--- title: Restricted Projective Dimension url: https://www.emergentmind.com/topics/restricted-projective-dimension type: topic --- # Restricted Projective Dimension Searching arXiv for recent and foundational papers on restricted projective dimension and closely related formulations. Restricted projective dimension denotes a family of research problems in which projective dimension is studied only after imposing auxiliary constraints. In commutative algebra, the prototype is a Stillman-type question: for an ideal \(I\subseteq R=K[x_1,\dots,x_N]\) generated by finitely many forms of bounded degree, can \(\operatorname{pd}_R(R/I)\) be bounded independently of the ambient number of variables \(N\)? In more relative settings, the phrase also refers to the large restricted projective dimension \(Rpd_R(M)\), obtained by testing \(\operatorname{Ext}\) only against modules of finite injective dimension, or to projective dimension after restricting representable functors to a rigid or cluster-tilting subcategory, or after specializing equivariant modules from infinite-variable objects to finite polynomial rings [1106.0839] [2508.20281] [1111.3077] [2207.05860]. ## 1. Terminology and principal viewpoints The literature represented here does not use a single universal definition. Rather, “restricted projective dimension” appears in several closely related forms. | Setting | Object whose projective dimension is studied | Restriction | |---|---|---| | Stillman-type commutative algebra | \(R/I\) | fixed number and degree of generators | | Relative homological algebra | \(M\) via \(Rpd_R(M)\) | test \(\operatorname{Ext}\) only against modules of finite injective dimension | | Triangulated and cluster-tilting categories | \(\mathrm{H}X=\operatorname{Hom}_{\mathcal C}(-,X)|_{\mathcal T}\) | restrict representable functors to \(\mathcal T\) | | Equivariant asymptotics | \(M(\mathbf C^n)\) over \(A(\mathbf C^n)\) | evaluate an equivariant infinite-variable module at finite rank | In the module-theoretic sense recalled from Christensen–Foxby–Frankild, the large restricted projective dimension is \[ Rpd_R(M)=\sup\{n\ge 0\mid \operatorname{Ext}_R^n(M,I)\neq 0\text{ for some }I\in I_{<\infty}\}, \] with \(\mathcal{R}P_n(R)=\{M\in \mathrm{Mod}\,R\mid Rpd_R(M)\le n\}\). The same paper records \(P_n\subseteq \mathcal{R}P_n\), \(\mathcal{R}P_n^{<\omega}=\mathcal{R}F_n^{<\omega}\), and \(Rpd_R(M)<\infty\) for every finitely generated \(M\in \mathrm{mod}\,R\) [2508.20281]. A different but closely allied usage occurs in triangulated categories: if \(\mathcal T\) is a contravariantly finite rigid subcategory of a Hom-finite Krull–Schmidt triangulated category \(\mathcal C\), then one studies the projective dimension of the restricted representable functor \[ \mathrm{H}X=\operatorname{Hom}_{\mathcal C}(-,X)\big|_{\mathcal T} \] in \(\mathrm{mod}\text{-}\mathcal T\). In cluster-tilting situations this produces a sharp trichotomy between projective dimension \(0\), \(1\), and \(\infty\) [1111.3077] [1111.2013]. ## 2. Quadratic Stillman bounds and the structural containment theorem The foundational positive result for the commutative-algebraic form of restricted projective dimension is Ananyan–Hochster’s theorem on ideals generated by linear and quadratic forms. Let \[ R=K[x_1,\dots,x_N] \] and let \(I=(F_1,\dots,F_{m+n})\) be generated by forms of degree at most \(2\). If the linear forms among the \(F_i\) span a \(K\)-vector space of dimension \(m\), and \(h\) is the height of the ideal generated by the images of the quadratic \(F_j\) after modding out by the ideal generated by all linear \(F_i\), then the paper defines \[ B(m,n,0)=m(n+1), \] and for \(h\ge 1\), \[ B(m,n,h) =(m+h)(n^3+n^2+n+1)+h(n+1)+B((m+h)n^2,n,h-1). \] It also defines \[ C(s)=\max\{\,B(m,n,h)+h:\ m+n=s,\ 0\le h\le n-1\,\}, \] and \[ C_0(s)=\max\{\,B(0,n,h)+m+h:\ m+n=s,\ 0d\). This is the device that either lowers the height in the induction or forces the remaining nonzero tail terms