Quasi-Global Dimension in Noetherian Rings

• Quasi-global dimension is a homological invariant defined by replacing projective dimension with quasi-projective dimension to measure the complexity of finitely generated modules.
• It parallels classical global dimension, equating to it when finite but remaining finite in cases where projective resolutions are infinite, as seen in Nakayama algebras.
• The invariant enjoys robust formal properties such as Morita invariance and behavior under ring products, providing deeper insights into module theory.

Quasi-global dimension is a homological invariant of a left Noetherian ring defined in analogy with classical global dimension by replacing projective dimension with quasi-projective dimension. For a left Noetherian ring $R$, it is given by

$\mathrm{qgldim}(R):=\sup\{\qpd_R(M)\mid M\in R\text{-mod}\},$

where $R\text{-mod}$ denotes the category of finitely generated left $R$-modules (Chen et al., 24 Sep 2025). The invariant is designed to measure homological complexity through quasi-projective resolutions rather than genuine projective resolutions, and in the examples analyzed so far it can remain finite even when $\mathrm{gldim}(R)=\infty$ (Chen et al., 24 Sep 2025). Earlier work on quasi-projective and quasi-injective dimensions had already isolated the module-theoretic mechanisms underlying such a notion, and explicitly observed that a corresponding ring-level “quasi-global dimension” is the natural analogue of global dimension, although not yet formalized there (Jorge-Pérez et al., 2024).

1. Definition and resolution-theoretic foundation

The definition of quasi-global dimension depends on quasi-projective dimension. In an abelian category $\mathcal A$ with enough projectives, the quasi-projective dimension of a nonzero object $M$ is

$\qpd_{\mathcal A}(M):=\inf\left\{ \sup(P_\bullet)-\mathrm{hsup}(P_\bullet)\ \middle|\ P_\bullet \text{ is a finite quasi-projective resolution of } M \right\},$

and $\qpd_{\mathcal A}(0)=0$ (Chen et al., 24 Sep 2025). A finite quasi-projective resolution is a bounded complex $P_\bullet$ of projectives such that, for all $\mathrm{qgldim}(R):=\sup\{\qpd_R(M)\mid M\in R\text{-mod}\},$0, one has $\mathrm{qgldim}(R):=\sup\{\qpd_R(M)\mid M\in R\text{-mod}\},$1 for integers $\mathrm{qgldim}(R):=\sup\{\qpd_R(M)\mid M\in R\text{-mod}\},$2, not all zero, and all but finitely many zero (Chen et al., 24 Sep 2025). In the case of $\mathrm{qgldim}(R):=\sup\{\qpd_R(M)\mid M\in R\text{-mod}\},$3, both projective and quasi-projective resolutions are built from finitely generated projectives (Chen et al., 24 Sep 2025).

This definition shifts attention from acyclic projective resolutions to bounded complexes whose homology is controlled by repeated copies of the same module. A plausible implication is that quasi-global dimension records how far finitely generated modules are from admitting short quasi-projective resolutions, even when their genuine projective resolutions are arbitrarily long or infinite.

2. Comparison with classical global dimension

For a left Noetherian ring $\mathrm{qgldim}(R):=\sup\{\qpd_R(M)\mid M\in R\text{-mod}\},$4, classical global dimension is

$\mathrm{qgldim}(R):=\sup\{\qpd_R(M)\mid M\in R\text{-mod}\},$5

The basic comparison is modulewise: $\mathrm{qgldim}(R):=\sup\{\qpd_R(M)\mid M\in R\text{-mod}\},$6 and therefore

$\mathrm{qgldim}(R):=\sup\{\qpd_R(M)\mid M\in R\text{-mod}\},$7

(Chen et al., 24 Sep 2025).

The relation becomes exact whenever global dimension is finite. If $\mathrm{qgldim}(R):=\sup\{\qpd_R(M)\mid M\in R\text{-mod}\},$8, then for all $\mathrm{qgldim}(R):=\sup\{\qpd_R(M)\mid M\in R\text{-mod}\},$9 one has $R\text{-mod}$0, and consequently

$R\text{-mod}$1

(Chen et al., 24 Sep 2025). The same paper also identifies quasi-global dimension with finitistic dimension whenever the former is finite: $R\text{-mod}$2 (Chen et al., 24 Sep 2025).

A further comparison criterion is module-theoretic rather than ring-theoretic: if $R\text{-mod}$3 and $R\text{-mod}$4 for all $R\text{-mod}$5, then $R\text{-mod}$6 (Chen et al., 24 Sep 2025). This situates quasi-global dimension between global dimension and finitistic dimension, with exact agreement in the finite-global-dimension regime and strict separation possible when $R\text{-mod}$7.

3. Local formulas and quasi-homological structure

The theory of quasi-global dimension is rooted in the local behavior of quasi-projective dimension and its dual, quasi-injective dimension. For a local Noetherian ring $R\text{-mod}$8 and an $R\text{-mod}$9-module $R$0, the quasi-projective and quasi-injective dimensions satisfy

$R$1

with equality when the right-hand side is finite (Jorge-Pérez et al., 2024).

