--- title: Quasi-Projective Dimension Overview url: https://www.emergentmind.com/topics/quasi-projective-dimension type: topic --- # Quasi-Projective Dimension Overview Quasi-projective dimension is a homological invariant that generalizes projective dimension by replacing exact projective resolutions with more flexible complexes whose homology is built from repeated copies of the object under study. The modern literature records this notion in several related settings. Gheibi–Jorgensen–Takahashi define quasi-projective dimension for modules by means of bounded quasi-projective resolutions [1912.10421], Sharif extends an older quasi-deformation-based notion to homologically finite complexes [1708.04455], and subsequent work develops the invariant for complexes over associative rings, for objects in abelian categories, and relative to semidualizing modules [2604.09279] [2509.20137] [2508.15064]. Across these settings, quasi-projective dimension is designed to retain depth-theoretic and rigidity properties usually associated with finite projective dimension while applying to substantially larger classes of modules and complexes. ## 1. Definitions and ambient frameworks In the module-theoretic framework of Gheibi–Jorgensen–Takahashi, a quasi-projective resolution of an \(R\)-module \(M\) is a bounded below complex \(P\) of projective \(R\)-modules such that for all \(i \ge \inf P\) there exist non-negative integers \(a_i\), not all zero, with \[ H_i(P)\cong M^{a_i}. \] The quasi-projective dimension is then \[ \operatorname{qpd}_R M = \inf\{\sup P-\operatorname{hsup}P \mid P \text{ is a bounded quasi-projective resolution of }M\}, \] with \(\operatorname{qpd}_R 0=-\infty\) in the module papers [1912.10421]. Over local rings, finite minimal quasi-projective resolutions exist, and the infimum is attained by such a resolution [1912.10421]. Chen–Chen–Liu formulate the same basic idea in an abelian category \(\mathcal A\) with enough projectives. For a nonzero object \(M\in\mathcal A\), a quasi-projective resolution is a bounded below complex \(P_\bullet\) of projectives such that above some bound every homology group is isomorphic to a finite direct sum of copies of \(M\), and \[ \qpd_{\mathcal A}(M) = \inf\{\sup(P_\bullet)-\mathrm{hsup}(P_\bullet)\mid P_\bullet \text{ is a finite quasi-projective resolution of }M\}. \] In that paper the convention is \(\qpd_{\mathcal A}(0)=0\) [2509.20137]. Sharif’s earlier complex-theoretic usage is different in construction. For a homologically finite complex \(X\) over a commutative Noetherian local ring, and a quasi-deformation \(R\to R' \leftarrow Q\), one sets \[ \operatorname{qpd}_R X = \inf\{\operatorname{pd}_Q(X\otimes_R R')\mid R\to R' \leftarrow Q \text{ is a quasi-deformation}\}. \] In that setting, \[ \operatorname{qpd}_R(X)=\operatorname{CI\mbox{-}dim}_R(X)+\operatorname{cx}_R(X), \] so quasi-projective dimension decomposes as complete intersection dimension plus complexity [1708.04455]. This suggests a terminological bifurcation: the same name is used both for a quasi-deformation invariant and for the later quasi-projective-resolution invariant. A further generalization to complexes over associative rings defines a quasi-projective resolution of a complex \(M\in \mathrm D(R)\) as a semi-projective complex \(P\) whose homology is, in graded form, a finite direct sum of shifts of \(\mathrm H(M)\). The resulting invariant is \[ \mathrm{qpd}_R M = \inf\{\mathrm{sup}P-\mathrm{hsup}P \mid P \text{ is a bounded above quasi-projective resolution of }M\}, \] with shift invariance \(\mathrm{qpd}_R(\Sigma^s M)=\mathrm{qpd}_R M\) [2604.09279]. ## 2. Structural properties