to form a regular sequence. The resulting bounds are explicit and asymptotically satisfy \[ C(s)\sim C_0(s)\sim 2s^{2s}, \] so the theorem proves the quadratic case of Stillman’s conjecture together with a recursive bound of order \(2n^{2n}\) [1106.0839]. ## 3. Positive and negative boundary results in commutative algebra Later work showed that not every natural restriction controls projective dimension. For any integers \(h,e\ge 2\) with \((h,e)\neq (2,2)\) and any integer \(p\ge 5\), there exists an unmixed ideal \(I_{h,e,p}\) in a polynomial ring such that \[ \operatorname{ht}(I_{h,e,p})=h,\qquad e(R/I_{h,e,p})=e,\qquad \operatorname{pd}_R(R/I_{h,e,p})\ge p, \] and \(\sqrt{I_{h,e,p}}\) is a linear prime. In the homogeneous formulation, the ideal may be chosen primary to a linear prime \((x_1,\dots,x_h)\). Thus fixed height, fixed Hilbert–Samuel multiplicity, unmixedness or Serre’s \((S_1)\), and support on a linear subspace do not bound projective dimension. The single exceptional pair is \((h,e)=(2,2)\), where Engheta’s classification gives \(\operatorname{pd}(R/I)\le 3\) [1301.4147]. By contrast, highly specific generator restrictions can produce sharp small bounds. For ideals generated by exactly three cubic forms, \[ I=(f_1,f_2,f_3)\subseteq R, \] one has \[ \operatorname{pd}(R/I)\le 5, \] and the bound is sharp. The paper records that Engheta had previously shown \(\operatorname{pd}(R/I)\le 36\), while the example \[ I=(x^3,\ y^3,\ x^2a+xyb+y^2c)\subseteq K[x,y,a,b,c] \] satisfies \(\operatorname{pd}(R/I)=5\) [1801.08195]. A different restriction problem concerns tensor products. The paper constructs modules \(M,N\) with \[ \operatorname{pd}_R(M)=\infty=\operatorname{pd}_R(N), \qquad \operatorname{pd}_R(M\otimes_R N)<\infty, \] and also examples with infinite \(G\)-dimension and infinite complexity whose tensor product still has finite projective dimension. It then proves that if \[ \operatorname{pd}_R(M\otimes_R N)\le n \] and \[ \operatorname{Tor}_i^R(M,N)=0\quad\text{for }i=1,\dots,n, \] then \[ \operatorname{pd}_R(M)\le n,\qquad \operatorname{pd}_R(N)\le n. \] If one factor is totally reflexive and \(M\otimes_R N\neq 0\) has finite projective dimension, then that factor is projective and the other has finite projective dimension. If \(R\) is Cohen–Macaulay local and \(M=\Omega_RL\) for a nonfree maximal Cohen–Macaulay module \(L\), then \[ \operatorname{pd}_R(M\otimes_R N)<\infty \iff N=0. \] These results isolate classes in which finite projective dimension of a tensor product becomes genuinely restrictive [2304.04490]. ## 4. Monomial, graph, clutter, and hypergraph restrictions For edge ideals and related square-free monomial ideals, restricted projective dimension becomes a combinatorial invariant. If \(G\) is a finite simple graph with edge ideal \(I(G)\subseteq S=\mathbf{k}[x_1,\dots,x_n]\), the notation \[ \operatorname{pd}(G):=\operatorname{pd}(S/I(G)) \] is used. A central recursive inequality is \[ \operatorname{pd}(G)\le \max\{\operatorname{pd}(G-\operatorname{st}x)+\deg x,\ \operatorname{pd}(G-x)+1\}. \] The paper then derives several domination-theoretic bounds, including \[ \operatorname{pd}(G)\le n-\epsilon(G),\qquad \operatorname{pd}(G)\le n-\tau(G),\qquad \operatorname{pd}(G)\ge n-i(G). \] For chordal graphs one gets the exact formula \[ \operatorname{pd}(G)=|V(G)|-i(G), \] and for a subgraph \(G\subseteq \mathbb Z^\ell\), \[ \operatorname{pd}(G)\le n\left(1-\frac1{2\ell+1}\right) \] [1110.2841]. The clutter formulation is parallel. For a clutter \(\mathcal C\) with associated square-free monomial ideal \(I(\mathcal C)\), the convention is \[ \operatorname{pd}(\mathcal C)=\operatorname{pd}(S/I(\mathcal C)). \] If \(\epsilon(\mathcal C)\) is the edgewise domination number, then \[ \operatorname{pd}(\mathcal C)\le |V(\mathcal C)|-\epsilon(\mathcal C). \] The same paper develops the recursive operations \(\mathcal C+A\) and \(\mathcal C:A\), satisfying \[ (I(\mathcal C),x^A)=I(\mathcal C+A),\qquad