Quasi-projective dimensions of complexes over rings

Abstract: Quasi-projective dimension of modules over associative rings is generalized in this paper to the one of complexes of modules. Basic properties of this dimension are established, including a comparison result with projective dimension and a derived Auslander-Buchsbaum formula for complexes of finite quasi-projective dimension. Several sufficient conditions are provided for a commutative noetherian local ring to be a complete intersection under the assumption that each finitely generated module has finite quasi-projective dimension. This provides some positive answers to an open question on quasi-projective dimension proposed by Gheibi-Jorgensen-Takahashi. Moreover, the behavior of quasi-projective dimension under taking the quotient of a commutative ring modulo a regular sequence is investigated, and some partial results toward the change-of-rings question on quasi-projective dimension are given.

• The paper introduces a quasi-projective dimension invariant for complexes by extending classical invariants like projective and G-dimensions.
• It establishes key results including a derived Auslander–Buchsbaum formula and change-of-rings theorems that quantify homological properties.
• The findings connect finite qpd with complete intersection properties, offering new criteria for classifying ring singularities.

Quasi-Projective Dimensions of Complexes: Theory and Applications

Introduction and Context

The paper "Quasi-projective dimensions of complexes over rings" (2604.09279) develops a homological invariant—quasi-projective dimension (qpd)—for complexes of modules over associative rings, generalizing previous work for modules. This concept was first formulated for modules by Gheibi, Jorgensen, and Takahashi, and is inspired by classical invariants such as projective dimension, G-dimension, and complete intersection dimension, all of which play central roles in measuring the singularity and homological complexity of modules or complexes over commutative Noetherian rings. The quasi-projective dimension is significant due to its intricate relationship with the structure of rings (especially complete intersections) and virtually small complexes in the derived category.

Main Results

Extension of Quasi-Projective Dimension to Complexes

The authors provide a formal definition of quasi-projective dimension for complexes within the unbounded derived category $\mathrm{D}(R)$ and explore its properties. For an $R$-complex $M$, $\mathrm{qpd}_R M$ is defined via the existence of a quasi-projective resolution, which is a special type of semi-projective complex whose homology consists of finite direct sums of (shifts of) the homology of $M$.

Key properties include:

• If $\mathrm{pd}_R M < \infty$, then $$\mathrm{qpd}_R M + \mathrm{hsup} M = \mathrm{pd}_R M$$.
• If $\mathrm{qpd}_R M < \infty$ and high-degree $\mathrm{Ext}$ vanishes, then $\mathrm{pd}_R M < \infty$.
• The quasi-projective dimension of a complex is bounded above by the supremum of the quasi-projective dimensions of its homology modules.
• For $R$0 of finite quasi-projective dimension over a local noetherian ring, a derived Auslander-Buchsbaum formula is established:

$R$1

This latter formula rigorously extends the classical module result to the derived setting.

Connections with Complete Intersections

A central line of inquiry is the following: If every finitely generated $R$2-module has finite quasi-projective dimension, is $R$3 a complete intersection? This is an open question in general but has been resolved in several special cases. The authors supply sufficient conditions (Theorem 1.2), constructing new results:

• If $R$4 with $R$5 regular and (i) $R$6, (ii) $R$7 is a Burch ideal, or (iii) $R$8 is generated by at most two elements, then $R$9 is a complete intersection if all finitely generated modules have finite quasi-projective dimension.
• In the artinian case, these criteria provide characterizations for when $M$0 is a complete intersection or a hypersurface based on finiteness of quasi-projective dimension for all finitely generated modules.

The paper further connects finiteness of qpd for all modules with the theory of virtually small complexes, thick subcategories, and the classification of dominant and isolated singularities. These connections illuminate the structural properties of the derived category and its subcategories over rings of interest.

Change-of-Rings Theorems

Another major thread is analysis of $M$1 under passage to quotients by regular sequences. The authors establish reduction formulas and inequalities for qpd:

• For a commutative ring $M$2, an $M$3-regular sequence $M$4, and a complex $M$5 over $M$6,

$M$7

with equality under vanishing $M$8 or if $M$9 is von Neumann regular.

• For any nonzero, finitely generated $\mathrm{qpd}_R M$0-module $\mathrm{qpd}_R M$1, if $\mathrm{qpd}_R M$2, then $\mathrm{qpd}_R M$3 if and only if $\mathrm{qpd}_R M$4.

These results systematically clarify the behavior of quasi-projective dimension in the context of regular and complete intersection sequences, generalizing previous results for modules and refining the change-of-rings framework.

Technical Consequences and Numerical Phenomena

• The derived Auslander-Buchsbaum formula for quasi-projective dimension extends the classical depth/projective dimension relation for complexes of finite qpd.
• The comparison results between qpd and projective dimension identify scenarios where the two coincide (e.g., over von Neumann regular rings or under strong vanishing of $\mathrm{qpd}_R M$5).
• The characterization results provide numerical thresholds, such as the embedding dimension or number of generators, where equivalence to complete intersections arises given module-level qpd finiteness.
• The authors exhibit that the inequality between qpd of a complex and qpd of its homologies may be strict, underlining the distinct information captured about complexes.

Implications and Future Directions

The theoretical advances in the paper deepen the understanding of homological dimensions in complex settings, supplying new invariants that retain sensitivity to subtle ring-theoretic and categorical properties. The results on sufficient conditions for complete intersection ring structure contribute partial answers to a significant open question in homological commutative algebra, with consequences for the classification of singularities and for representation theory (such as progress on the Auslander-Reiten and Tachikawa's conjectures).

From the category-theoretic perspective, the connection to thick subcategories and virtually small complexes opens avenues for further structural theorems and possibly for new invariants relevant to higher algebraic $\mathrm{qpd}_R M$6-theory, singularity categories, and support theory. The explicit treatment of change-of-rings phenomena suggests further analysis of behavior under deformation and for more general localizations and quotient constructions.

In view of open questions, most notably the general converse (i.e., whether finite qpd for all modules implies a complete intersection without further ring-theoretic constraints), the techniques developed here provide essential tools for approaching difficult cases and constructing potential counterexamples or new characterizations.

Conclusion

The paper systematically builds the quasi-projective dimension theory for complexes over arbitrary associative and commutative rings, establishing foundational results, comparison theorems, and deep applications to the study of complete intersection rings and the structure of derived categories. The derived Auslander-Buchsbaum formula for complexes of finite qpd is a notable generalization. The interplay between module and complex dimensions, virtually small complexes, and ring-theoretic properties enriches both homological algebra and commutative algebra, with implications for broader categorical and geometric investigations.