Singular Nakajima Category

• Singular Nakajima category is an algebraic framework defined via framed repetition quivers with mesh relations, connecting affine graded quiver varieties to representation theory.
• It realizes a triangulated equivalence between its Gorenstein–projective stable category and the bounded derived category of finite-dimensional modules over Dynkin quivers.
• Its construction enables explicit stratification of affine quiver varieties and extends to n-fold tensor product varieties, bridging geometric and categorical insights.

The singular Nakajima category, denoted $\mathcal{S}$, provides an algebraic framework for the study of affine graded Nakajima quiver varieties associated with Dynkin quivers. Constructed as a full subcategory of a quiver with mesh relations, $\mathcal{S}$ connects the geometry of quiver varieties to the representation theory of finite-dimensional algebras. Its modules correspond to points of quiver varieties, and its Gorenstein–projective stable category is triangulated equivalent to the bounded derived category of the original quiver, establishing $\mathcal{S}$ as a crucial object linking geometric and categorical approaches within representation theory (Canesin, 14 Jan 2026).

1. Definition and Construction

Given a fixed acyclic quiver $Q = (Q_0, Q_1)$, the singular Nakajima category $\mathcal{S}$ is defined via the framed repetition quiver $ZQ^\infty$. Its construction is as follows:

• Vertices: The vertex set is $$Q_0^{\text{sing}} = \{ (i,p) \mid i \in Q_0,\, p\in\mathbb{Z} \}$$.
• Arrows: For each arrow $\alpha: i \to j$ in $Q_1$ and every $p \in \mathbb{Z}$, a horizontal arrow $\alpha_p: (i,p) \to (j,p)$; for each $i \in Q_0$ and $p \in \mathbb{Z}$, a vertical arrow $\tau_{i,p}:(i,p-1)\to(i,p)$.
• Relations (mesh relations): For each non-frozen vertex $(i, p)$,

$$\sum_{\alpha\in Q_1,\, t(\alpha)=i} \tau_{i,p} \circ \alpha_{p-1} = \sum_{\alpha\in Q_1,\, s(\alpha)=i} \alpha_p \circ \tau_{s(\alpha),p}$$

in the path algebra.

Let $R$ be the $k$-linear category obtained by quotienting the path category $kZQ^\infty$ by the ideal generated by the mesh relations. The singular Nakajima category $\mathcal{S}$ is then the full subcategory of $R$ with objects all $(i,p)$, $i \in Q_0$, $p \in \mathbb{Z}$—these are referred to as "frozen" vertices.

This construction is equivalent to its presentations in works by Hernandez–Leclerc, Leclerc–Plamondon, and Keller–Scherotzke, wherein horizontal arrows $\alpha_p$ encode time-like shifts of $Q$'s arrows, and vertical arrows $\tau_{i,p}$ provide an additional infinite framing in both directions (Canesin, 14 Jan 2026).

2. Representation-Theoretic Realization of Quiver Varieties

Assigning a dimension vector $w:Q_0\times\mathbb{Z}\to\mathbb{N}$ with finite support, a $k$-point of the affine graded quiver variety $\mathcal{M}_0(w)$ corresponds precisely to a representation of $\mathcal{S}$ with graded dimension $w$, i.e., a $k$-linear functor

$$M:\,\mathcal{S}^{\text{op}}\to \operatorname{Vect}_k\qquad \text{with} \qquad \dim M(i,p) = w(i,p).$$

This identification provides an isomorphism of algebraic varieties:

$$\mathcal{M}_0(w) \cong \operatorname{Rep}_{\mathcal{S}}(w) := \{ M \in \mathcal{S}\text{-Mod} \mid \dim M(i,p)=w(i,p) \}.$$

The coordinate ring of $\mathcal{M}_0(w)$ is generated by the matrix-coefficient functions $M\mapsto M(a)$ for all arrows $a$ in $Q_1(\mathcal{S})$.

Nakajima’s framed quiver variety $\mathcal{M}(v, w)$ can be obtained as a GIT quotient of the space of $R$-modules of dimension $(v, w)$ by the gauge group $G_v$, with the map $\mathcal{M}(v,w) \to \mathcal{M}_0(w)$ realized by restriction along the inclusion $\mathcal{S} \to R$ (Canesin, 14 Jan 2026).

3. Gorenstein–Projective Modules and Derived Equivalence

A finite-dimensional $\mathcal{S}$-module $M$ is Gorenstein–projective if it possesses a complete projective resolution

$\cdots \to P_1 \to P_0 \to P_{-1} \to \cdots$

with $\operatorname{Hom}(P, -)$ exact on projectives; equivalently, $\operatorname{Ext}_\mathcal{S}^p(M, P) = 0$ for all $p > 0$ and all projective $P$. The category $\operatorname{GPrj}(\mathcal{S})$ of such modules forms a Frobenius exact category, with projective–injective objects the ordinary projectives.

