Beilinson Algebras: Definitions and Representations

• Beilinson algebras are finite-dimensional k-algebras defined via quiver presentations with commutativity relations that yield derived equivalence with coherent sheaves on projective spaces.
• They provide a framework for analyzing graded modules over truncated polynomial rings and categorizing modules into equal images, equal kernels, and constant Jordan type classes.
• Their structure underpins advances in noncommutative projective geometry, Auslander–Reiten theory, and derived equivalences in modular representation theory.

A Beilinson algebra is a finite-dimensional $k$-algebra, central to the study of coherent sheaves on projective spaces, noncommutative projective geometry, and the modular representation theory of elementary abelian $p$-groups. The classical construction, developed by Beilinson, provides an explicit endomorphism algebra whose module category is derived equivalent to that of coherent sheaves on $\mathbb{P}^n$. Generalizations, such as the generalized Beilinson algebras $B(n,r)$, arise as path algebras of quivers with relations reflecting the structure of truncated polynomial or quantum polynomial rings. These algebras exhibit rich homological and representation-theoretic properties, organizing key categories of modules—such as those of constant Jordan type, equal images, and equal kernels—through precise homological criteria and Auslander–Reiten theory.

1. Definition and Quiver Presentation

Generalized Beilinson algebras $B(n,r)$ are defined for integers $n\ge 2$, $r\ge 2$ over an algebraically closed field $k$. Let $Q(n,r)$ denote the quiver with vertices $0,1,\ldots,n-1$ and $r$ parallel arrows $\gamma_1^{(i)}, \ldots, \gamma_r^{(i)}$ from vertex $i$ to $i+1$ for $0\leq i \leq n-2$: $$Q(n,r):\quad 0 \overset{r}{\longrightarrow} 1 \overset{r}{\longrightarrow} 2 \longrightarrow \cdots \overset{r}{\longrightarrow} n-1.$$ The path algebra $E(n,r) = k Q(n,r)$ is then factored by the ideal $\mathcal{I}$ generated by the commutativity relations

$$\gamma_s^{(i+1)} \gamma_t^{(i)} - \gamma_t^{(i+1)} \gamma_s^{(i)}\,, \quad 0 \leq i \leq n-2,\ 1 \leq s, t \leq r,$$

yielding

$B(n,r) := E(n,r)/\mathcal{I}.$

For $r=1$, $B(n,1)$ recovers the classical Beilinson algebra associated to $\mathbb{P}^{n-1}$, while for $n=2$ the algebra is the $r$-Kronecker algebra $\mathcal{K}_r$ (Worch, 2012).

2. Module Categories and Graded Interpretations

The category $\mathrm{mod}\,B(n,r)$ admits a natural interpretation as the category of graded modules over the truncated polynomial algebra $k[X_1,\dots,X_r]$ supported in degrees $0$ through $n-1$: $$\mathcal{C}_{[0, n-1]} \subset \mathrm{mod}_{\mathbb{Z}}(k[X_1,\dots,X_r]/(X_1^p, ..., X_r^p)).$$ Under this correspondence, each module $M=\bigoplus_{i=0}^{n-1} M_i$ has $e_i$ (primitive idempotent) acting as a projection onto $M_i$, and each arrow $\gamma_j^{(i)}$ as an operator mapping $M_i \rightarrow M_{i+1}$. This identification allows one to study categories of finite length $n$ graded modules for truncated polynomial rings, and provides a bridge to the category of modules for the group algebra of the elementary abelian group $E_r=(\mathbb{Z}/p)^r$ (Worch, 2012).

3. Homological Characterization of Special Module Categories

For $$\alpha=(\alpha_1, \dots, \alpha_r) \in k^r \setminus \{0\}$$, the operator

$$\tilde{\alpha} = \sum_{i=0}^{n-2} \sum_{j=1}^r \alpha_j \gamma_j^{(i)}$$

acts on any $B(n,r)$-module $M$. The following subcategories are distinguished:

• Equal Images (EIP$(n,r)$): $M$ such that for all nonzero $\alpha$, $$\operatorname{im}(\tilde{\alpha}) = \bigoplus_{i=1}^{n-1} M_i$$.
• Equal Kernels (EKP$(n,r)$): $M$ such that for all nonzero $\alpha$, $\ker(\tilde{\alpha}) = M_{n-1}$.
• Constant Jordan Type (CJT$(n,r)$): $M$ such that, for all $j\geq 1$, the rank of $(\tilde{\alpha})^j$ acting from $M_i \to M_{i+j}$ is independent of $\alpha$.
• These categories satisfy $$\mathrm{EIP}(n,r) \cup \mathrm{EKP}(n,r) \subseteq \mathrm{CJT}(n,r)$$ (Worch, 2014, Worch, 2012).

A projective-dimension-1 family $X_\alpha^j$ of $B(n,r)$-modules parameterized by $\mathbb{P}^{r-1}$ provides a homological characterization: $$\begin{align*} \mathrm{EIP}(n,r) &= \{ M ~|~ \operatorname{Ext}^1(X_\alpha^1, M) = 0\;\forall \alpha \}, \ \mathrm{EKP}(n,r) &= \{ M ~|~ \operatorname{Hom}(X_\alpha^1, M) = 0\;\forall \alpha \}, \ \mathrm{CR}^j(n,r) &= \{ M ~|~ \dim \operatorname{Ext}^1(X_\alpha^j, M) \text{ constant in } \alpha\}. \end{align*}$$ This underpins torsion theories for $\mathrm{mod}\,B(n,r)$: EIP is a torsion class closed under extensions and images, containing preinjectives; EKP is a torsion-free class closed under submodules, containing preprojectives (Worch, 2012, Worch, 2014).

