On Noetherian algebras, Schur functors and Hemmer-Nakano dimensions

TL;DR

This paper explores the relationships between Noetherian algebras and their module categories via Schur functors, introducing the Hemmer-Nakano dimension to measure the faithfulness of these connections, with applications to Schur algebras and category O.

Contribution

It introduces the Hemmer-Nakano dimension for standard modules and provides methods to compute it, extending results to integral setups and deformations of category O.

Findings

Hemmer-Nakano dimension is bounded by the number of simple modules.

Reduction techniques connect integral and finite-dimensional cases.

Deformations of category O have improved homological properties.

Abstract

Important connections in representation theory arise from resolving a finite-dimensional algebra by an endomorphism algebra of a generator-cogenerator with finite global dimension; for instance, Auslander's correspondence, classical Schur--Weyl duality and Soergel's Struktursatz. Here, the module category of the resolution and the module category of the algebra being resolved are linked via an exact functor known as Schur functor. In this paper, we investigate how to measure the quality of the connection between module categories of (projective) Noetherian algebras, $B$, and module categories of endomorphism algebras of generators-relative cogenerators over $B$ which are split quasi-hereditary Noetherian algebras. In particular, we are interested in finding, if it exists, the highest degree $n$ so that the endomorphism algebra of a generator-cogenerator provides an $n$-faithful cover, in the sense of Rouquier, of $B$. The degree $n$ is known as Hemmer-Nakano dimension of the standard modules. We prove that the Hemmer-Nakano dimension of standard modules with respect to a Schur functor from a split highest weight category over a field to the module category of a finite-dimensional algebra $B$ is bounded above by the number of non-isomorphic simple modules of $B$. We establish methods how to reduce computations of Hemmer-Nakano dimensions in the integral setup to computations of Hemmer-Nakano dimensions over finite-dimensional algebras. This theory allows us to derive results for Schur algebras and the BGG category O in the integral setup from the finite-dimensional case using relative dominant dimension. We exhibit several structural properties of deformations of the blocks of the BGG category O establishing an integral version of Soergel's Struktursatz. We show that deformations of the combinatorial Soergel's functor have better homological properties than the classical one.