It turns out that a simple extension of the well-known groupoid underlying the rotation algebra, which itself does not yield a non-trivial projection in the algebra since its object space is connected, has a disconnected space of objects. One of its clopen components corresponds to the unit of the rotation algebra, and its complement corresponds to a projection in a larger, Morita equivalent, C*-algebra with the same K$_0$-class as the Rieffel projection.

This groupoid, consisting of a disconnected version of the cut-down of the groupoid of the Kronecker flow to the figure eight consisting of the union of the two generating circles on the torus, can also be used to construct the Rieffel module directly. One notices that it has a (clopen) copy of the real line on which the cut-down subgroupoid corresponding to the rotation algebra acts, and then the natural module action of the algebra of continuous functions of compact support on the groupoid on the linear space of continuous functions of compact support on the line, arising from (finite sum) convolution, constitutes on completion the Rieffel Hilbert module over the rotation C*-algebra. (This construction yields in a natural way that the module is finitely generated and projective.)

I will show that the space $\Omega_n^{\leq 1}$ of formal differential $\leq1$-forms on $\mathbb{R}^n$ has an (induced) SAYD module structure on the Connes-Moscovici Hopf algebra $\mathcal{H}_n$. Then we will see that the Hopf-cyclic cohomology $\mathcal{H}_n$ with coefficients in formal differential forms is identified with the Gelfand-Fuks cohomology of the Lie algebra $W_n$ of formal vector fields on $\mathbb{R}^n$. Furthermore, I will introduce a multiplicative structure on the Hopf-cyclic bicomplex, and we will see that this van Est type isomorphism is multiplicative. I finally show the whole machinery in the case $n=1$; by pulling back the multiplicative generators of $H^\ast(W_1,\Omega_1^{\leq1})$ to $H^\ast(\mathcal{H}_1,\Omega_{1\delta}^{\leq1})$.