Initial growth of the Betti sequence of the canonical module: when is $\beta_n(\omega_R)>\beta_{n-1}(\omega_R)$, and is $\beta_n(\omega_R)$ determined by the preceding Betti numbers?

For a Cohen–Macaulay local ring $R$ with canonical module $\omega_R$, Problem 1.1 of Asgharzadeh's preprint "On the initial Betti numbers" — generalizing Question 2.6 of Jorgensen–Leuschke (2007) and the questions of Christensen–Striuli–Veliche (2010) — asks two coupled questions about the initial segment of the minimal free resolution of $\omega_R$: (i) characterize precisely when $\beta_n(\omega_R)>\beta_{n-1}(\omega_R)$ (in particular, must this strict inequality hold for all $n\ge 1$ whenever $R$ is not Gorenstein?), and (ii) determine $\beta_n(\omega_R)$ from the finite initial data $\{\beta_i(\omega_R): i<n\}$, or show that no such determination exists in general. The state of the art is partial: $\beta_1(\omega_R)\ge\beta_0(\omega_R)$ holds for all such rings; strict growth is known only for special families (codimension two, embedding codepth at most 3, Golod rings, fiber products, rings with $\mathfrak m^3=0$, Teter rings, rings of minimal multiplicity, and He's 2025 Artinian conditions). The general characterization and the determination problem remain open, as does even the $n=1$ strict inequality (Jorgensen–Leuschke Question 2.6) in full generality.

Background

Let $(R,\mathfrak m,k)$ be a (Noetherian) local ring. Every finitely generated $R$-module $M$ has a minimal free resolution, and its $i$-th Betti number $\beta_i^R(M)=\dim_k\operatorname{Tor}_i^R(k,M)$ is the rank of the free module in position $i$ of that resolution; it is finite for each $i$. A Cohen–Macaulay local ring is one whose depth equals its dimension; a canonical module $\omega_R$ is a maximal Cohen–Macaulay module of finite injective dimension such that the natural map $R\to\operatorname{Hom}_R(\omega_R,\omega_R)$ is an isomorphism. It exists, for example, whenever $R$ is a homomorphic image of a Gorenstein ring, and is unique up to isomorphism. The ring is Gorenstein precisely when $\omega_R$ is free, i.e. when the Betti sequence of $\omega_R$ is finite (it is then $\beta_0=1$, $\beta_i=0$ for $i>0$). The zeroth Betti number $\beta_0(\omega_R)$ is the type $r(R)$ of $R$, a classical measure of distance from being Gorenstein; for a Cohen–Macaulay $R$ the Betti numbers of $\omega_R$ coincide with the Bass numbers of $R$, that is, the multiplicities of the injective hull of $k$ in the minimal injective resolution of $R$.

