Graded Double Ore Extension

• Graded double Ore extension is a connected ℕ-graded algebra constructed by adjoining two degree‑1 variables governed by matrix-controlled twisting and a specific quadratic relation.
• It preserves properties like Koszulity and Artin–Schelter regularity while increasing the global dimension by 2 and ensuring a PBW-type basis under suitable freeness conditions.
• The construction facilitates explicit computations of Nakayama automorphisms, Calabi–Yau criteria, and derivation-quotient structures via twisted superpotentials and homological invariants.

A graded double Ore extension is a connected $\mathbb N$-graded algebra obtained from a graded algebra by adjoining two degree-$1$ variables subject to a matrix-controlled twisting action and a single quadratic relation between the new variables. In the literature it appears in the general form $R_P[x_1,x_2;\sigma,\delta,\tau]$ and in the trimmed form $A_P[y_1,y_2;\sigma]$. For Koszul Artin–Schelter regular input algebras, the construction preserves Koszulity and increases global dimension by $2$; more recent work further gives a minimal free resolution in the presence of nontrivial skew derivations, introduces the $\sigma$-divergence as a homological invariant, computes the Nakayama automorphism explicitly, and realizes the algebra as a derivation-quotient algebra of a twisted superpotential (1810.06778, Cao et al., 21 Jul 2025).

1. Defining data and basic variants

Let $R=\bigoplus_{m\ge 0} R_m$ be an $\mathbb N$-graded algebra. A graded right double Ore extension of $R$ in two new variables is an algebra

$$A \;=\; R_P[x_1,x_2;\sigma,\delta,\tau] \;=\; R\langle x_1,x_2\rangle/I$$

with parameters $1$0, tail data $1$1, a $1$2 matrix of graded $1$3-linear maps

$1$4

and a pair of graded $1$5-derivations

$1$6

satisfying the degree conditions

$1$7

together with the relations

$1$8

and

$1$9

The grading is specified by $R_P[x_1,x_2;\sigma,\delta,\tau]$0 and $R_P[x_1,x_2;\sigma,\delta,\tau]$1, and all defining relations are required to be homogeneous. In the graded context one further requires freeness as a left $R_P[x_1,x_2;\sigma,\delta,\tau]$2-module on the standard monomials, and under mild conditions also as a right $R_P[x_1,x_2;\sigma,\delta,\tau]$3-module with the obvious monomial bases (1810.06778).

In the formulation used for Koszul AS-regular algebras, one takes $R_P[x_1,x_2;\sigma,\delta,\tau]$4, assumes that $R_P[x_1,x_2;\sigma,\delta,\tau]$5 is Koszul Artin–Schelter regular of dimension $R_P[x_1,x_2;\sigma,\delta,\tau]$6, fixes $R_P[x_1,x_2;\sigma,\delta,\tau]$7, a graded algebra homomorphism

$R_P[x_1,x_2;\sigma,\delta,\tau]$8

and a degree-$R_P[x_1,x_2;\sigma,\delta,\tau]$9 linear map

$A_P[y_1,y_2;\sigma]$0

satisfying the $A_P[y_1,y_2;\sigma]$1-derivation rule

$A_P[y_1,y_2;\sigma]$2

The resulting graded right double Ore extension

$A_P[y_1,y_2;\sigma]$3

is generated by $A_P[y_1,y_2;\sigma]$4 and degree-$A_P[y_1,y_2;\sigma]$5 variables $A_P[y_1,y_2;\sigma]$6 with mixed relations

$A_P[y_1,y_2;\sigma]$7

and quadratic relation

$A_P[y_1,y_2;\sigma]$8

It is required that $A_P[y_1,y_2;\sigma]$9 be free as a left $2$0-module with basis $2$1. When $2$2, $2$3 must be invertible in the matrix sense, and then $2$4 is also a left double Ore extension (Cao et al., 21 Jul 2025).

A central special case is the trimmed graded double Ore extension. Here one sets $2$5 and $2$6, and writes

$2$7

This is the setting of the Nakayama-automorphism formulas in Zhu–Van Oystaeyen–Zhang and of much of the explicit PBW theory for regular algebras of type $2$8 (Zhu et al., 2014, Rubiano, 7 Jun 2026).

2. PBW structure and relation to graded skew PBW extensions

The defining freeness condition already imposes a PBW-type monomial structure. In the trimmed setting over a quadratic base algebra $2$9, one fixes the order

$\sigma$0

and a monomial is PBW normal if it has the form

$\sigma$1

Bergman’s Diamond Lemma together with the trimmed-double-extension hypotheses implies that

$\sigma$2

is a $\sigma$3-basis, and every word has a unique normal form obtained by repeatedly rewriting the forbidden adjacent pairs $\sigma$4, $\sigma$5, and $\sigma$6 via the quadratic and mixed relations (Rubiano, 7 Jun 2026).

