Non-Noetherian Bass and Betti numbers

Abstract: This paper investigates the vanishing and non-vanishing of Betti and Bass numbers for non-finitely generated modules. We prove that for (d)-dimensional Cohen--Macaulay local rings, every non-zero (\mathfrak{m})-torsion module satisfies (βd(M)\neq 0), and we establish the Betti number behavior of the injective hull (E_R(k)). We study tor-rigidity for (H^d{\mathfrak{m}}(R)). We also provide partial positive answers to Schoutens' question on whether the vanishing of some Betti number of a big Cohen--Macaulay algebra forces the Cohen--Macaulay property of (R). For the absolute integral closure (R^+), we establish both Tor and Ext results. On the Tor side, we prove that (β_i(R^+)=0) for some (i>0) implies regularity in a series cases. On the Ext side, we prove that (μ_i(R^+)=0) for some (i> d) forces regularity for Gorenstein domains of prime characteristic, and we obtain analogous results for graded normal domains of dimension (2) and also for quotient and isolated singularities in any dimension. Also $μ_i(R<sup>\infty)=0$ forces regularity for F-pure rings.

• The paper shows that vanishing Bass and Betti numbers in non-Noetherian modules imply regularity in rings like quotient singularities and Cohen–Macaulay types.
• It employs advanced homological tools, including Tor and spectral sequences, to analyze invariants in absolute integral closures and big Cohen–Macaulay algebras.
• The study constructs explicit counterexamples to classical rigidity assumptions, deepening insights into module depth, local cohomology, and singularity theory.

Homological Invariants in Non-Noetherian Contexts: Bass and Betti Numbers

Introduction

The paper "Non-Noetherian Bass and Betti numbers" (2606.20391) investigates the vanishing and non-vanishing phenomena of Betti and Bass numbers, particularly for non-finitely generated modules and non-Noetherian rings. It establishes new results concerning module invariants in the absolute integral closure, local cohomology, injective hulls, and big Cohen–Macaulay algebras, with a focus on connections between homological vanishing and ring regularity. The paper addresses classical questions about Betti and Bass number vanishing in non-Noetherian settings, providing partial positive answers to the types of rigidity and purity conjectures raised by Schoutens, Bhatt, Iyengar, and Ma. The results impact singularity theory, homological dimensions, and the structural properties of commutative rings in both equicharacteristic and mixed characteristic cases.

Non-Vanishing and Rigidity of Betti Numbers

The authors extend classical results concerning the non-vanishing of top Betti numbers for $\mathfrak{m}$-torsion modules. For a $d$-dimensional Cohen–Macaulay local ring, every nonzero $\mathfrak{m}$-torsion module satisfies $\beta_d(M) \neq 0$. Furthermore, they generalize Grothendieck's non-vanishing results to injective hulls, showing $\beta_d(E_R(k)) \neq 0$ and $\beta_i(E_R(k)) = 0$ for $i < d$. For modules of finite length and artinian modules, the paper proves inductively that high-degree Tor groups do not vanish unless the module itself is zero.

With respect to rigidity, the authors show that the classical rigidity of Tor does not hold for local cohomology modules. Specifically, $H^d_{\mathfrak{m}}(R)$ is generally not tor-rigid, as the relation $$\operatorname{Tor}_i^R(M, H^d_{\mathfrak{m}}(R)) \cong H^{d-i}_{\mathfrak{m}}(M)$$ exposes cases where vanishing in one degree does not propagate. Consequently, this provides explicit counterexamples to overly broad rigidity conjectures and demonstrates subtle connections between Tor, depth, and local cohomology.

Betti and Bass Numbers and Regularity of Absolute Integral Closures

A central focus is the absolute integral closure $R^+$ and its homological invariants. The paper addresses whether vanishing Betti (or Bass) numbers of $d$0 force regularity of the base ring:

• If $d$1 for some $d$2, the ring $d$3 must be regular in several important classes: quotient singularities, semigroup rings, Cohen–Macaulay rings of finite type, and homogeneous domains over $d$4 of multiplicity at most two.
• For Gorenstein domains of prime characteristic, $d$5 for $d$6 forces regularity; analogous results are shown for graded normal domains of dimension 2, isolated singularities, and F-pure rings via perfect closures.