The stable category $$\mathcal{S}\backslash \text{Proj} = \operatorname{GPrj}(\mathcal{S})/\text{Proj}$$ admits a triangulated structure with suspension given by the syzygy functor $\Omega(M) = \ker(P_0\to M)$. For a Dynkin quiver $Q$, there is a canonical triangle equivalence

$\mathcal{S}\backslash\text{Proj} \simeq D^b(kQ),$

where $D^b(kQ)$ denotes the bounded derived category of finite-dimensional $kQ$-modules. The equivalence is constructed by associating indecomposable $\mathcal{S}$-modules to objects in $D^b(kQ)$ using projective resolutions and by analyzing mesh relations, following methods of Happel, Keller–Scherotzke, and others (Canesin, 14 Jan 2026).

4. Stratification Functor and Affine Strata

Every finite-dimensional $\mathcal{S}$-module $M$ admits a minimal projective cover $P \to M$ in $\operatorname{GPrj}(\mathcal{S})$. The stratification functor

$$\Phi(M) := \Omega(M) \in \mathcal{S}\backslash\text{Proj} \simeq D^b(kQ)$$

sends $M$ to its first syzygy and then applies the derived equivalence. For Dynkin $Q$ and $w$ as above, two $\mathcal{S}$-modules $M, N$ of dimension $w$ satisfy

$\Phi(M) \cong \Phi(N)\quad \text{in}~ D^b(kQ)$

if and only if $M, N$ lie in the same Nakajima stratum of $\mathcal{M}_0(w)$. Thus, the fibers of $\Phi$ correspond to the affine stratification of the quiver variety, and the triangulated structure on $D^b(kQ)$ encodes incidence relations among strata (Canesin, 14 Jan 2026).

5. The Type $\mathbf{A_2}$ Example

For $Q$ the type $A_2$ quiver $1 \to 2$, the vertices of $\mathcal{S}$ are $(1,k)$ and $(2,k)$ for $k \in \mathbb{Z}$. Arrows are:

• Horizontal: $\alpha_k: (1,k) \to (2,k)$
• Vertical: $\tau_{1,k}: (1,k-1) \to (1,k)$ and $\tau_{2,k}: (2,k-1) \to (2,k)$

The mesh relations are $$\tau_{2,k} \circ \alpha_{k-1} - \alpha_k \circ \tau_{1,k} = 0$$ at each $(i,k)$. The simple modules $S_{1,k}, S_{2,k}$ are supported at frozen vertices. Minimal projective resolutions are:

$$0 \to P(1,k-1) \to P(2,k-1) \oplus P(1,k) \to P(2,k) \to S_{2,k} \to 0$$

$0 \to P(1,k-1) \to P(1,k) \to S_{1,k} \to 0$

The syzygy $\Omega S_{2,k}$ realizes the image of the stalk complex $k_{1\to2}[0]$ under the derived equivalence. In this case, one recovers the two simple objects of $D^b(kA_2)$, and $\Phi$ identifies the one-parameter families of $\mathcal{S}$-modules over the two strata of the $A_2$ graded variety (Canesin, 14 Jan 2026).

6. Relations to Graded Quiver Varieties and Extensions

The category $\mathcal{S}$ encapsulates the algebraic structure underlying affine graded Nakajima varieties for Dynkin quivers, as articulated in works by Hernandez–Leclerc, Leclerc–Plamondon, and Keller–Scherotzke. For higher tensor product constructions, the framework extends to categories of filtrations with splitting, leading to a category $\mathcal{S}^{n\text{-filt}}$, whose module category is equivalent to that of a triangular matrix category and parametrizes points of $n$-fold tensor product varieties. The stable category of finitely generated Gorenstein projective $\mathcal{S}^{n\text{-filt}}$-modules is triangulated equivalent to the derived category of the algebra of $n \times n$ upper triangular matrices over $kQ$, augmented by a corresponding stratification functor (Canesin, 14 Jan 2026).

7. Significance and Further Directions

The singular Nakajima category forms a conceptual and technical bridge between the geometry of graded quiver varieties and the homological algebra of derived categories. Its realization allows for explicit categorical and functorial realizations of geometric stratifications, projective resolutions, and derived equivalences central to modern representation theory. Generalizations to $n$-fold tensor product varieties further enhance the reach of this framework, highlighting its foundational status in the study of quiver varieties and representation categories associated with quantum affine algebras (Canesin, 14 Jan 2026).