Decomposition of a higher degree form

We give another proof of Harrison's decomposition result,([2]) Prop. 2.3 for higher degree forms over a noetherian ring, exploiting an earlier introduction of the centre. We generalise to higher degree forms over a noetherian scheme: we extend the notion of centre; we prove a decomposition result; we extend Harrison's result,([2]) Prop. 4.3 on the behaviour of the centre under a flat base extension; and we improve his result,([2]) Prop. 4.2, giving conditions on the base scheme under which the centre of the tensor product of two higher degree forms is isomorphic to the tensor product of their centres.