Categories of split filtrations and graded quiver varieties

TL;DR

This paper extends the algebraic and geometric framework of Nakajima quiver varieties to n-fold tensor products, introducing a filtration category that links modules over a matrix algebra to derived categories.

Contribution

It introduces a new filtration category for modules over Nakajima's singular category, generalizing existing frameworks to n-fold tensor product varieties.

Findings

Modules over the new filtration category are parametrized by n-fold tensor product varieties.

The stable category of Gorenstein projective modules is equivalent to the derived category of an upper triangular matrix algebra.

A stratification functor is constructed for the extended framework.

Abstract

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.