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The purpose of this paper is to provide a detailed description of the spacesthat can be specified as $L^2$ domains for the operators of a first orderelliptic complex on a compact manifold with conical singularities. This entailsan analysis of the nature of the minimal domain and of a complementary space inthe maximal domain of each of the operators. The key technical result is thenondegeneracy of a certain pairing of cohomology classes associated with theindicial complex. It is further proved that the set of choices of domainsleading to Hilbert complexes in the sense of Br\"uning and Lesch form avariety, as well as a theorem establishing a necessary and sufficient conditionfor the operator in a given degree to map its maximal domain into the minimaldomain of the next operator.

Elliptic complexes of first-order cone operators: ideal boundary conditions