Algebraic Geometry Seminar
Local properties of an invariant ring are determined by inertia groups (plus an application)
Speaker: Ben Blum-Smith, Eugene Lang College / The New School
Location: Warren Weaver Hall 317
Date: Tuesday, November 27, 2018, 3:30 p.m.
An important general principle in geometry is that the local structure of a quotient of a space by the action of a discrete group is determined by the point stabilizers. An analogous principle holds in commutative algebra in great generality: when a finite group acts on an arbitrary commutative, unital ring, the local structure of the invariant ring is determined by the inertia groups. The correct statement of this principle involves the concept of the strict henselization.
We give a novel application of this principle to a problem of longstanding interest in the invariant theory of finite groups: when is an invariant ring free as a module over a polynomial subring? (In other words, when is it Cohen-Macaulay?) Specifically, we show that for a permutation group acting on a polynomial ring over Z, the invariant ring is free over a polynomial subring if and only if the group is generated by its transpositions, double-transpositions, and 3-cycles. The “only if” direction rests on finding a specific inertia subgroup of the action that inhibits Cohen-Macaulayness. This is joint work with Sophie Marques.