Notably, the paper provides partial positive answers to Schoutens' question: vanishing of Betti numbers for big Cohen–Macaulay algebras can enforce Cohen–Macaulayness for $d$7 in several cases, including weakly tor-rigid points and isolated singularities. For example, for a big Cohen–Macaulay algebra $d$8 with $d$9 for $\mathfrak{m}$0, regularity is forced.

Homological Behavior for Tensor Products and Non-Finite Generation

The paper establishes preservation and failure results for the Cohen–Macaulay property under tensor products of modules:

• The tensor product of two generalized Cohen–Macaulay modules need not be generalized Cohen–Macaulay, and counterexamples are explicitly constructed.
• If $\mathfrak{m}$1 is generalized Cohen–Macaulay (over certain regular/hypersurface/isolated singularity rings), then both $\mathfrak{m}$2 and $\mathfrak{m}$3 are generalized Cohen–Macaulay.
• For balanced big Cohen–Macaulay modules over regular rings, tensor products retain balanced big Cohen–Macaulayness if and only if both factors possess it.

Regarding non-finitely generated modules, the paper studies the behavior of Betti numbers for artinian and Matlis reflexive modules. It demonstrates non-vanishing of top Betti numbers, relations to local cohomology, and the consequences of zero Betti numbers for such modules, particularly in determining depth and dimension constraints. It also analyzes the failure of rigidity in specific cases and delineates conditions where high-degree Betti and Bass numbers necessarily do not vanish.

Positive and Negative Results for Weak Tor-Rigidity

The study of weak tor-rigidity and its implications for Cohen–Macaulayness is highly technical:

• For big Cohen–Macaulay algebras that are weakly tor-rigid, vanishing of $\mathfrak{m}$4 rapidly enforces the Cohen–Macaulay property of $\mathfrak{m}$5.
• The paper proves vanishing implications propagate via spectral sequences for Tor and Ext, using homological algebra machinery to relate vanishing properties in tensor and Hom complexes to module and ring structure.
• It establishes sharp numerical bounds and vanishing thresholds for Betti numbers, connecting them to depth, projective dimension, and the structure of artinian, Matlis reflexive, and torsionless modules.

Explicit counterexamples are constructed to show the limitations of weak rigidity. For instance, the classical example due to Lichtenbaum demonstrates that vanishing of $\mathfrak{m}$6 does not always propagate to higher degrees, even when modules are maximal Cohen–Macaulay.

Purity, F-Purity, and Ext Vanishing in Prime Characteristic

For rings of positive characteristic, the authors derive strong results relating purity and F-purity to homological vanishing:

• If a $\mathfrak{m}$7-dimensional F-pure ring $\mathfrak{m}$8 satisfies $\mathfrak{m}$9 for some $\beta_d(M) \neq 0$0, regularity is enforced.
• Purity is translated, via spectral sequences and Ext functors, to powerful consequences for local cohomology vanishing and injective dimension for absolute integral closures and their limit objects.

These results leverage technical algebraic machinery such as Matlis duality, local duality, and properties of module-finite integral extensions, splitting arguments, and Hochster’s canonical element theorems.

Implications and Directions for Future Research

The theoretical implications are substantial for homological conjectures in commutative algebra concerning singularities, integral extensions, and homological dimensions:

• The paper provides effective homological criteria for regularity in both equicharacteristic and mixed characteristic cases, reducing many deep questions about class groups, UFD structure, and multiplicities to vanishing of Betti or Bass numbers.
• The results suggest further exploration of absolute integral closures, perfect closures, and big Cohen–Macaulay algebras in relation to homological invariants and purity, especially in the context of singularity theory and tight closure.
• Open conjectures remain about full extension of Ext vanishing implications for regularity in prime characteristic, motivating deeper study via local cohomology and homological algebra techniques.

Conclusion

This paper systematically explores the vanishing and non-vanishing of Betti and Bass numbers in non-Noetherian and non-finitely generated module contexts, connecting these properties to structure, depth, rigidity, and regularity of rings. By establishing precise vanishing thresholds and their consequences for module and ring theory, it clarifies the limitations of classical homological theory in broader settings and presents new pathways for deeper understanding of singularities and homological purity in commutative algebra (2606.20391).