I(\mathcal C):x^A=I(\mathcal C:A), \] and the bound \[ \operatorname{pd}(\mathcal C)\le \max\{\operatorname{pd}(\mathcal C+A),\operatorname{pd}(\mathcal C:A)\}. \] For connected graph clutters \(C_k(P_n)\) and \(C_k(\Gamma_n)\), the paper records explicit formulas such as \[ \operatorname{pd}(C_k(P_n))=\left\lfloor \frac{n}{k+1}\right\rfloor+\left\lfloor \frac{n+1}{k+1}\right\rfloor, \] and \[ \operatorname{pd}(C_k(\Gamma_n))=\left\lfloor \frac{n}{k+1}\right\rfloor+\left\lceil \frac{n}{k+1}\right\rceil \] [1301.2665]. A hypergraph version sharpens this restricted behavior for strings and cycles. If \(\mathcal H\) is an open string on \(\mu\) vertices, then \[ \operatorname{pd}(\mathcal H)=\mu-\left\lfloor \frac{\mu}{3}\right\rfloor, \] while for an open cycle, \[ \operatorname{pd}(\mathcal H)=\mu-1-\left\lfloor \frac{\mu-2}{3}\right\rfloor. \] If one adds a single nonredundant higher-dimensional edge to an open string, the projective dimension either stays the same or increases by exactly \(1\); for an open cycle with one higher-dimensional edge, it never changes at all [1910.01053]. ## 5. Hyperplane arrangements and deformation-theoretic restrictions For a central hyperplane arrangement \(\mathcal A\subseteq V=\Bbb K^\ell\), the logarithmic derivation module \[ D(\mathcal A)=\{\theta\in \operatorname{Der}S\mid \theta(\alpha_H)\in S\alpha_H\ \forall H\in\mathcal A\} \] is a reflexive \(S\)-module of rank \(\ell\), and one writes \[ \operatorname{pd}\mathcal A:=\operatorname{pd}_S D(\mathcal A). \] Because \(D(\mathcal A)\) is reflexive, \[ 0\le \operatorname{pd}\mathcal A\le \ell-2. \] Abe’s general theory studies how \(\operatorname{pd}\mathcal A\) behaves under deletion and restriction. For \(H\in\mathcal A\), the Euler restriction map \[ \rho^H:D(\mathcal A)\to D(\mathcal A^H) \] and the Ziegler restriction map \[ \pi^H:D_H(\mathcal A)\to D(\mathcal A^H,m^H) \] govern the comparison. Under the NMPD hypothesis and local codimension-three surjectivity, the restriction theorem states that if \(\operatorname{pd}\mathcal A'=k\) for \(\mathcal A'=\mathcal A\setminus\{H\}\), then: \[ \operatorname{pd}\mathcal Ak \implies \operatorname{pd}\mathcal A^H=\operatorname{pd}\mathcal A-1. \] The same framework includes a Yoshinaga-type theorem and a division theorem for projective dimensions [2009.04101]. Graphic arrangements provide a sharp graph-theoretic classification at the first nonfree level. If \(G\) is a graph and \[ A(G)=\{\ker(x_i-x_j)\mid \{i,j\}\in E(G)\}, \] then \[ \operatorname{pd} D(A(G))\le 1 \iff G \text{ is weakly chordal}, \] and \[ \operatorname{pd} D(A(G))=1 \] exactly when \(G\) is weakly chordal but not chordal. Induced cycles \(C_m\) force \[ \operatorname{pd}(A(G))\ge m-3, \] and antiholes satisfy \[ \operatorname{pd}(A(C_\ell^C))=2\qquad (\ell\ge 6) \] [2307.06021]. For cones of deformations of Weyl arrangements, a different restricted parameter range appears. If \(\Phi\) is simply laced and the deformation interval is \[ [-k,k+2], \] then \[ \operatorname{pd}_S\bigl(D(c_{\Phi^+}^{[-k,k+2]})\bigr)=1. \] In type \(A_3\), the reduced summand \(D_0(\mathcal A)\) has minimal free resolution \[ 0 \rightarrow S[-4k-8]^3 \rightarrow S[-4k-7]^6 \rightarrow D_0(\mathcal A) \rightarrow 0. \] For type \(B_2\), the paper proves projective-dimension-one resolutions for all intervals \([-k,k+j]\), and shows that modules with the same graded Betti numbers may still be non-isomorphic because their associated vector bundles have different maximal jumping lines [2601.10466]. ## 6. Relative and categorical formulations In the module-theoretic formulation, restricted projective dimension is itself an invariant. For a commutative noetherian ring \(R\), \[ Rpd_R(M) = \sup \{n \geq 0 \mid \operatorname{Ext}_R^n(M,I) \neq 0 \text{ for some } I \in I_{<\infty}\}, \] and \[ \mathcal{R}P_n(R)=\{M\in \mathrm{Mod}\,R\mid Rpd_R(M)\le n\}. \] Over Gorenstein rings one has \[ Rpd_R = Gpd_R, \] while over any finite-dimensional noetherian