As in the case of differential forms on a manifold, we can define differential forms on a differential space S either as alternating multilinear maps from the space of derivations of C℞(S), called Koszul forms, or pointwise, as maps associating to each point of x ∈ S an alternating multilinear form on TxS, which we call Zariski forms. These deinitions are inequivalent on the singular part of S. Moreover, Koszul forms admit an exterior differential but not a pullback, whereas Zariski forms admit a pull-back but not an exterior differential. There is a third deinition, given by Marshall, which leads to forms that admit both pull-backs and exterior differentials. All three deinitions agree on the level of 1-forms. It is of interest to see how these forms appear in applications.

Koszul forms

Recall that, for a differential space S, the space of all derivations Der C℞(S) is a module over C℞(S).

Definition 5.1.1 For each, a Koszul k-form is an alternating map from (Der to that is k-linear over.

We denote the space of Koszul k-forms on S by For the value of on Der is denoted by By definition, for every permutation.