The quasi-projective dimension satisfies an Auslander–Buchsbaum type formula. If $R$2 has finite quasi-projective dimension over a local ring, then

$R$3

hence

$R$4

(Jorge-Pérez et al., 2024). The same work establishes Ischebeck-type formulas: if $R$5 is local, $R$6 are nonzero, and $R$7, then

$R$8

whenever $R$9 has finite quasi-projective dimension, or $\mathrm{gldim}(R)=\infty$0 has finite Gorenstein dimension and $\mathrm{gldim}(R)=\infty$1 has finite quasi-projective dimension, or $\mathrm{gldim}(R)=\infty$2 has finite quasi-injective dimension (Jorge-Pérez et al., 2024).

The grade inequalities

$\mathrm{gldim}(R)=\infty$3

provide another structural constraint (Jorge-Pérez et al., 2024). A module is called quasi-perfect if $\mathrm{gldim}(R)=\infty$4 and

$\mathrm{gldim}(R)=\infty$5

(Jorge-Pérez et al., 2024). Over a Cohen–Macaulay ring, a nonzero module with finite quasi-projective dimension is Cohen–Macaulay if and only if it is quasi-perfect (Jorge-Pérez et al., 2024). The residue field $\mathrm{gldim}(R)=\infty$6 of a local ring always has finite quasi-projective and quasi-injective dimension, and $\mathrm{gldim}(R)=\infty$7 is always quasi-perfect but only perfect if $\mathrm{gldim}(R)=\infty$8 is regular (Jorge-Pérez et al., 2024). These local facts explain why the supremum defining quasi-global dimension can remain finite in settings where projective dimensions do not.

4. Formal properties of the invariant

Quasi-global dimension satisfies several functorial and structural properties parallel to those of global dimension. If $\mathrm{gldim}(R)=\infty$9 and $\mathcal A$0 are Morita equivalent left Noetherian rings, then

$\mathcal A$1

(Chen et al., 24 Sep 2025). For products,

$\mathcal A$2

(Chen et al., 24 Sep 2025). For finite-dimensional $\mathcal A$3-algebras,

$\mathcal A$4

(Chen et al., 24 Sep 2025).

The invariant can also be recovered from its finite values: $\mathcal A$5 (Chen et al., 24 Sep 2025). Over quasi-Frobenius rings, the behavior is dichotomic: quasi-global dimension is either $\mathcal A$6 or $\mathcal A$7, and representation-finite self-injective algebras have $\mathcal A$8 (Chen et al., 24 Sep 2025). For finite-dimensional self-injective algebras, quasi-global dimension is invariant under stable equivalence of Morita type and derived equivalence (Chen et al., 24 Sep 2025). Under an equivalence of stable module categories for quasi-Frobenius rings, the quasi-global dimensions compare by

$\mathcal A$9

(Chen et al., 24 Sep 2025).

The existence of symmetric algebras for which all non-projective modules have infinite projective and quasi-projective dimension shows that quasi-global dimension need not improve finiteness in every self-injective setting (Chen et al., 24 Sep 2025). This suggests that the invariant is especially responsive to periodicity and short quasi-projective patterns, but not universally finite in representation-theoretic contexts.

5. Finite-dimensional Nakayama algebras as a test case

The principal examples exhibiting separation between quasi-global dimension and global dimension are finite-dimensional Nakayama algebras $M$0 (Chen et al., 24 Sep 2025). The paper gives the following values.

Algebra	$M$1	$M$2
$M$3	$M$4	$M$5
$M$6	$M$7	$M$8
$M$9, $\qpd_{\mathcal A}(M):=\inf\left\{ \sup(P_\bullet)-\mathrm{hsup}(P_\bullet)\ \middle|\ P_\bullet \text{ is a finite quasi-projective resolution of } M \right\},$0, $\qpd_{\mathcal A}(M):=\inf\left\{ \sup(P_\bullet)-\mathrm{hsup}(P_\bullet)\ \middle|\ P_\bullet \text{ is a finite quasi-projective resolution of } M \right\},$1	$\qpd_{\mathcal A}(M):=\inf\left\{ \sup(P_\bullet)-\mathrm{hsup}(P_\bullet)\ \middle|\ P_\bullet \text{ is a finite quasi-projective resolution of } M \right\},$2	$\qpd_{\mathcal A}(M):=\inf\left\{ \sup(P_\bullet)-\mathrm{hsup}(P_\bullet)\ \middle|\ P_\bullet \text{ is a finite quasi-projective resolution of } M \right\},$3

For $\qpd_{\mathcal A}(M):=\inf\left\{ \sup(P_\bullet)-\mathrm{hsup}(P_\bullet)\ \middle|\ P_\bullet \text{ is a finite quasi-projective resolution of } M \right\},$4, one has

$\qpd_{\mathcal A}(M):=\inf\left\{ \sup(P_\bullet)-\mathrm{hsup}(P_\bullet)\ \middle|\ P_\bullet \text{ is a finite quasi-projective resolution of } M \right\},$5