4. Generalized $W$-Modules and Iterated One-Point Extensions

Let $R = k[X_1, \dots, X_r]$ and $I = (X_1, \dots, X_r)$. For integers $m \geq n$, define

$$M_{m,n}^{(r)} := (I^{m-n} / I^n)[n-m], \qquad W_{m,n}^{(r)} := D(M_{m,n}^{(r)}),$$

where $D = \operatorname{Hom}_k(-, k)$ denotes duality. These are the generalized $M$-modules (in EKP) and $W$-modules (in EIP): $$M_{m,n}^{(r)} \in \mathrm{EKP}(n,r), \qquad W_{m,n}^{(r)} \in \mathrm{EIP}(n,r).$$ These modules are indecomposable, $\operatorname{GL}_r(k)$-stable, and have local, commutative endomorphism rings. For $r=2$ they exhaust all indecomposable objects in EIP$(2,2)$, for $r>2$, there are infinitely many bricks per torsion class (Worch, 2012).

More structurally, $B(n,r)$ admits an inductive construction as iterated one-point extensions: $$B(2,r) = \mathcal{K}_r,\quad B(n,r)\cong B(n-1,r)[M_{n, n-1}^{(r)}],$$ yielding

$$B(n,r) \cong \mathcal{K}_r [M_{3,2}^{(r)}]\,[M_{4,3}^{(r)}]\,\cdots\,[M_{n, n-1}^{(r)}].$$

This construction controls the lifting of almost split sequences (Auslander–Reiten theory) between module categories (Worch, 2014).

5. Auslander–Reiten Theory and $\mathbb{Z}A_{\infty}$-Components

For $n\ge 3$, the Auslander–Reiten quiver $\Gamma(n,r)$ of $B(n,r)$ contains regular components of tree class $A_\infty$ (so-called $\mathbb{Z}A_{\infty}$-components). The central theorem (Worch, 2014) establishes:

• Each generalized $W$-module $W_{m,n}^{(r)}$ ($m>n$) is quasi-simple in a unique regular $\mathbb{Z}A_{\infty}$-component $\mathcal{C}_m\subset\Gamma(n,r)$.
• $\mathcal{C}_m$ is the disjoint union

$$\mathcal{C}_m = \big( \mathrm{EIP}(n,r) \cap \mathcal{C}_m \big) \; \dot{\cup} \; \big( \mathrm{EKP}(n,r) \cap \mathcal{C}_m \big),$$

where every module in $\mathcal{C}_m$ has constant Jordan type, i.e., lies in $\mathrm{CJT}(n,r)$.

• For $r>2$, the EIP and EKP cones in $\mathcal{C}_m$ meet along a single mesh, while for $r=2$, there is one mesh where modules are neither EIP nor EKP.
• The almost-split (Auslander–Reiten) sequences in $B(n-1,r)$ lift to those in $B(n,r)$, allowing inductive control over the components.
• Computation of the Auslander–Reiten translates $\tau(M_{m,n}^{(r)})$, $\tau^{-1}(W_{m,n}^{(r)})$ describes the mesh structure, confirming the CJT property throughout $\mathcal{C}_m$.

The structure of $\Gamma(n,r)$ links the homological and combinatorial representation theory of $B(n,r)$, with organizing subcategories and homological invariants determined via the projective-dimension-1 modules $X_{\alpha}^j$ (Worch, 2014).

6. Beilinson Algebras in Noncommutative Projective Geometry

The Beilinson algebra framework extends to noncommutative projective geometry, especially for quantum polynomial algebras. For $A$ a 3-dimensional quantum polynomial algebra, its Beilinson algebra $\nabla A$ is

$$\nabla A = \begin{pmatrix} A_0 & A_1 & A_2 \ 0 & A_0 & A_1 \ 0 & 0 & A_0 \end{pmatrix}$$

with multiplication induced by $A$'s ring structure (Itaba, 2023).

A crucial result for Type S' quantum polynomial algebras relates the following:

• (i) The noncommutative projective plane $\mathrm{Proj}_{nc}\,A$ is finite over its center.
• (ii) $\nabla A$ is $2$-representation tame (its preprojective algebra $\Pi(\nabla A)$ is right Noetherian and finite over its center).
• (iii) The isomorphism classes of simple $2$-regular $\nabla A$-modules are parametrized by the points of $\mathbb{P}^2$.

This equivalence links geometric properties of noncommutative projective schemes to representation-theoretic properties of Beilinson algebras. The quiver of $\nabla A$ is directly determined by the generators and relations of $A$, and the preprojective center structure enables explicit parametrization of simple modules (Itaba, 2023).

7. Extensions, Derived Equivalences, and Infinite Generalizations

Generalizations include the construction of locally $\Phi$-Beilinson-Green algebras $G^{\Phi}_F(X)$ for various admissible index sets $\Phi \subset \mathbb{Z}$, in the setting of triangulated or $n$-exangulated categories. In this context: $$G^{\Phi}_F(X) = \big( \mathrm{Hom}_{\mathcal{T}}(X, F^{j-i} X)\big)_{i,j \in \Phi}$$ is an upper-triangular (possibly infinite) matrix algebra with a locally finite system of orthogonal idempotents. When $\Phi = \{0,1,\ldots, n\}$, the classical Beilinson algebra is recovered (Pan, 2019).

Symmetric approximation sequences and higher exact sequences supply a machinery for producing derived equivalences between (quotients of) such algebras. Notably, this generalizes and unifies results for tilting complexes, D-split sequences, and graded Morita theory, allowing derived equivalences across Beilinson-Green algebras constructed from semi-Gorenstein modules and group graded algebras (Pan, 2019).

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References:

(Worch, 2014, Worch, 2012, Itaba, 2023, Pan, 2019)