Inertial algebras, inertial bimodules, and projective covers of algebras

Abstract

For a henselian local ring R, Azurnaya [3] proved that every module-finite R-algebra R with R/Rad A separable admits an inertial algebra, that is. a separable subalgebra B with B + Rad A = A, and he showed that B is unique up to conjugation. This fundamental theorem has been a starting point for intensive research on inertial algebras, with major contributions by Ingraham [10, 16. 17, 18], Brown [7, 8, 9, 10], Kirkman [19], Wehlen [25, 26, 27], Cipolla [11, 12], Deneen [14]. and others.

In the present paper, we shall introduce an additive counterpart of an inertial algebra B of A, namely, a subbimodule M of _BA_B, unique up to isomorphism, with analogous properties like inertial algebras (see the introduction below). We then show that both invariants B and M together are equivalently given by the projective cover of A, which exists for all algebras of a suitably defined category.