Erik Leffler: Bimodule Derivations and the Canonical Sequence

Abstract: The two functions \(f \mapsto f(x) - f(-x)\) and \(f \mapsto f'(0)\) both have kernels which are subalgebras of \(k[x]\). These functions are examples of bimodule derivations: derivations into a bimodule satisfying a version of the Leibniz rule where the left and right actions may differ.

Given a ring extension \(A \subset B\), the intersection of the kernels of all \(A\)-linear bimodule derivations out of \(B\) is a subring \(D_A(B)\) known as the dominion. By iterating this construction with \(A\) fixed one obtains a sequence \(B \supseteq D_A(B) \supseteq D_A(D_A(B)) \supseteq D_A^3(B) \supseteq \ldots\) known as the canonical sequence. There are several natural questions attached to this sequence: When does it stabilize? When does it reach \(A\)? A result proved by Grothendieck in the Noetherian case, and by Sanders in greater generality, states that if \(B\) is a finite \(A\)-module then the canonical sequence reaches \(A\) in finitely many steps. What happens for ring extensions that are merely of finite type?

The main goal of this talk is to introduce the canonical sequence by way of examples and explore techniques for calculating it. I will conclude by presenting some new results concerning stabilization.