The motivation parallels the classical homological characterizations of singularities: by the Auslander–Buchsbaum–Serre theorem $R$ is regular if and only if the Betti sequence of the residue field $k$ is finite, and by a theorem of Gulliksen $R$ is a complete intersection if and only if that sequence grows polynomially. A question first heard from C. Huneke asks the analogous thing one level up: if $R$ is Cohen–Macaulay but not Gorenstein, must the Betti numbers of $\omega_R$ grow exponentially? Jorgensen and Leuschke (2007) proved exponential growth for a large class of rings, showed the growth is in general not extremal, and isolated the initial behavior as the hard part: their Question 2.6 asks whether $\beta_1(\omega_R)\le\beta_0(\omega_R)$ forces $R$ to be Gorenstein (equivalently, whether $\beta_1(\omega_R)>\beta_0(\omega_R)$ for every non-Gorenstein such $R$); they proved it in codimension two via the Hilbert–Burch theorem (the classical structure theorem for minimal free resolutions of codimension-two perfect ideals). Christensen, Striuli and Veliche (2010) proved that $\beta_1(\omega_R)\ge\beta_0(\omega_R)$ for every Cohen–Macaulay local ring with canonical module, and established strict increase with exponential growth of the whole sequence for several families: Golod rings (a class of rings whose homology algebra has trivial multiplication, detected by an equality between the Poincaré series of the residue field — the generating function $\sum_i\beta_i^R(k)t^i$ of its Betti numbers — and its universal coefficient-wise lower bound) of embedding dimension minus depth at least 2 or type at least 2, fiber products of local rings (rings of the form $R_1\times_k R_2$, two local rings glued over a common residue field), Artinian rings whose socle (the annihilator $\operatorname{Soc}R=(0):\mathfrak m$ of the maximal ideal) is not contained in $\mathfrak m^2$, rings with $\mathfrak m^3=0$, and Teter rings ($R\cong Q/\operatorname{Soc}Q$ with $Q$ Artinian Gorenstein). Avramov (2012) proved $\beta_{n-1}(\omega_R)<\beta_n(\omega_R)$ for all $n$ over Cohen–Macaulay rings of embedding codepth at most 3 (embedding dimension minus depth, the smallest non-hypersurface cases), and over rings of minimal multiplicity (rings whose Hilbert–Samuel multiplicity takes the smallest value $\operatorname{edim}R-\dim R+1$ allowed by the embedding dimension) the whole sequence $\{\beta_i(\omega_R)\}$ is known in closed form.

Asgharzadeh's preprint (arXiv:1912.10864, 2019) collects further progress and poses the general problem. His Observation A gives, for every $n$, alternating-sum lower bounds on $\beta_n(\omega_R)$ in terms of $\{\beta_i(\omega_R): i<n\}$ valid for all Cohen–Macaulay $R$ (e.g. $\beta_2(\omega_R)\ge\beta_1(\omega_R)$ when $R$ has type 2), and for a homogeneous Cohen–Macaulay ideal $I$ in a graded ring $S$ with $q$-linear resolution (a graded resolution whose $i$-th syzygy module is generated entirely in degree $q+i$) he proves $\beta_1(\omega_{S/I})-\beta_0(\omega_{S/I})\ge\operatorname{ht}(I)-1$. For the truncated polynomial rings $R_{d,n}=k[X_1,\dots,X_d]/(X_1,\dots,X_d)^n$ he extends a theorem of Gover and Ramras (1980), showing $\beta_{i+1}(M)-\beta_i(M)\ge d-1$ for every nonfree module and all $n>1$, $i\ge2$, with a stronger combinatorial bound that is sharp for $i\ge2$ but can fail at $i=1$. The same preprint records companion open questions that are adjacent but distinct threads: the Dutta–Griffith question of when the strict length inequality $\ell(M\otimes N)>\ell(\operatorname{Tor}_1^R(M,N))$ holds (the non-strict version with $N=\omega_R$ always holds for finite-length $M$), and Huneke's questions on products of ideals (whether $\beta_0(\omega_{S/JK})\ge\operatorname{ht}(JK)$ when $JK$ is Cohen–Macaulay, and whether the type of $S/JK$ is at least the height), which are settled only in special cases. The most recent partial progress on the initial-growth question is due to He (2025), who answers Jorgensen–Leuschke Question 2.6 affirmatively under additional Artinian structural hypotheses (e.g. $\operatorname{Soc}(R)=\mathfrak m^2$ with $\mu(\mathfrak m)\ge4$, where $\mu(\mathfrak m)=\dim_k\mathfrak m/\mathfrak m^2$ is the minimal number of generators of the maximal ideal, or $\mathfrak m^2 I=0$ with Tor-vanishing conditions). All known positive results proceed family by family; no argument is known that handles an arbitrary Cohen–Macaulay local ring.

Problem Statement

Let $(R,\mathfrak m,k)$ be a Cohen–Macaulay local ring possessing a canonical module $\omega_R$; the Gorenstein case is trivial ($\omega_R$ free), so assume $R$ is not Gorenstein. This is Problem 1.1 of Asgharzadeh's "On the initial Betti numbers", generalizing Question 2.6 of Jorgensen–Leuschke and the questions of Christensen–Striuli–Veliche.

