Homological dimensions for co-rank one idempotent subalgebras

TL;DR

This paper investigates the relationship between the homological dimensions of a Noetherian algebra and its co-rank one idempotent subalgebra, revealing conditions under which their global dimensions are finite and related.

Contribution

It establishes new criteria connecting the homological properties of an algebra and its idempotent subalgebra via the simple module's extension properties.

Findings

Global dimension finiteness of A and Γ are equivalent if S_e has no self-extensions.

Finite global dimensions of A and Γ imply S_e has no higher self-extensions.

Conditions depend on the algebra being positively graded or semiperfect.

Abstract

Let $k$ be an algebraically closed field and $A$ be a (left and right) Noetherian associative $k$-algebra. Assume further that $A$ is either positively graded or semiperfect (this includes the class of finite dimensional $k$-algebras, and $k$-algebras that are finitely generated modules over a Noetherian central Henselian ring). Let $e$ be a primitive idempotent of $A$, which we assume is of degree $0$ if $A$ is positively graded. We consider the idempotent subalgebra $\Gamma = (1-e)A(1-e)$ and $S_e$ the simple right $A$-module $S_e = eA/e{\rm rad}A$, where ${\rm rad}A$ is the Jacobson radical of $A$, or the graded Jacobson radical of $A$ if $A$ is positively graded. In this paper, we relate the homological dimensions of $A$ and $\Gamma$, using the homological properties of $S_e$. First, if $S_e$ has no self-extensions of any degree, then the global dimension of $A$ is finite if and only if that of $\Gamma$ is. On the other hand, if the global dimensions of both $A$ and $\Gamma$ are finite, then $S_e$ cannot have self-extensions of degree greater than one, provided $A/{\rm rad}A$ is finite dimensional.