Categories of split filtrations and graded quiver varieties

By the work of Hernandez-Leclerc, Leclerc-Plamondon, and Keller-Scherotzke, affine graded Nakajima quiver varieties associated with a Dynkin quiver $Q$ admit an algebraic description in terms of modules over the singular Nakajima category $\mathcal{S}$ and a stratification functor to the derived category of $Q$. In this paper, we extend this framework to Nakajima’s $n$-fold affine graded tensor product varieties, which allow one to geometrically realize $n$-fold tensor products of standard modules over the quantum affine algebra. We introduce a category of filtrations with splitting of length $n$ of modules over a category and show that it is equivalent to the module category of a triangular matrix category. Applied to the singular Nakajima category, this yields a category $\mathcal{S}^{n\operatorname{-filt}}$ whose modules are parametrized by the points of the $n$-fold tensor product varieties. Generalizing the results of Keller-Scherotzke from $\mathcal{S}$ to $\mathcal{S}^{n\operatorname{-filt}}$, we prove that the stable category of pseudo-coherent Gorenstein projective $\mathcal{S}^{n\operatorname{-filt}}$-modules is triangle equivalent to the derived category of the algebra of $n \times n$ upper triangular matrices over the path algebra of $Q$, and we obtain a corresponding stratification functor.

💡 Research Summary

This paper extends the algebraic framework that links affine graded Nakajima quiver varieties to modules over the singular Nakajima category 𝒮, originally developed by Hernandez–Leclerc, Leclerc–Plamondon, and Keller–Scherotzke, to the setting of Nakajima’s n‑fold affine graded tensor product varieties. The main innovation is the introduction of a “category of split filtrations of length n” for modules over a small k‑category A with respect to a subcategory B. An object consists of an A‑module M equipped with a chain of submodules
(0=M_0\subset M_1\subset\cdots\subset M_n=M)
together with B‑linear retractions (r_i:M_i\to M_{i-1}) that split each inclusion when restricted to B. The authors prove (Proposition 3.3) that this category, denoted Filt(_n^B)(A), is equivalent to the module category of a triangular matrix category (T_n^B(A)). The latter is built from A, B, and the bimodule of morphisms between them, generalising the classical triangular matrix algebra to an arbitrary number of layers. This equivalence is a categorical analogue of the well‑known description of modules over a triangular matrix ring as a comma category.

Specialising to A = 𝒮, the singular Nakajima category associated with a Dynkin quiver Q, and B = k𝒮₀ (the smallest subcategory containing all objects), the authors define the “singular Nakajima category of split‑filtrations of length n” as
(\mathcal{S}^{n\text{-filt}}:=T_n^{k\mathcal{S}_0}(\mathcal{S})).
They show that modules over (\mathcal{S}^{n\text{-filt}}) are precisely parametrised by points of Nakajima’s n‑fold affine graded tensor product varieties. In other words, the geometric objects that realise the tensor product of n standard modules over the quantum affine algebra admit an algebraic description as representation varieties of (\mathcal{S}^{n\text{-filt}}). The splitting over k𝒮₀ encodes the group action appearing in Nakajima’s construction and is essential for the equivalence.

The paper then turns to homological aspects. The authors prove that (\mathcal{S}^{n\text{-filt}}) is weakly Gorenstein and that its stable category of pseudo‑coherent Gorenstein projective modules, denoted gpr((\mathcal{S}^{n\text{-filt}})), is triangle‑equivalent to the bounded derived category of the algebra
(k!!\mathsf{A}_n\otimes_k kQ),
where (k!!\mathsf{A}_n) is the path algebra of a linearly oriented Dynkin quiver of type Aₙ. This result (Theorem 4.22) generalises Keller–Scherotzke’s theorem for n = 1. The proof exploits the triangular matrix description of (\mathcal{S}^{n\text{-filt}}) and interprets its singularity category as a gluing of two simpler categories, following the approach of Kalck–Lenzing. Consequently, the derived equivalence is obtained by analysing projective and injective resolutions inside the triangular matrix framework.

With this homological machinery in place, the authors define a stratification functor
(\Phi_n:\operatorname{mod}\mathcal{S}^{n\text{-filt}}\to D^b(\operatorname{mod}k!!\mathsf{A}_n\otimes_k kQ))
as the composition
(\operatorname{mod}\mathcal{S}^{n\text{-filt}}\xrightarrow{\Omega}\operatorname{gpr}(\mathcal{S}^{n\text{-filt}})\xrightarrow{\sim} D^b(\operatorname{mod}k!!\mathsf{A}_n\otimes_k kQ)),
where Ω is the syzygy functor sending a module to the kernel of a projective cover. For n = 1 this recovers Keller–Scherotzke’s functor Φ. Theorem 4.32 shows that if two points M, N in an n‑fold tensor product variety satisfy Φₙ(M) ≅ Φₙ(N), then for every pair 0 ≤ i < j ≤ n the subquotients M_j/M_i and N_j/N_i lie in the same stratum of an ordinary affine graded quiver variety. The converse fails in general (Example 4.33), and the authors discuss conditions under which it might hold, suggesting a rich geometric partition induced by Φₙ.

The paper is organised as follows. Section 2 reviews modules over categories, bimodules, tensor products, and triangular matrix categories. Section 3 introduces split filtrations, proves their equivalence with triangular matrix module categories, and analyses the associated functors. Section 4 applies the theory to the singular Nakajima category, describes (\mathcal{S}^{n\text{-filt}}) via mesh categories, proves Gorenstein properties, establishes the derived equivalence, and constructs the stratification functor Φₙ. Throughout, the authors provide explicit descriptions of projective and injective resolutions, demonstrate weak Gorensteinness, and verify that many arguments from Keller–Scherotzke’s work extend verbatim to the higher‑length setting.

In summary, the authors have introduced a new categorical construction—split filtrations of length n—that unifies the representation‑theoretic description of Nakajima’s n‑fold affine graded tensor product varieties with the homological algebra of triangular matrix categories. By proving a derived equivalence with (k!!\mathsf{A}_n\otimes_k kQ) and defining a natural stratification functor, they substantially generalise the existing framework for n = 1 and open new avenues for studying higher‑tensor‑product geometry, Gorenstein homological properties, and categorical actions on quiver varieties.