The tetrahedron \(T\) is a surface with 4 vertices, 6 edges, and 4 faces of dimension 2, and its homology is the homology of the complex \begin{align*} C_{-}\coloneqq(\cdots \to 0 \to R^4 \to R^6 \to R^4 \to 0 \to \cdots) .\end{align*} Write down the matrices in this complex and computationally verify that \begin{align*} H_*(T) = [R, 0, R, 0, \cdots] .\end{align*}