Equivariant periodic cyclic homology for ample groupoids

We develop an equivariant version of bivariant periodic cyclic homology for actions of Hausdorff ample groupoids, extending the classical bivariant theory of Cuntz and Quillen and its equivariant refinement for groups. For an ample groupoid G, we construct a monoidal category of modules over its convolution algebra and study structural features of its objects, the G-modules. In parallel, we present an equivalent comodule formulation and prove the equivalence between the module and comodule pictures. We introduce G-algebras and give some important examples. After reviewing pro-categories, we define the equivariant X-complex, which is central to the construction of the bivariant equivariant periodic cyclic homology for G-algebras. In analogy with the classical and group-equivariant settings, we establish homotopy invariance, stability, and excision for the resulting theory.