ring the class \(\mathcal{R}P_0\) is the left half of the hereditary cotorsion pair \[ (\mathcal{R}P_0, I_{<\infty}), \] hence finitely deconstructible. If \(R\) is Cohen–Macaulay with a pointwise dualizing module, then for every \(n\ge 0\) the class \(\mathcal{R}P_n\) is finitely deconstructible [2508.20281]. A categorical version replaces modules over \(R\) by restricted representable functors. If \(\mathcal T\) is a contravariantly finite rigid subcategory of a Hom-finite Krull–Schmidt triangulated category \(\mathcal C\), then for \[ \mathrm H X=\operatorname{Hom}_{\mathcal C}(-,X)\big|_{\mathcal T}, \] one has, for \(X\in \mathcal T*\mathcal T[1]\) with no direct summands in \(\mathcal T[1]\), \[ \operatorname{pd}\mathrm H X \le 1 \iff \mathcal I_X(\mathcal T[1])=0, \] where \(\mathcal I_X(\mathcal T[1])\) is the ideal of morphisms between objects of \(\mathcal T[1]\) factoring through \(X\). In the cluster-tilting case, \(\mathrm{mod}\text{-}\mathcal T\) is Gorenstein of dimension at most one, so \[ \operatorname{pd}\mathrm H X=\infty \iff \mathcal I_X(\mathcal T[1])\neq 0 \] [1111.3077]. For cluster-tilted algebras \(C=\operatorname{End}_{\mathcal C}(T)\), the same phenomenon appears in object form. Since \(C\) is Gorenstein of dimension at most \(1\), every finitely generated \(C\)-module has projective dimension in \[ \{0,1,\infty\}. \] If \(M\notin \operatorname{add}T[1]\), then \[ \operatorname{pd}_C\operatorname{Hom}_{\mathcal C}(T,M)=\infty \iff I_M\neq 0, \] where \(I_M\) is the ideal of endomorphisms of \(T[1]\) factoring through \(M\) [1111.2013]. ## 7. Asymptotic growth under symmetry and specialization A different restricted projective-dimension program studies asymptotics in families with symmetry. For the free twisted commutative algebra \[ A=\operatorname{Sym}(\mathbf V\otimes E), \qquad d=\dim(E), \] and a finitely generated \(A\)-module \(M\), Sam and Snowden prove that \[ \pdim_M(n):=\operatorname{pd}_{A(\mathbf C^n)}M(\mathbf C^n) \] is eventually linear in \(n\), with slope at most \(d\). The key formula is \[ \pdim_M(n)=\max_k\big(\gamma(F_k(M);n)-k\big), \] where the \(F_k(M)\) come from the linear strands of the minimal free resolution of \(M\). For the determinantal quotient \(M=A/\mathfrak a_r\), \[ \operatorname{pd}_{A(\mathbf C^n)}M(\mathbf C^n)=(d-r)(n-r)=(d-r)n-(d-r)r \] whenever \(\min(n,d)\ge r\) [2207.05860]. A parallel asymptotic framework appears for \(\operatorname{Inc}^i\)-invariant chains of ideals \[ I_1\subseteq I_2\subseteq \cdots \subseteq I_n\subseteq \cdots \] in polynomial rings \[ R_n=K[x_{k,j}\mid 1\le k\le c,\ 1\le j\le n]. \] The paper conjectures eventual linearity \[ \operatorname{pd}(R_n/I_n)=an+b\qquad (n\gg 0), \] and proves that if the chain is eventually perfect, then the conjecture holds. In general one has, for \(n\gg0\), \[ cn \ge \operatorname{pd}(R_n/I_n)\ge \gamma_i(I)\,n + D(I), \] and for monomial chains a stronger lower bound \[ \operatorname{pd}(R_n/I_n)\ge \Gamma_i(I)\,n + D(I). \] When \(c=1\), the projective dimension is either eventually constant or satisfies \[ n-D\le \operatorname{pd}(R_n/I_n)\le n \] for all large \(n\). The same paper proves eventual linearity of codimension, \[ \operatorname{codim}(I_n)=\gamma_i(I)\,n + D(I), \] which furnishes the baseline asymptotic lower bound for projective dimension [1809.06877]. These asymptotic results suggest that symmetry can force affine-linear homological growth even when a full projective-dimension theorem remains conjectural. Across Stillman-type problems, combinatorial models, arrangement theory, categorical restriction, and equivariant asymptotics, restricted projective dimension functions as a unifying strategy: one seeks projective-dimension control not in complete generality, but inside classes where algebraic, geometric, or combinatorial structure can replace ambient size by a finite and often explicit controlling datum. Source: https://www.emergentmind.com/topics/restricted-projective-dimension