next up previous
Next: The Discriminant Valuation Up: Discriminants of Hecke Algebras Previous: Introduction

Discriminants of Hecke Algebras

Let $ R$ be a ring and let $ A$ be an $ R$ algebra that is free as an $ R$ module. The trace of an element of $ A$ is the trace, in the sense of linear algebra, of left multiplication by that element on $ A$.

Definition 2.1 (Discriminant)   Let $ \omega_1,\ldots,\omega_n$ is a $ R$-basis for $ A$. Then the discriminant of $ A$, denoted $ \disc(A)$, is the determinant of the $ n\times n$ matrix $ (\tr(\omega_i\omega_j))$, which is well defined modulo squares of units in $ A$.

When $ R=\mathbb{Z}$ the discriminant is well defined, since the only units are $ \pm 1$.

Proposition 2.2   Suppose $ R$ is a field. Then $ A$ has discriminant 0 if and only if $ A$ is separable over $ R$, i.e., for every extension $ R'$ of $ R$, the ring $ A\otimes R'$ contains no nilpotents.

The following proof is summarized from Section 26 of Matsumura. If $ A$ contains a nilpotent then that nilpotent is in the kernel of the trace pairing. If $ A$ is separable then we may assume that $ R$ is algebraically closed. Then $ A$ is an Artinian reduced ring, hence isomorphic as a ring to a finite product of copies of $ R$, since $ R$ is algebraically closed. Thus the trace form on $ A$ is nondegenerate.


Subsections
next up previous
Next: The Discriminant Valuation Up: Discriminants of Hecke Algebras Previous: Introduction
William A Stein 2002-09-30