Grade 3 Perfect Ideal Theory

• Grade three perfect ideals are codimension 3 ideals in Noetherian rings with projective dimension 3 that yield Cohen–Macaulay quotient rings.
• The Buchsbaum–Eisenbud theorem and ADE classification organize these ideals into finite families distinguished by their graded-commutative differential graded algebra structures.
• Linkage theory and universal complexes reveal rigidity and deformation properties, offering key insights into free resolutions and algebraic structures.

A grade three perfect ideal is a codimension 3 ideal $I \subseteq R$ in a Noetherian local or graded ring, such that $\operatorname{pd}_R(R/I)=3$ and $R/I$ is Cohen–Macaulay. The theory of grade three perfect ideals intertwines homological algebra, commutative algebra, and algebraic geometry, with the structure theory encapsulated in the Buchsbaum–Eisenbud theorem, the ADE (Dynkin) classification, and the paper of linkage, Tor-algebra structure, and rigidity. These ideals govern broad classes of projective and local algebraic phenomena, underpinning implicitization, canonical module theory, and the structure of free resolutions.

1. Definitions, Homological Invariants, and Structure

A perfect ideal $I\subseteq R$ of grade 3 satisfies:

• $\operatorname{ht}(I)=3$,
• $\operatorname{pd}_R(R/I)=3$,
• $R/I$ is Cohen-Macaulay, i.e., $\operatorname{depth}(R/I) = \dim(R) - 3$.

The minimal graded free resolution of $R/I$ is: $$0 \to F_3 \xrightarrow{d_3} F_2 \xrightarrow{d_2} F_1 \xrightarrow{d_1} F_0 \to R/I \to 0$$ with Betti vector $f=(1, m, m+n-1, n)$ where $m = \mu(I)$ and $n$ is the type $$r(R/I) = \operatorname{rank}_R \operatorname{Ext}_R^3(R/I, R)$$. The deviation is $d(I) = m-3$.

The minimal free resolution admits a graded-commutative differential graded algebra (DG-algebra) structure, and $A = \operatorname{Tor}^R_*(R/I,k)$ realizes the (co)homological invariants that underlie the ADE classification (Guerrieri et al., 2 Jul 2024).

2. Buchsbaum–Eisenbud Theorem and ADE Classification

The Buchsbaum–Eisenbud structure theorem asserts:

• For Gorenstein ideals ($n=1$), $I$ is generated by submaximal Pfaffians of an odd-sized skew-symmetric matrix.
• General (non-Gorenstein) grade 3 perfect ideals fall into a finite number of families, classified by ADE Dynkin diagrams. The Betti format $(1,m,m+n-1, n)$ corresponds to a graph $\Gamma_f$; the format is Dynkin if and only if $\Gamma_f$ is one of $A_n$, $D_n$, $E_6$, $E_7$, $E_8$ (Guerrieri et al., 2 Jul 2024, Christensen et al., 2017).

This classification further reflects in the algebraic structure on $\operatorname{Tor}^R_*(R/I,k)$: the DG-algebra structure allows for only five types of multiplicative behavior (categories $\mathbf{C}(3)$, $\mathbf{B}$, $\mathbf{G}(r)$, $\mathbf{H}(p,q)$, $\mathbf{T}$), realized in different formats and distinguished by products in $$\operatorname{Tor}_1 \otimes \operatorname{Tor}_1 \to \operatorname{Tor}_2$$ (Christensen et al., 2018, Hardesty, 1 Mar 2024).

3. Linkage Theory and Golod Ideals

A core result of modern linkage theory is that every grade 3 perfect ideal in a regular local ring is linked by a finite sequence of direct links (using height-3 complete intersections) to an ideal $J$ such that $R/J$ is either a complete intersection (CI) or a Golod ring (Christensen et al., 2018). In the Golod case, the multiplication and all Massey operations vanish on the Koszul homology, and the Tor algebra is trivial.

For any grade 2 perfect ideal, linkage always reaches a CI, but grade 3 exhibits genuinely new phenomena: the possibility of landing on a Golod ring. The proof uses homological classification (Buchsbaum–Eisenbud plus the finer structure of Tor algebras) and explicit control of numerical invariants $(m,n,p,q)$ under linkage.

4. Tor-Algebra Structure and Distinguished Families

The structure of $\operatorname{Tor}^R_*(R/I,k)$ distinguishes families of grade 3 perfect ideals with the same Betti numbers but different homotopy multiplicative structures. The case $(1,5,6,2)$ (type $E_6$) provides two universal such families:

• The Celikbas–Laxmi–Kraśkiewicz–Weyman (CLKW) family with trivial multiplication (Golod), where no minimal Koszul relation among the generators survives to the second syzygy.
• Brown’s family (1987), where at least one Koszul relation lifts to a minimal second syzygy, giving nontrivial cup product in Tor and non-Golod structure (Kustin, 2020, Celikbas et al., 2018).

Specialization from the universal complex yields both families; any Brown ideal arises by suitable (non-homogeneous) specialization from the CLKW family. When these ideals are defined over power series rings, both are rigid in the sense of Lichtenbaum–Schlessinger—their deformations are strongly unobstructed (Kustin, 2020).

5. Universal Complexes and Explicit Constructions

Two coordinate-free “universal” complexes parameterize all perfect grade 3 ideals of format $(2,6,5,1)$:

• The generic complex $Q$ constructed from free modules $F$, $G$ with maps $\varphi$, $\ell$, $y$, $\delta$, and scalars, specializes to
• The CLKW (trivial Tor-multiplication) and Brown (nontrivial Tor-multiplication) complexes via explicit substitutions of indeterminates.

Each complex yields an explicit acyclic free resolution when the maps satisfy genericity and depth conditions, and the detailed matrices specify minimal resolutions of all such grade 3 perfect ideals with these Betti numbers (Kustin, 2020).

6. Rigidity, Deformation Theory, and Linkage

Both CLKW and Brown families, once completed, yield rigid $k$-algebras—having vanishing $T^1$ and $T^2$ obstructions in the tangent space of the functor of Artin rings deformations—provided the quotient is licci (linked to a complete intersection) and generically a CI (Kustin, 2020). The proof hinges on properties of semigeneric linkage and the inheritance of rigidity through such links.

Moreover, there is an explicit chain of linkages from classical determinantal rings (e.g., $2\times2$ minors), through Brown’s ideal, to the CLKW ideal, with rigidity preserved at each linkage step. This identifies all such algebras as lying in the linkage class of a complete intersection in the Dynkin case (Christensen et al., 2017, Kustin, 2020).

7. Applications and Broader Context

Grade three perfect ideals with relevant formats appear in the construction of Artinian Gorenstein algebras via doubling, describing all such algebras of socle degree three in embedding dimension four (Marques et al., 2022). They realize the $E_6$ case of the ADE-correspondence for Cohen–Macaulay rings, and their classification aligns with the behavior of higher structure maps on generic resolutions (Guerrieri et al., 2 Jul 2024). In addition, their canonical module theory, implied by the structure of the last module in the minimal free resolution, connects to broader phenomena in duality and residual intersection theory.

The ideals with format $(1,5,6,2)$, $E_6$-type, thus serve as key test objects for the paper of the interplay between free resolutions, linkage, deformation theory, and the structure of Tor algebras, and are central in ongoing efforts to fully realize and classify all ADE-type perfect ideals—especially for those with nontrivial Tor algebra structures (Guerrieri et al., 2 Jul 2024, Hardesty, 1 Mar 2024, Christensen et al., 2018).