Gómez–Suárez gives necessary and sufficient conditions for a graded double Ore extension to be a graded skew PBW extension. If

$\sigma$7

is a connected graded double Ore extension of a connected graded algebra $\sigma$8, then $\sigma$9 is a graded skew PBW extension of $R=\bigoplus_{m\ge 0} R_m$0 if and only if the following hold:

1. $R=\bigoplus_{m\ge 0} R_m$1 and $R=\bigoplus_{m\ge 0} R_m$2;
2. $R=\bigoplus_{m\ge 0} R_m$3, and $R=\bigoplus_{m\ge 0} R_m$4 are graded automorphisms of $R=\bigoplus_{m\ge 0} R_m$5;
3. $R=\bigoplus_{m\ge 0} R_m$6 are arbitrary graded $R=\bigoplus_{m\ge 0} R_m$7-derivations;
4. $R=\bigoplus_{m\ge 0} R_m$8 are homogeneous elements of positive degree.

Under these hypotheses the defining relations become exactly those of a skew PBW extension, the ordered monomials $R=\bigoplus_{m\ge 0} R_m$9 remain linearly independent over $\mathbb N$0, and the PBW property follows (1810.06778).

The explicit trimmed family

$\mathbb N$1

with

$\mathbb N$2

has defining relations \begin{align*} x_2x_1+x_1x_2&=0, & y_2y_1+y_1y_2&=0,\ y_1x_1-x_1y_1&=0, & y_1x_2-x_2y_2&=0,\ y_2x_1-x_1y_2&=0, & y_2x_2-\alpha x_2y_1&=0. \end{align*} A finite Gröbner–Shirshov calculation yields the PBW basis

$\mathbb N$3

and hence

$\mathbb N$4

The same example is neither an iterated Ore extension in the order $\mathbb N$5 nor $\mathbb N$6 (Rubiano, 2 Apr 2026).

3. Homological behavior, Koszulity, and minimal resolutions

For Koszul AS-regular input algebras, graded double Ore extensions preserve the main homological regularity properties. Zhu–Van Oystaeyen–Zhang state that if $\mathbb N$7 is Koszul then the trimmed extension $\mathbb N$8 is also Koszul, and if in addition $\mathbb N$9 is Artin–Schelter regular of global dimension $R$0, $R$1 is invertible, and $R$2, then $R$3 is AS-regular with

$R$4

Gómez–Suárez formulates the same increase-by-$R$5 phenomenon for connected graded skew PBW extensions of Artin–Schelter regular algebras and derives the skew Calabi–Yau consequence for connected skew Calabi–Yau bases (Zhu et al., 2014, 1810.06778).

For the nontrivial derivation case, let

$R$6

be the Koszul resolution of the trivial $R$7-module. To lift this to

$R$8

one introduces a quadruple of linear maps

$R$9

$$A \;=\; R_P[x_1,x_2;\sigma,\delta,\tau] \;=\; R\langle x_1,x_2\rangle/I$$0

satisfying compatibilities with $$A \;=\; R_P[x_1,x_2;\sigma,\delta,\tau] \;=\; R\langle x_1,x_2\rangle/I$$1, the differentials, and the Ore homotopies. From these data one defines maps

$$A \;=\; R_P[x_1,x_2;\sigma,\delta,\tau] \;=\; R\langle x_1,x_2\rangle/I$$2

$$A \;=\; R_P[x_1,x_2;\sigma,\delta,\tau] \;=\; R\langle x_1,x_2\rangle/I$$3

$$A \;=\; R_P[x_1,x_2;\sigma,\delta,\tau] \;=\; R\langle x_1,x_2\rangle/I$$4

forms two iterated mapping cones, and obtains a complex

$$A \;=\; R_P[x_1,x_2;\sigma,\delta,\tau] \;=\; R\langle x_1,x_2\rangle/I$$5

Theorem 2.14 asserts that $$A \;=\; R_P[x_1,x_2;\sigma,\delta,\tau] \;=\; R\langle x_1,x_2\rangle/I$$6 is exact and minimal; hence $$A \;=\; R_P[x_1,x_2;\sigma,\delta,\tau] \;=\; R\langle x_1,x_2\rangle/I$$7 is Koszul of global dimension $$A \;=\; R_P[x_1,x_2;\sigma,\delta,\tau] \;=\; R\langle x_1,x_2\rangle/I$$8 (Cao et al., 21 Jul 2025).

The pilot family $$A \;=\; R_P[x_1,x_2;\sigma,\delta,\tau] \;=\; R\langle x_1,x_2\rangle/I$$9 gives a fully explicit type-$1$00 resolution. Its trivial right module has a linear resolution

$1$01

with differentials written in matrix form and all entries in positive degree. The graded Betti numbers are

$1$02

equivalently

$1$03

The same paper also computes $1$04-step linear resolutions for the cyclic modules

$1$05

with Betti numbers

$1$06

for each module (Rubiano, 2 Apr 2026).

