Let (S, n, k) be a commutative noetherian local ring and M be a finitely generated S-module. Then the Poincaré series of M is rational if any of the following holds: edim S-depth S ≤ 3; edim S-depth S = 4 and S is Gorenstein; S is one link from a complete intersection; S is two links from a complete intersection and S is Gorenstein. To prove this result we assume, without loss of generality, that S = R I with (R, m) regular local and ⊆. The key to the argument is to produce a factorization of the canonical map R → S as a composition of a complete intersection R → C with a Golod map C → S. This is accomplished by invoking a theorem of Avramov and Backelin once one has existence of a DGΓ-algebra structure on a minimal R-free resolution of S together with detailed knowledge of the structure of the induced homology algebra TorR(S, k) = H(KS). Linkage theory provides the main technical tool for this analysis. © 1988.
Avramov, L. L., Kustin, A. R., & Miller, M. (1988). Poincaré series of modules over local rings of small embedding codepth or small linking number. Journal of Algebra, 118(1), 162–204. https://doi.org/10.1016/0021-8693(88)90056-7