and comparison with projective dimension A basic feature of quasi-projective dimension is that it extends projective dimension without altering the finite-projective-dimension regime. Every deleted projective resolution is a quasi-projective resolution, so \[ \qpd_{\mathcal A}(M)\le {\rm pd}_{\mathcal A}(M), \] and if \({\rm pd}_{\mathcal A}(M)<\infty\), then \[ \qpd_{\mathcal A}(M)={\rm pd}_{\mathcal A}(M) \] in the abelian-category setting [2509.20137]. The corresponding comparison for bounded complexes over rings is \[ \mathrm{qpd}_R M+\mathrm{hsup}M=\mathrm{pd}_R M \quad\text{whenever}\quad \mathrm{pd}_R M<\infty, \] which recovers the module equality when \(M\) is concentrated in degree \(0\) [2604.09279]. The quasi-projective-resolution definition behaves naturally under direct sums and syzygies. One has \[ \operatorname{qpd}_R(M\oplus N)\le \sup\{\operatorname{qpd}_R(M),\operatorname{qpd}_R(N)\}, \] and adding a projective summand does not increase quasi-projective dimension; over a local ring, equality holds for finitely generated modules [1912.10421]. In an abelian category, \[ \qpd_{\mathcal A}(\Omega(M))\le \qpd_{\mathcal A}(M), \] and if \(M\) is periodic, meaning \(\Omega^r(M)\cong M\) for some \(r>0\), then \[ \qpd_{\mathcal A}(M)=0, \] even though such an object has infinite projective dimension unless it is projective [2509.20137]. The distinction from projective dimension becomes especially sharp in Frobenius and self-injective contexts. If \(\mathcal A\) is Frobenius, then for every \(M\in\mathcal A\), \[ \qpd_{\mathcal A}(M)=0 \quad\text{or}\quad \qpd_{\mathcal A}(M)=\infty, \] and syzygies preserve quasi-projective dimension: \[ \qpd_{\mathcal A}(M\oplus P)=\qpd_{\mathcal A}(M)=\qpd_{\mathcal A}(\Omega^i(M)) \] for projective \(P\) and any integer \(i\) [2509.20137]. Vanishing conditions can force quasi-projective dimension back into the classical regime. If \(\qpd_{\mathcal A}(M)<\infty\) and \[ \operatorname{Ext}^n_{\mathcal A}(M,M)=0\quad\text{for all }n\ge 2, \] then \({\rm pd}_{\mathcal A}(M)<\infty\). Specializing to a left coherent ring \(R\), if \(M\) is finitely presented, \(\qpd_R(M)<\infty\), and \[ \operatorname{Ext}_R^n(M,M\oplus R)=0 \quad\text{for all }n\ge 1, \] then \(M\) is projective [2509.20137]. ## 3. Depth, grade, and derived formulas One of the principal reasons for introducing quasi-projective dimension is that it preserves depth formulas traditionally attached to finite projective dimension. In the module setting over a commutative Noetherian local ring, \[ \operatorname{qpd}_R M=\operatorname{depth}R-\operatorname{depth}_R M \] for every nonzero finitely generated \(M\) of finite quasi-projective dimension [1912.10421]. For complexes over local rings, the derived analogue is \[ \mathrm{qpd}_R M+\mathrm{hsup}M=\operatorname{depth}R-\operatorname{depth}_R M \] for \(0\not\simeq M\in \mathrm D^\mathrm f_\Box(R)\) with finite quasi-projective dimension [2604.09279]. This invariant also supports depth formulas beyond the Tor-independent case. Gheibi–Jorgensen–Takahashi proved that the classical depth formula remains valid for finitely generated Tor-independent modules when one module has finite quasi-projective dimension [1912.10421]. Jorge-Pérez, Martins, and Mendoza-Rubio then established a generalized Auslander-type formula: if \(M,N\) are nonzero modules over a local ring, \(q=\sup\{i\ge 0\mid \operatorname{Tor}_i^R(M,N)\neq 0\}<\infty\), \(\operatorname{qpd}_R M<\infty\), and either \(q=0\) or \(\operatorname{depth}(\operatorname{Tor}_q^R(M,N))\le 1\), then \[ \operatorname{depth} N = \operatorname{depth}(\operatorname{Tor}_q^R(M,N)) + \operatorname{qpd}_R