(1) Characterize precisely when $\beta_n(\omega_R)>\beta_{n-1}(\omega_R)$: give necessary and sufficient conditions on $R$ (and $n$) for the strict inequality — in particular, decide whether it holds for all $n\ge 1$ for every non-Gorenstein Cohen–Macaulay local ring with canonical module. The case $n=1$ is exactly Jorgensen–Leuschke Question 2.6: does $\beta_1(\omega_R)\le\beta_0(\omega_R)$ imply that $R$ is Gorenstein?

(2) Determine $\beta_n(\omega_R)$ from the finite initial data $\{\beta_i(\omega_R) : i<n\}$: provide a formula or a provable procedure computing $\beta_n(\omega_R)$ from that data, or prove that no such determination is possible in general by exhibiting Cohen–Macaulay rings with identical initial canonical-module Betti data but different next Betti numbers.

An answer must be valid for arbitrary Cohen–Macaulay local rings with canonical module, without restricting to a special family (such as Golod rings, fiber products, rings of bounded embedding codepth, or rings of minimal multiplicity); results for such families are the known partial cases, not the target.

References

1. Mohsen Asgharzadeh, On the initial Betti numbers, preprint arXiv:1912.10864 [math.AC] (2019). DOI: 10.48550/arXiv.1912.10864, https://arxiv.org/abs/1912.10864
2. David A. Jorgensen and Graham J. Leuschke, On the growth of the Betti sequence of the canonical module, Mathematische Zeitschrift 256 (2007), no. 3, 647–659. DOI: 10.1007/s00209-006-0096-x, https://doi.org/10.1007/s00209-006-0096-x
3. David A. Jorgensen and Graham J. Leuschke, Erratum: On the growth of the Betti sequence of the canonical module, Mathematische Zeitschrift 260 (2008), no. 3, 713–715. DOI: 10.1007/s00209-008-0382-x, https://doi.org/10.1007/s00209-008-0382-x
4. Lars Winther Christensen, Janet Striuli and Oana Veliche, Growth in the minimal injective resolution of a local ring, Journal of the London Mathematical Society (2) 81 (2010), no. 1, 24–44. DOI: 10.1112/jlms/jdp058, https://doi.org/10.1112/jlms/jdp058
5. Luchezar L. Avramov, A cohomological study of local rings of embedding codepth 3, Journal of Pure and Applied Algebra 216 (2012), no. 11, 2489–2506. DOI: 10.1016/j.jpaa.2012.03.012, https://doi.org/10.1016/j.jpaa.2012.03.012
6. Craig Huneke, Ideals defining Gorenstein rings are (almost) never products, Proceedings of the American Mathematical Society 135 (2007), no. 7, 2003–2005. DOI: 10.1090/s0002-9939-07-08758-8, https://doi.org/10.1090/s0002-9939-07-08758-8
7. Eugene Gover and Mark Ramras, Increasing sequences of Betti numbers, Pacific Journal of Mathematics 87 (1980), no. 1, 65–68. DOI: 10.2140/pjm.1980.87.65, https://doi.org/10.2140/pjm.1980.87.65
8. S. P. Dutta and Phillip Griffith, Intersection multiplicity, canonical element conjecture and the syzygy problem, Michigan Mathematical Journal 57 (2008), 227–247. DOI: 10.1307/mmj/1220879406, https://doi.org/10.1307/mmj/1220879406
9. Kaiyue He, Betti numbers for modules over Artinian local rings, Journal of Pure and Applied Algebra 230 (2026), no. 2, article 108172. DOI: 10.1016/j.jpaa.2026.108172, arXiv:2503.11866, https://arxiv.org/abs/2503.11866
10. Doan Trung Cuong and Toshinori Kobayashi, On direct summands of syzygies of the residue field of a local ring, preprint arXiv:2510.24220 (2025). https://arxiv.org/abs/2510.24220