(Chen et al., 24 Sep 2025). For the self-injective, representation-finite algebra $\qpd_{\mathcal A}(M):=\inf\left\{ \sup(P_\bullet)-\mathrm{hsup}(P_\bullet)\ \middle|\ P_\bullet \text{ is a finite quasi-projective resolution of } M \right\},$6,

$\qpd_{\mathcal A}(M):=\inf\left\{ \sup(P_\bullet)-\mathrm{hsup}(P_\bullet)\ \middle|\ P_\bullet \text{ is a finite quasi-projective resolution of } M \right\},$7

and all finitely generated modules are periodic and hence have quasi-projective dimension zero (Chen et al., 24 Sep 2025). In the intermediate case $\qpd_{\mathcal A}(M):=\inf\left\{ \sup(P_\bullet)-\mathrm{hsup}(P_\bullet)\ \middle|\ P_\bullet \text{ is a finite quasi-projective resolution of } M \right\},$8 and $\qpd_{\mathcal A}(M):=\inf\left\{ \sup(P_\bullet)-\mathrm{hsup}(P_\bullet)\ \middle|\ P_\bullet \text{ is a finite quasi-projective resolution of } M \right\},$9,

$\qpd_{\mathcal A}(0)=0$0

(Chen et al., 24 Sep 2025).

The paper identifies the mechanism for this discrepancy: many indecomposable modules are periodic or have projective syzygies after at most $\qpd_{\mathcal A}(0)=0$1 steps in a suitable quasi-projective resolution, even though their projective dimension is infinite (Chen et al., 24 Sep 2025). These examples show that finite quasi-global dimension does not imply finite global dimension, and that quasi-global dimension can detect a form of bounded quasi-projective complexity invisible to the classical invariant.

6. Terminological scope and neighboring notions

A recurrent source of confusion is that several areas use “quasi-” language while continuing to study classical global dimension. In the literature on Auslander–Dlab–Ringel algebras, the relevant invariant is global dimension itself, controlled by quasi-hereditary structure. For an artin algebra $\qpd_{\mathcal A}(0)=0$2 and a semilocal $\qpd_{\mathcal A}(0)=0$3-module $\qpd_{\mathcal A}(0)=0$4, with $\qpd_{\mathcal A}(0)=0$5, the ADR algebra is left-strongly quasi-hereditary, $\qpd_{\mathcal A}(0)=0$6 has an $\qpd_{\mathcal A}(0)=0$7-total left rejective chain of length $\qpd_{\mathcal A}(0)=0$8, and

$\qpd_{\mathcal A}(0)=0$9

(Tsukamoto, 2018). For original ADR algebras, the conditions “$P_\bullet$0 is strongly quasi-hereditary” and “$P_\bullet$1” are equivalent (Tsukamoto, 2018). The same source explicitly states that it does not define a separate invariant called quasi-global dimension; instead, global dimension is the key homological invariant in that setting (Tsukamoto, 2018).

An analogous distinction appears for quiver algebras. If $P_\bullet$2 is a finite quiver without loops, there exists an admissible ideal $P_\bullet$3 such that $P_\bullet$4 and $P_\bullet$5 is strongly quasi-hereditary; additional constructions yield strongly quasi-hereditary quotients with larger global dimension (Poettering, 2010). Here again, the subject is classical global dimension under quasi-hereditary constraints, not quasi-global dimension in the sense of quasi-projective resolutions.

In a different direction, the category of quasi-coherent Cartier crystals over an $P_\bullet$6-finite Noetherian ring $P_\bullet$7 is equivalent to the category of unit Cartier modules, and these equivalent categories have finite injective, hence global, dimension; the resolution length is uniformly bounded by a number depending only on $P_\bullet$8, and for a quotient $P_\bullet$9 of a regular $\mathrm{qgldim}(R):=\sup\{\qpd_R(M)\mid M\in R\text{-mod}\},$00-finite Noetherian ring $\mathrm{qgldim}(R):=\sup\{\qpd_R(M)\mid M\in R\text{-mod}\},$01 of dimension $\mathrm{qgldim}(R):=\sup\{\qpd_R(M)\mid M\in R\text{-mod}\},$02, the bound is $\mathrm{qgldim}(R):=\sup\{\qpd_R(M)\mid M\in R\text{-mod}\},$03 (Blickle et al., 2022). Although the summary of that work discusses “(quasi-)global dimension,” the invariant under study is again ordinary global or injective dimension of an abelian category, not the ring-theoretic $\mathrm{qgldim}(R):=\sup\{\qpd_R(M)\mid M\in R\text{-mod}\},$04 defined via quasi-projective dimension (Blickle et al., 2022).

The 2024 work on Ischebeck’s formula makes this terminological boundary explicit from the opposite side: quasi-global dimension is described there only as the natural analogue

$\mathrm{qgldim}(R):=\sup\{\qpd_R(M)\mid M\in R\text{-mod}\},$05

not as a formally adopted definition within that paper (Jorge-Pérez et al., 2024). In current usage, therefore, “quasi-global dimension” has a precise meaning in the 2025 ring-theoretic framework based on quasi-projective dimension, while earlier and neighboring literatures frequently invoke quasi-hereditary or quasi-coherent structures to obtain results about classical global dimension rather than a distinct quasi-global invariant.