Let $\Omega$ be a regular, finite, $n$-dimensional CW complex with chain modules $\mathscr{C}_k$ and boundary operators $\partial_k$. For many problems in computational geometry, a key operation is to take some representation of $\Omega$, compute a representation of the cone $C\Omega$ with respect to some external point $v$, and then delete some cells. For example, this is the key transformation required to compute convex hulls.
I was very pleased with myself when I figured out that the boundary matrices $\{\partial_k'\}$ for the cone space $C\Omega$ have an explicit representation in terms of the boundary matrices of $\Omega$. Let $\mathbf{1}$ denote the column vector of all 1s; we can define the 0th boundary matrix to be the row vector all 1s, i.e. $\partial_0 = \mathbf{1}^*$. We then have that, for $1 \le k < n$, $$\partial_k' = \begin{bmatrix}\partial_k & I \\ 0 & -\partial_{k - 1}\end{bmatrix}\tag{1}\label{eq:dd-k}.$$ For the top dimensional chains, $$\partial_n' = \begin{bmatrix}-\partial_n & \operatorname{diag}(\partial_n\cdot\mathbf{1}) \\ 0 & -\partial_{n - 1}\cdot\operatorname{diag}(\partial_n\cdot\mathbf{1})\end{bmatrix}.$$ Finally we can add $\partial_{n + 1}' = \mathbf{1}$. It's easiest to work this out by drawing it for the 2D case, noting that $\partial_0 = \mathbf{1}^*$, and then guessing the formula for higher dimensions.
Then, of course, I did more digging and realized that the formula isn't new; equation \eqref{eq:dd-k} appears on page 27 of Gelfand and Manin - Methods of Homological Algebra. The second equation is just a column operation on the original formula that I think has nicer properties in some respects. In any case, the Gelfand and Manin book doesn't cite any specific reference for the formula. Where can I find more information or discussion about equation \eqref{eq:dd-k} and where did it first appear? I've implemented it in a software package and it makes parts of some algorithms very easy to write.
