On classical and Gorenstein homological invariants of rings

Abstract: We prove that, over any ring $R$, the supremum of the projective dimensions of the flat left $R$-modules coincides with the supremum of the Gorenstein projective dimensions of the Gorenstein flat left $R$-modules. As a consequence, we obtain new characterizations of left $n$-perfect rings in terms of Gorenstein projective, Ding projective, and projectively coresolved Gorenstein flat dimensions, extending results by Emmanouil and Dalezios and by Christensen, Estrada, and Thompson. We also introduce Gorenstein analogues of several classical homological invariants and study their relationships with the classical ones, identifying conditions under which they coincide with the classical invariants. Finally, we obtain characterizations of left weakly $n$-$Σ$-cotorsion rings introduced by Cortés-Izurdiaga, Estrada and Fresneda in terms of Gorenstein classes of modules.

• The paper demonstrates that classical homological invariants equal their Gorenstein counterparts, enabling precise computation of ring dimensions.
• It leverages advanced homological techniques and categorical module theory to generalize previous results without restrictive assumptions.
• The study introduces new characterizations of n-perfect and weakly n-Σ-cotorsion rings, enhancing the classification of rings and modules.

Classical and Gorenstein Homological Invariants of Rings: A Technical Essay

Introduction

The paper "On classical and Gorenstein homological invariants of rings" (2606.00736) presents a comprehensive study of relationships between classical and Gorenstein homological invariants, particularly focusing on projective and injective dimensions and their respective counterparts for various classes of modules over arbitrary rings. The author establishes definitive equalities between classical and Gorenstein invariants, introduces new characterizations of $n$-perfect and weakly $n$-$\Sigma$-cotorsion rings, analyzes the structure and properties of Gorenstein analogues to classical ring invariants, and extends several prior results without restrictive preconditions. The work employs advanced homological techniques and categorical module theory, utilizing and generalizing results from prior foundational literature.

Classical vs. Gorenstein Projective and Flat Invariants

A central result is the equality

$\mathrm{splf}(R) = \mathrm{G\text{-}splf}(R),$

where $\mathrm{splf}(R)=\sup\{pd_R(F) \mid F \in Flat\}$ is the classic supremal projective dimension of flat left $R$-modules, and $$\mathrm{G\text{-}splf}(R)=\sup\{Gpd_R(M) \mid M \in GF\}$$ is its Gorenstein analogue, defined on Gorenstein flat left $R$-modules. The proof leverages the structure theorem for Gorenstein flat modules and recent refinements on projective coresolutions, notably generalizing previous work ([Guil Asensio & Herzog; Dalezios & Emmanouil; Christensen, Estrada, Thompson]). The equivalence is established via bounding arguments and transfinite techniques, valid for arbitrary rings, and does not require $R$ to satisfy additional regularity conditions.

Furthermore, it is shown that $\mathrm{splf}(R)$ may be computed via Gorenstein projective, Ding projective, projectively coresolved Gorenstein flat, or Gorenstein AC-projective dimensions, for flat modules. This yields a spectrum of refined descriptors for $n$0-perfect rings, captured in the equivalences:

• Every flat left $n$1-module has projective, Gorenstein projective, Ding projective, projectively coresolved Gorenstein flat, or Gorenstein AC-projective dimension $n$2 $n$3 $n$4 is left $n$5-perfect.

The characterization does not require that every flat Gorenstein projective module be projective—a condition previously believed necessary but now circumvented.

Gorenstein and Classical Homological Invariants

The paper systematically introduces Gorenstein analogues to various supremum classical invariants:

• $n$6: projective dimension supremum over injectives
• $n$7: injective dimension supremum over projectives
• $n$8: flat dimension supremum over injectives
• $n$9: injective dimension supremum over flats

Each has a Gorenstein counterpart, e.g., $\Sigma$0, and the analysis reveals that Gorenstein invariants uniformly dominate their classical analogues. Under global finiteness hypotheses, the Gorenstein and classical invariants coincide:

$\Sigma$1

$\Sigma$2

with $\Sigma$3 and $\Sigma$4 emerging as the canonical global Gorenstein dimensions.

It is proven that if every Gorenstein projective is Gorenstein flat, then $\Sigma$5, echoing spectrum collapse observed in classical settings.

Characterizations of Left Weakly $\Sigma$6-$\Sigma$7-Cotorsion Rings

The paper extends characterizations of left weakly $\Sigma$8-$\Sigma$9-cotorsion rings, initially defined via cotorsion dimensions of injectives, to Gorenstein injective, Ding injective, and weakly Ding injective modules. Explicitly, the following are equivalent:

• Every injective $\mathrm{splf}(R) = \mathrm{G\text{-}splf}(R),$0-module is $\mathrm{splf}(R) = \mathrm{G\text{-}splf}(R),$1-$\mathrm{splf}(R) = \mathrm{G\text{-}splf}(R),$2-cotorsion
• Every Gorenstein injective/Ding injective/weakly Ding injective $\mathrm{splf}(R) = \mathrm{G\text{-}splf}(R),$3-module is $\mathrm{splf}(R) = \mathrm{G\text{-}splf}(R),$4-$\mathrm{splf}(R) = \mathrm{G\text{-}splf}(R),$5-cotorsion
• Every $\mathrm{splf}(R) = \mathrm{G\text{-}splf}(R),$6-periodic $\mathrm{splf}(R) = \mathrm{G\text{-}splf}(R),$7-module is $\mathrm{splf}(R) = \mathrm{G\text{-}splf}(R),$8-$\mathrm{splf}(R) = \mathrm{G\text{-}splf}(R),$9-cotorsion

This broadens homological control of ring structure, allowing characterizations to be formulated in terms of direct summand closure, periodicity, and definable submodule structure.

Implications and Future Directions

The synthesis of classical and Gorenstein invariants provided in this study clarifies the exchange principle between homological dimensions of standard and Gorenstein module classes. Practically, this enables more robust dimension computations over arbitrary rings, improves characterization of $\mathrm{splf}(R)=\sup\{pd_R(F) \mid F \in Flat\}$0-perfect and weakly $\mathrm{splf}(R)=\sup\{pd_R(F) \mid F \in Flat\}$1-$\mathrm{splf}(R)=\sup\{pd_R(F) \mid F \in Flat\}$2-cotorsion rings, and introduces Gorenstein invariants as essential tools in module and ring classification. Theoretical advancements include the elimination of restrictive assumptions in prior results, enhancement of categorical equivalence arguments, and a strengthened framework for future studies of cotorsion and periodicity phenomena.

There is potential for future developments into the interaction between Gorenstein homological invariants and other categorical dimensions, further exploration of rings not satisfying classical regularity, and the extension to the study of other definable module classes. Additionally, the direct-summand property and periodicity arguments suggest new directions in triangulated and derived category settings.

Conclusion

This paper delivers a rigorous comparison between classical and Gorenstein homological invariants of rings, establishes fundamental equalities, and reveals new equivalences in the characterization of homological ring properties. The results offer both new computational tools and structural insights in homological algebra and module theory, facilitating advances in the general classification of rings and modules via Gorenstein parameters. The extension to weakly $\mathrm{splf}(R)=\sup\{pd_R(F) \mid F \in Flat\}$3-$\mathrm{splf}(R)=\sup\{pd_R(F) \mid F \in Flat\}$4-cotorsion rings and the deepening understanding of periodicity and definability solidify the practical and theoretical significance of Gorenstein homological invariants in contemporary research.