chapter  5
20 Pages

Kemer’s Capelli Theorem

Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 5.2 Second Proof (Pumping Plus Representation Theory) . . . . . . . . . . 180

5.2.1 Sparse systems and d-identities . . . . . . . . . . . . . . . . . . . . . . . . . 180 5.2.2 Pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 5.2.3 Affine algebras satisfying a sparse system . . . . . . . . . . . . . . . 184 5.2.4 The Representation Theoretic Approach . . . . . . . . . . . . . . . . 184

5.2.4.1 The characteristic 0 case . . . . . . . . . . . . . . . . . . . 185 5.2.4.2 Actions of the group algebra on sparse

systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 5.2.4.3 Simple Specht modules in characteristic

p > 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 5.2.4.4 Capelli identities in characteristic p . . . . . . . 189

5.2.5 Kemer’s Capelli Theorem over Noetherian base rings . . 189 Exercises for Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

So far we know that any affine algebra satisfying a Capelli identity has nilpotent radical. In order to conclude the proof of the Razmyslov-Kemer-Braun Theorem, we need to prove what we have called Kemer’s Capelli Theorem. In fact, in characteristic p Kemer obtained the following stronger result, even for non-affine algebras:

Theorem 5.0.1 ([Kem95]). Any PI-algebra over a field F of characteristic p > 0 satisfies a Capelli identity cn, for large enough n.