M - q. \] Equivalently, \[ \operatorname{depth}N+\operatorname{depth}M = \operatorname{depth}R+\operatorname{depth}(\operatorname{Tor}_q^R(M,N))-q \] under the same hypotheses [2409.08996]. Ferraro–Lyle extend this depth-theoretic picture to the derived category. If \(M\) is a finitely generated \(R\)-module of finite quasi-projective dimension and \(N\) is an \(R\)-complex such that \(N\) and \(M\otimes_R^{\mathbf L}N\) both have bounded homology, then \[ \operatorname{depth}(M\otimes_R^{\mathbf L}N) = \operatorname{depth}M+\operatorname{depth}N-\operatorname{depth}R. \] The same paper proves a derived width formula, Ischebeck-type formulas, and a dependency formula in the vein of Jorgensen [2605.07004]. The theory has also developed a substantial grade calculus. If \(M\neq 0\), then \[ \operatorname{grade}M\le \operatorname{qpd}_R M, \] and this motivates the definition of a quasi-perfect module by the equality \[ \operatorname{qpd}_R M=\operatorname{grade}M. \] Over a Cohen–Macaulay local ring, a finitely generated module of finite quasi-projective dimension is Cohen–Macaulay if and only if it is quasi-perfect [2412.06659]. In parallel, Ischebeck’s formula holds under several quasi-homological hypotheses: if \(P_R(M,N)=\sup\{i\ge 0\mid \operatorname{Ext}_R^i(M,N)\neq 0\}<\infty\), then \[ P_R(M,N)=\operatorname{depth}R-\operatorname{depth}M \] 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]. ## 4. Dual, relative, and complete-intersection-oriented refinements The dual notion, quasi-injective dimension, is defined by bounded quasi-injective resolutions of injectives whose homology consists of copies of the module. For a nonzero finitely generated module over a local ring, finite quasi-injective dimension satisfies the Bass-type identity \[ \operatorname{qid}_R M=\operatorname{depth}R, \] and \(\operatorname{qid}_R M\le \operatorname{id}_R M\), with equality when \(\operatorname{id}_R M<\infty\). Over a Gorenstein local ring, finite quasi-injective dimension and finite quasi-projective dimension are equivalent for finitely generated modules [2207.06170]. Dey–Ferraro–Gheibi extend quasi-projective and quasi-injective dimensions relative to a semidualizing module \(C\). A \(C\)-quasi-projective resolution is a bounded complex \(P_\bullet\) of projective modules such that \(P_\bullet\otimes_R C\) is not acyclic and every homology module of \(P_\bullet\otimes_R C\) is either zero or a finite direct sum of copies of \(M\). The resulting invariant \(\operatorname{qpd}_C^R(M)\) satisfies \[ \operatorname{qpd}_C^R(M)=\operatorname{depth}R-\operatorname{depth}_R M \] for nonzero finitely generated \(M\) over a local ring, and it transfers to the classical invariant through Bass and Auslander classes: \[ \operatorname{qpd}_C^R(M) = \operatorname{qpd}_R(\operatorname{Hom}_R(C,M)) \quad\text{for }M\in B_C(R), \] and \[ \operatorname{qpd}_R(M) = \operatorname{qpd}_C^R(C\otimes_R M) \quad\text{for }M\in A_C(R). \] The same framework yields relative versions of Bass’ formula, Ischebeck’s formula, Auslander's depth formula, Jorgensen’s dependency formula, rigidity bounds for Ext and Tor, and a special case of the Auslander–Reiten conjecture [2508.15064]. Sharif’s quasi-deformation-based complex invariant sits closer to complete intersection theory. For a homologically finite complex \(X\) over a local ring, \[ \operatorname{qpd}_R(X)=\operatorname{CI\mbox{-}dim}_R(X)+\operatorname{cx}_R(X), \] and if \(R\) is a complete intersection, then \(\operatorname{CI\mbox{-}dim}_R(X)=0\), so \[ \operatorname{qpd}_R(X)=\operatorname{cx}_R(X). \] This version satisfies an intersection theorem \[ \dim_R Y \le \dim_R(X\otimes_R^{\mathbf L}Y)+\operatorname{qpd}_R(X) \] and derived grade inequalities controlled by complete intersection dimension and complexity [1708.04455]. The newer complex theory over associative rings reconnects the quasi-projective-resolution approach to complete intersection questions. It provides derived Auslander–Buchsbaum formulas and shows that, if every finitely generated module over a commutative Noetherian local ring has finite quasi-projective dimension, then \(R\) is a complete intersection under additional hypotheses such as \(\dim Q\le \dim R+2\), \(I\) being a Burch ideal in a Cohen presentation \(\widehat R\cong Q/I\), or \(\mu(I)\le 2\) [2604.09279]. ## 5. Abelian categories and quasi-global dimension Chen–Chen–Liu place quasi-projective dimension in a general abelian-category setting and show that it admits a useful reduction to ordinary projective dimension. If \(M\) has finite quasi-projective dimension, then there exists a bounded quasi-projective resolution \(P_\bullet\) with \(\mathrm{hsup}(P_\bullet)=0\) and with \[ \qpd_{\mathcal A}(M)={\rm pd}_{\mathcal A}(N), \qquad N=\operatorname{Im}(d_1:P_1\to P_0). \] Moreover, for any right exact covariant functor \(F:\mathcal A\to \mathcal B\) with finite nonvanishing derived functors, the highest-degree nonzero \(L^iF(M)\) equals that for \(N\), and the top derived functor values coincide in positive degree [2509.20137]. This reduction makes quasi-projective dimension a bridge between self-referential resolutions and classical finite projective dimension. The same paper introduces the quasi-global dimension of a left Noetherian ring \(R\): \[ \operatorname{qgl.dim}R = \sup\{\qpd_R(M)\mid M\in R\text{-mod}\}. \] It always satisfies \[ \operatorname{qgl.dim}R\le \operatorname{gl.dim}R, \] and when \(\operatorname{qgl.dim}R<\infty\), \[ \operatorname{fd}(R)=\operatorname{qgl.dim}R, \] so finite quasi-global dimension forces coincidence with finitistic dimension [2509.20137]. Quasi-global dimension is Morita invariant, \[ R \text{ Morita equivalent to } S \Longrightarrow \operatorname{qgl.dim}R=\operatorname{qgl.dim}S, \] behaves predictably on products, \[ \operatorname{qgl.dim}(R\times S)=\max\{\operatorname{qgl.dim}R,\operatorname{qgl.dim}S\}, \] and for finite-dimensional \(K\)-algebras satisfies \[ \operatorname{qgl.dim}(R\otimes_K S)\ge \max\{\operatorname{qgl.dim}R,\operatorname{qgl.dim}S\}. \] If \(R\) is quasi-Frobenius, then \[ \operatorname{qgl.dim}R=0 \quad\text{or}\quad \operatorname{qgl.dim}R=\infty, \] and for finite-dimensional self-injective algebras it is invariant under stable equivalence of Morita type and under derived equivalence [2509.20137]. ## 6. Examples, computations, and current direction of the subject Concrete examples show that quasi-projective dimension can remain small when projective dimension is infinite. Gheibi–Jorgensen–Takahashi prove that over a quotient of a regular local ring by a regular sequence, every finitely generated module has finite quasi-projective dimension [1912.10421]. They also show that periodic modules have quasi-projective dimension \(0\), and the residue field \(k\) over a local ring always has finite quasi-projective dimension; in fact, \[ \operatorname{qpd}_R k=\operatorname{depth}R \] in the local setting [2409.08996]. Nakayama algebras provide especially sharp ring-theoretic examples. For the family \[ A_{n,m}=KQ/\langle \rho_{\lambda_1},\dots,\rho_{\lambda_m}\rangle, \] Chen–Chen–Liu prove: \[ \operatorname{qgl.dim}A_{n,1} = \operatorname{gl.dim}A_{n,1} = 2, \] \[ \operatorname{gl.dim}A_{n,n}=\infty, \qquad \operatorname{qgl.dim}A_{n,n}=0, \] and for \(n>2\) and \(1