 0 if and only if it is semisimple (every module is projective).

Question 2

The projective dimension of Z/nZ as a Z-module is 1 for any n ≥ 2.

• T. True
• F. False

Answer: True

The short exact sequence 0 → Z →^n Z → Z/nZ → 0 is a projective (in fact free) resolution of Z/nZ of length 1. Since Z/nZ is not projective (it is not a direct summand of a free Z-module, being finite), its projective dimension is exactly 1. Over a PID, every module has projective dimension at most 1.

Question 3

State the Auslander-Buchsbaum formula and explain its significance.

Model answer: For a finitely generated module M of finite projective dimension over a Noetherian local ring (R, m): pd(M) + depth(M) = depth(R). It says projective dimension and depth are complementary measures of complexity, summing to a constant (the depth of the ring).

The formula connects homological algebra (projective dimension, computed from resolutions) to commutative algebra (depth, computed from regular sequences). For instance, if R is Cohen-Macaulay of dimension d, and M has pd(M) = r, then depth(M) = d - r. The formula also proves that regular local rings have finite global dimension, since depth(R) = dim(R) bounds pd(M) for all M.

Question 4

A Noetherian local ring is regular if and only if it has finite global dimension.

• T. True
• F. False

Answer: True

This is Serre's theorem, proved using homological methods in the 1950s. The 'only if' direction: a regular local ring of dimension d has global dimension d (the residue field has a Koszul resolution of length d). The 'if' direction is deeper: finite global dimension implies the residue field has finite projective dimension, which forces the ring to be regular via the Auslander-Buchsbaum formula.

Question 5

What is the global dimension of the polynomial ring k[x_1, ..., x_n] over a field?

Model answer: n. The Hilbert syzygy theorem states that every finitely generated module over k[x_1, ..., x_n] has a free resolution of length at most n, and the residue field k = k[x_1,...,x_n]/(x_1,...,x_n) achieves this bound.

The Koszul complex on (x_1, ..., x_n) provides an explicit free resolution of k of length n. The Hilbert syzygy theorem is the global version: no module needs a longer resolution. This matches the Krull dimension n, consistent with the general fact that gl.dim(R) = dim(R) for regular rings.