---
title: Quasi-Global Dimension in Noetherian Rings
url: https://www.emergentmind.com/topics/quasi-global-dimension
type: topic
---

# Quasi-Global Dimension in Noetherian Rings

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 [2509.20137]. 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 \(\mathrm{gldim}(R)=\infty\) [2509.20137]. 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 [2412.06659].

## 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 \(\qpd_{\mathcal A}(0)=0\) [2509.20137]. A finite quasi-projective resolution is a bounded complex \(P_\bullet\) of projectives such that, for all \(j\), one has \(H_j(P_\bullet)\cong M^{n_j}\) for integers \(n_j\ge 0\), not all zero, and all but finitely many zero [2509.20137]. In the case of \(R\text{-mod}\), both projective and quasi-projective resolutions are built from finitely generated projectives [2509.20137].

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 \(R\), classical global dimension is
\[
\gldim(R):=\sup\{\pd_R(M)\mid M\in R\text{-mod}\}.
\]
The basic comparison is modulewise:
\[
\qpd_R(M)\le \pd_R(M),
\]
and therefore
\[
\mathrm{qgldim}(R)\le \gldim(R)
\]
[2509.20137].

The relation becomes exact whenever global dimension is finite. If \(\gldim(R)<\infty\), then for all \(M\) one has \(\qpd_R(M)=\pd_R(M)\), and consequently
\[
\mathrm{qgldim}(R)=\gldim(R)
\]
[2509.20137]. The same paper also identifies quasi-global dimension with finitistic dimension whenever the former is finite:
\[
\mathrm{qgldim}(R)=\mathrm{findim}(R)=\sup\{\pd_R(M)\mid M\in R\text{-mod},\ \pd_R(M)<\infty\}
\]
[2509.20137].

A further comparison criterion is module-theoretic rather than ring-theoretic: if \(\qpd_{\mathcal A}(M)<\infty\) and \(\Ext_{\mathcal A}^n(M,M)=0\) for all \(n\ge 2\), then \(\pd_{\mathcal A}(M)<\infty\) [2509.20137]. 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 \(\gldim(R)=\infty\).

## 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 \(R\) and an \(R\)-module \(M\), the quasi-projective and quasi-injective dimensions satisfy
\[
\operatorname{qpd}_R M \leq \operatorname{pd}_R M,\qquad \operatorname{qid}_R M \leq \operatorname{id}_R M,
\]
with equality when the right-hand side is finite [2412.06659].

The quasi-projective dimension satisfies an Auslander–Buchsbaum type formula. If \(M\) has finite quasi-projective dimension over a local ring, then
\[
\operatorname{qpd}_R M=\operatorname{depth}R-\operatorname{depth}M,
\]
hence
\[
\operatorname{depth}M\le \operatorname{depth}R,\qquad \operatorname{qpd}_R M\le \operatorname{depth}R
\]
[2412.06659]. The same work establishes Ischebeck-type formulas: if \(R\) is local, \(M,N\) are nonzero, and \(\operatorname{PR}(M,N)<\infty\), then
\[
\operatorname{PR}(M,N)=\operatorname{depth}R-\operatorname{depth}M
\]
whenever \(M\) has finite quasi-projective dimension, or \(M\) has finite Gorenstein dimension and \(N\) has finite quasi-projective dimension, or \(N\) has finite quasi-injective dimension [2412.06659].

The grade inequalities
\[
\operatorname{grade}M\le \operatorname{qpd}_R M\le \operatorname{pd}_R M
\]
provide another structural constraint [2412.06659]. A module is called quasi-perfect if \(\operatorname{qpd}_R M<\infty\) and
\[
\operatorname{grade}M=\operatorname{qpd}_R M
\]
[2412.06659]. Over a Cohen–Macaulay ring, a nonzero module with finite quasi-projective dimension is Cohen–Macaulay if and only if it is quasi-perfect [2412.06659]. The residue field \(k\) of a local ring always has finite quasi-projective and quasi-injective dimension, and \(k\) is always quasi-perfect but only perfect if \(R\) is regular [2412.06659]. 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 \(R\) and \(S\) are Morita equivalent left Noetherian rings, then
\[
\mathrm{qgldim}(R)=\mathrm{qgldim}(S)
\]
[2509.20137]. For products,
\[
\mathrm{qgldim}(R\times S)=\max\{\mathrm{qgldim}(R),\,\mathrm{qgldim}(S)\}
\]
[2509.20137]. For finite-dimensional \(K\)-algebras,
\[
\mathrm{qgldim}(R\otimes_K S)\ge \max\{\mathrm{qgldim}(R),\,\mathrm{qgldim}(S)\}
\]
[2509.20137].