4. $1$07-divergence and explicit Nakayama automorphisms

When $1$08 is AS-regular and $1$09, choose a basis $1$10. Then the top-degree components of the auxiliary maps satisfy

$1$11

which define vectors $1$12. Although the quadruple of auxiliary maps is non-unique, the combination

$1$13

is independent of choices. This invariant is called the $1$14-divergence of $1$15. Homologically, it measures the failure of the obvious part of the Ore homotopies to be a chain-map at the top degree and plays the same rôle as the usual divergence in the $1$16-variable Ore extension case (Cao et al., 21 Jul 2025).

Since a graded double Ore extension of a Koszul AS-regular algebra is again AS-regular, it has a Nakayama automorphism. In the nontrivial derivation case, Theorem 3.9 gives a closed formula. On the embedded copy of $1$17,

$1$18

On the column vector $1$19,

$1$20

where

$1$21

and $1$22 is the homological determinant of $1$23 defined by its action on $1$24 (Cao et al., 21 Jul 2025).

In the trimmed case, Zhu–Van Oystaeyen–Zhang obtain the corresponding formula without the derivation term. If $1$25 denotes the Nakayama automorphism of $1$26, then

$1$27

and

$1$28

Thus the restriction to $1$29 is governed by $1$30, while the action on the adjoined variables is governed by $1$31 and the parameter matrix $1$32 (Zhu et al., 2014).

5. Twisted superpotentials and Calabi–Yau criteria

For Koszul AS-regular algebras, the derivation-quotient viewpoint is intrinsic. The detailed formulation in the nontrivial derivation case states that

$1$33

for a single $1$34-form

$1$35

constructed from $1$36, the matrix $1$37, $1$38, and the four families

$1$39

Moreover $1$40 is $1$41-twisted in the sense that

$1$42

and

$1$43

Its leading term is

$1$44

while the lower-order corrections involve the $1$45’s and $1$46’s and encode the commutation rules between the $1$47’s and $1$48 (Cao et al., 21 Jul 2025).

In the trimmed case, the Nakayama formula yields a concrete Calabi–Yau criterion. A connected graded AS-regular algebra is Calabi–Yau exactly when its Nakayama automorphism is inner, and in positive degrees the only inner automorphisms of a connected graded algebra are trivial. Therefore

$1$49

This criterion is given explicitly in Theorem 3.12 of Zhu–Van Oystaeyen–Zhang (Zhu et al., 2014).

A worked example with nontrivial skew derivation appears for

$1$50

with

$1$51

$1$52

In this case

$1$53

$1$54

and Theorem 3.9 yields $1$55, so $1$56 is Calabi–Yau (Cao et al., 21 Jul 2025).

6. Explicit families, normal ordering, and combinatorial structure

The Zhang–Zhang classification contains $1$57 families

$1$58

of trimmed double extension regular algebras of type $1$59, each Artin–Schelter regular of global dimension $1$60 with resolution length pattern $1$61–$1$62–$1$63–$1$64–$1$65. Up to relabeling, the internal data are summarized by pairs

$1$66

through the relations

$1$67

The internal regimes include quantum-plane type when $1$68 or $1$69, and Jordan type for exactly two families: $1$70

$1$71

Thus in families $1$72 and $1$73 one of the two internal systems is a Jordan plane (Rubiano, 7 Jun 2026).

The normal-ordering theory is explicit. For all $1$74,

$1$75

$1$76

where

$1$77

For the mixed relations one has

$1$78

and the corresponding recursive coefficient systems $1$79, $1$80, $1$81, $1$82, and $1$83 control the normal form of a single crossing, mixed blocks, products of PBW monomials, powers of a normal block, and noncommutative multinomials (Rubiano, 7 Jun 2026).

The coefficient behavior separates into distinct combinatorial regimes. When $1$84 or $1$85, the internal $1$86-coefficients reduce to scalar quantum-plane factors. If $1$87 or $1$88, one obtains skew-commutative signs. In the Jordan subcases one has

$1$89

identified in the paper as Lah–Whitney numbers, with triangular recurrence

$1$90

The mixed $1$91- and $1$92-arrays admit a weighted lattice-path interpretation in which local crossings contribute weights from the crossing kernels and internal reorderings contribute quantum or Whitney factors (Rubiano, 7 Jun 2026).

For computational purposes, the symbolic PBW machinery for all $1$93 families is implemented in the ancillary SageMath file double_ore_pbw.sage, which applies the six rewriting rules under the fixed order $1$94 and computes normal forms, products, powers, commutators, and center-tests in specified bidegrees (Rubiano, 7 Jun 2026). The explicit family $1$95 shows that even when the total Betti numbers are forced by AS-regularity, the matrices in the minimal resolution detect the parameter $1$96 through the mixing blocks of $1$97, $1$98, and $1$99, so the defining double-extension data remain visible in the highest syzygies (Rubiano, 2 Apr 2026).