---
title: 'Beilinson Algebras: Definitions and Representations'
url: https://www.emergentmind.com/topics/beilinson-algebra
type: topic
---

# Beilinson Algebras: Definitions and Representations

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 $\mathcal{K}_r$ [1206.1556].

## 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 $E_r=(\mathbb{Z}/p)^r$ [1206.1556].

## 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 $\mathrm{EIP}(n,r) \cup \mathrm{EKP}(n,r) \subseteq \mathrm{CJT}(n,r)$ [1401.5947, 1206.1556].

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 [1206.1556, 1401.5947].

## 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 [1206.1556].

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 [1401.5947].

## 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 [1401.5947] 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 $X_{\alpha}^j$ [1401.5947].

## 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 [2304.02242].

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 [2304.02242].

## 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 [1905.11296].

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 [1905.11296].

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**References**:  
[1401.5947], [1206.1556], [2304.02242], [1905.11296]

Source: https://www.emergentmind.com/topics/beilinson-algebra