Finite projective dimension and a question of Jorgensen

Abstract: This paper studies finite projective dimension of finitely generated modules over a Noetherian local ring, by means of spectral sequence methods related to generalized local cohomology. Our main goal is to address a question raised by D. Jorgensen over fifteen years ago, concerning a prescribed bound (via Ext vanishing) for projective dimension over a complete intersection local ring. We obtain similar results involving other homological dimensions as well. Also we make use of weakly full ideals to derive further criteria for prescribed bound on projective dimension.

• The paper presents vanishing criteria and depth conditions that guarantee strict bounds on projective dimension.
• It applies spectral sequences and generalized local cohomology to relate Ext vanishing to module invariants over Cohen-Macaulay, Gorenstein, and complete intersection rings.
• The work extends results to modules over weakly full ideals, offering practical insights into rigidity and criteria for achieving freeness.

Finite Projective Dimension, Ext Vanishing, and Jorgensen's Question

Introduction and Context

This paper (“Finite projective dimension and a question of Jorgensen” (2604.24500)) addresses fundamental questions in homological algebra concerning the detection and bounding of projective dimension for finitely generated modules over Noetherian local rings, focusing especially on complete intersection and Gorenstein rings. The core of the study is a question posed by Jorgensen in 2008 regarding whether the vanishing of $\Ext^n_R(M, M)$ for a module $M$ implies $\pd_R M \leq n-1$, where $R$ is a complete intersection local ring and $M$ has finite projective dimension.

The authors employ spectral sequence techniques and generalized local cohomology to identify conditions under which vanishing Ext modules provide upper bounds for projective, injective, or Gorenstein injective dimensions. Criteria involving depth conditions on Ext modules and properties of weakly full ideals are developed to extend and partially resolve Jorgensen's question across a broader class of rings.

Main Contributions: Vanishing Criteria and Projective Dimension Bounds

The principal technical development is the identification of two conditions—$({\bf V}_{n, s})$ and $({\bf D}_s)$—which, when satisfied, guarantee the desired bound on projective dimension. Explicitly, for a Cohen-Macaulay local ring $R$ with a canonical module $\omega_R$, a finitely generated $R$-module $M$0 with $M$1 and satisfying these two conditions for positive integers $M$2 yields $M$3.

• $M$4 Condition: Requires that $M$5 for $M$6.
• $M$7 Condition: Imposes that $M$8 for $M$9 where $\pd_R M \leq n-1$0.

The main theorem affirms that under these hypotheses, the projective dimension is strictly less than $\pd_R M \leq n-1$1. This result generalizes known partial solutions to Jorgensen's question and extends the result beyond complete intersection rings to Gorenstein and Cohen-Macaulay rings with canonical modules. The necessity of the depth condition is illustrated, capturing cases where the ring is Gorenstein as a special instance, thus offering a broader affirmative answer to Jorgensen’s question.

The paper also announces explicit corollaries:

• Over Gorenstein local rings, given a module of finite projective dimension with $\pd_R M \leq n-1$2 and suitable depth on Ext modules, one gets $\pd_R M \leq n-1$3.
• Over complete intersections, if $\pd_R M \leq n-1$4 is even and Ext vanishing and depth conditions are met, the same conclusion follows.

Strong numerical results include characterizations for freeness (when $\pd_R M \leq n-1$5 and the conditions hold, $\pd_R M \leq n-1$6 is free) and explicit bounds in illustrative examples.

Extensions to Other Homological Dimensions

Beyond projective dimension, the authors generalize the vanishing criteria to bounds on injective and Gorenstein injective dimensions. For modules with finite injective dimension, similar spectral sequence arguments and duality results (via Matlis duality) are applied. New conditions ($\pd_R M \leq n-1$7) involving vanishing of Ext modules for $\pd_R M \leq n-1$8 are introduced, and depth conditions are formulated. Theorems provide criteria for depth of $\pd_R M \leq n-1$9, bounds on injective dimension, and when Gorenstein injective dimension is finite, explicit computation of $R$0.

Further, the results yield explicit criteria for modules to be isomorphic to direct sums of canonical modules when depth and Ext vanishing conditions are met, and for freeness when $R$1. These findings link the vanishing of Ext modules to structural properties of modules over Cohen-Macaulay and Gorenstein rings.

Connections to Weakly Full Ideals and Strong Rigidity

A substantial contribution is the extension of projective dimension bounding to modules associated with weakly $R$2-full ideals. Such ideals, especially integrally closed ideals in rings of positive depth, provide strongly rigid modules. Leveraging rigidity properties, the authors derive vanishing criteria which, when satisfied alongside spectral sequence depth conditions, ensure strict bounds on projective dimension.

Proposition 3.14 (in the paper) formalizes this: If $R$3 is an $R$4-primary ideal, weakly $R$5-full with respect to some power of $R$6, and $R$7 is a module with vanishing Exts and depth conditions, then any of several additional vanishing or Tor conditions guarantee $R$8. This advances projective dimension bounding methodology for modules determined via strongly rigid structures and weakly full ideals.

The paper also details obstructions, particularly the necessity to avoid modules whose structure would force regularity in $R$9, thus providing guidance on feasible classes for applying the criteria.

Illustrative Examples and Methodological Implications

Comprehensive examples demonstrate the applicability and limitations of the $M$0 depth condition and other criteria, including modules arising from hypersurface rings, periodic resolutions, and those constructed via regular sequences. The examples clarify the triviality of the depth condition in some cases (such as $M$1), and its power in others.

Methodologically, the approach underscores the power of spectral sequences in relating vanishing of Ext modules to global homological invariants. Generalized local cohomology and local duality serve as central tools, and the paper’s results draw a tighter connection between cohomological vanishing, module depth, and classical homological dimensions.

Implications and Future Directions

The theoretical implications of this work are multifaceted. The results provide a systematic method for bounding projective (and other homological) dimensions by vanishing of Ext modules and depth conditions, extending traditional bounds and offering partial resolutions for extant open questions such as Jorgensen's. Practically, these criteria could be utilized in computational algebra systems to detect finite projective dimension and module freeness within broader classes of local rings, especially for applications in algebraic geometry, singularity theory, and representation theory.

Future developments may include tightening vanishing criteria, classifying classes of rings where depth conditions are automatically satisfied, generalizing results to higher-order syzygies or non-commutative settings, and exploring connections to support varieties and cohomology operators as in Avramov-Buchweitz theory. Further progress towards a complete resolution of Jorgensen’s question, potentially via new rigidity phenomena or deeper duality results, remains an open avenue.

Conclusion

The paper provides a substantial advancement in the detection and bounding of projective dimension for finitely generated modules over Noetherian local rings by means of local cohomology, spectral sequences, and vanishing criteria. The innovative use of depth conditions on Ext modules and weakly full ideals expands the set of known cases where Jorgensen’s question receives an affirmative answer, and situates the results within a broader homological algebra framework encompassing injective and Gorenstein injective dimensions. The criteria and examples developed have implications for both theoretical investigations of module invariants and practical detection of homological properties in algebraic structures.