Poincaré series of compressed local Artinian rings with odd top socle degree

Abstract

We define a notion of compressed local Artinian ring that does not require the ring to contain a field. Let (R, m) be a compressed local Artinian ring with odd top socle degree s, at least five, and socle(R) ∩ m s−1 = m s . We prove that the Poincaré series of all finitely generated modules over R are rational, sharing a common denominator, and that there is a Golod homomorphism from a complete intersection onto R.

Detecting Koszulness and Related Homological Properties From the Algebra Structure of Koszul Homology

“…Compressed Gorenstein artinian local rings, and, more generally, compressed level artinian local rings are defined in terms of an extremal condition involving the length, embedding dimension, and the socle of the ring. We refer to [24] and [15] for the precise definitions. Such rings can be viewed as being "generic", in a sense explained in more detail in [ [24].…”

“…Certain higher order homology operations on Koszul homology, introduced by Golod [10], can be used to characterize extremality in the growth of the minimal free resolution of k over R. If the ring R is Golod, then it has the property that for all finitely generated R-modules M the Poincaré series i≥0 rank k (Tor R i (M, k))z i are rational and share a common denominator, see [9]. This property is also satisfied by other large classes of rings, and the recent papers [24] and [15] provide insight into the fact that the multiplicative structure of Koszul homology plays a role in establishing such results. In this paper we further explore how the structure of H R can be used to derive rationality of Poincaré series and other homological properties of R. In particular, we give special attention in the graded case to the Koszul property.…”

“…For appropriate values of t, the property P t holds for the class of compressed Gorenstein artinian local rings discussed in [24] and also for the class of compressed level local artinian rings of odd socle degree, see [15]. In particular, our results in Section 4 can be used to recover the results of [24] and [15] regarding the fact that when the socle degree is different than three, these rings can be obtained as homomorphic images of a hypersurface, via a Golod homomorphism.…”

A survey on the Koszul homology algebra

“…As a direct application of Theorem 4.10, Rossi and S ¸ega calculate the Poincaré series of a class of artinian Gorenstein rings in [56, Proposition 6.2] and show that they are indeed rational. For similar applications of the Koszul homology algebra structure in calculating Poincaré series, see [56,Theorem 5.1] and [43,Theorem 7.1].…”

“…Complete intersection rings, Gorenstein rings, and Golod rings are also characterized by their Koszul homology algebras; we discuss these classical characterizations in Section 4. Such characterizations have been quite useful in a wide range of applications in commutative algebra, particularly so in the study of Golod rings, and very recently in calculating Poincaré series over certain classes of rings; see for example [56] and [43]. We highlight a few such applications at the end of Section 5.…”