The invariant can also be recovered from its finite values:
\[
\mathrm{qgldim}(R)=\sup\{\qpd_R(M)\mid M\in R\text{-mod},\,\qpd_R(M)<\infty\}
\]
[2509.20137]. Over quasi-Frobenius rings, the behavior is dichotomic: quasi-global dimension is either \(0\) or \(\infty\), and representation-finite self-injective algebras have \(\mathrm{qgldim}=0\) [2509.20137]. For finite-dimensional self-injective algebras, quasi-global dimension is invariant under stable equivalence of Morita type and derived equivalence [2509.20137]. Under an equivalence of stable module categories for quasi-Frobenius rings, the quasi-global dimensions compare by
\[
\mathrm{qgldim}(B)\le \mathrm{qgldim}(A)
\]
[2509.20137].

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 [2509.20137]. 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 \(A_{n,m}\) [2509.20137]. The paper gives the following values.

| Algebra | \(\gldim\) | \(\mathrm{qgldim}\) |
|---|---:|---:|
| \(A_{n,1}\) | \(2\) | \(2\) |
| \(A_{n,n}\) | \(\infty\) | \(0\) |
| \(A_{n,m}\), \(n>2\), \(1<m<n\) | \(\infty\) | \(2\) |

For \(A_{n,1}\), one has
\[
\mathrm{qgldim}(A_{n,1})=\gldim(A_{n,1})=2
\]
[2509.20137]. For the self-injective, representation-finite algebra \(A_{n,n}\),
\[
\gldim(A_{n,n})=\infty,\qquad \mathrm{qgldim}(A_{n,n})=0,
\]
and all finitely generated modules are periodic and hence have quasi-projective dimension zero [2509.20137]. In the intermediate case \(n>2\) and \(1<m<n\),
\[
\gldim(A_{n,m})=\infty,\qquad \mathrm{qgldim}(A_{n,m})=2
\]
[2509.20137].

The paper identifies the mechanism for this discrepancy: many indecomposable modules are periodic or have projective syzygies after at most \(2\) steps in a suitable quasi-projective resolution, even though their projective dimension is infinite [2509.20137]. 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 \(A\) and a semilocal \(A\)-module \(M\), with \(B=\mathrm{End}_A(M)\), the ADR algebra is left-strongly quasi-hereditary, \(\add M\) has an \(A\)-total left rejective chain of length \(n_M\), and
\[
\mathrm{gldim}\,\mathrm{End}_A(M)\le n_M
\]
[1805.08085]. For original ADR algebras, the conditions “\(B\) is strongly quasi-hereditary” and “\(\mathrm{gldim}\,B=2\)” are equivalent [1805.08085]. 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 [1805.08085].

An analogous distinction appears for quiver algebras. If \(Q\) is a finite quiver without loops, there exists an admissible ideal \(I\) such that \(\mathrm{gldim}(kQ/I)\le 2\) and \(kQ/I\) is strongly quasi-hereditary; additional constructions yield strongly quasi-hereditary quotients with larger global dimension [1010.3871]. 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 \(F\)-finite Noetherian ring \(R\) 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 \(R\), and for a quotient \(R\) of a regular \(F\)-finite Noetherian ring \(S\) of dimension \(d\), the bound is \(d+1\) [2211.11466]. 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 \(\mathrm{qgldim}\) defined via quasi-projective dimension [2211.11466].

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
\[
\mathrm{qgldim}\,R:=\sup\{\operatorname{qpd}_R M\mid M\text{ finitely generated}\},
\]
not as a formally adopted definition within that paper [2412.06659]. 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.

Source: https://www.emergentmind.com/topics/quasi-global-dimension