What is the "combinatorial coboundary" in Cech cohomology? In David R. Morrison's The Clemens-Schmid exact sequence and applications, we have a singular variety $\mathfrak{X}_0$ which is the union of irreducible components $\{X_i\}_i$ with each $X_i$ smooth.
For some collection of indices $i_0,\ldots,i_p$, write $X_{i_0\ldots i_p} := X_{i_0}\cap \cdots\cap X_{i_p}$, and $\mathfrak{X}^{[p]} := \bigsqcup_{i_0 < i_1 < \cdots < i_p} X_{i_0\ldots i_p}$. Let $\iota_p : \mathfrak{X}^{[p]}\rightarrow\mathfrak{X}$ be the natural map. For a suitable open cover $\mathfrak{U}$ of $\mathfrak{X}_0$, consider the groups of Cech cochains
$$E_0^{p,q} := \check{C}^q(\iota_p^{-1}(\mathfrak{U}),\mathbb{Q})$$
Now he denotes by $d : E_0^{p,q}\rightarrow E_0^{p,q+1}$ the Cech coboundary. He denotes by $\delta : E_0^{p,q}\rightarrow E_0^{p-1,q}$ the "combinatorial coboundary induced by"
$$\delta\phi(V\cap X_{j_0\ldots j_{q+1}}) = \sum_a (-1)^q\phi(V\cap X_{j_0\ldots\hat{j_a}\ldots j_{q+1}})$$
Is this a typo? I'm failing to see how the right hand side defines an element of $E_0^{p-1,q}$. What is $V$? What am I missing? (see here for a partial scan of the article containing the exact text I'm asking about).
 A: I will discuss a more general situation than the particular situation you are asking about, so the notation will not match up exactly.
First things first.
Whenever we have a space $X$ that is a union of subspaces $Y_i$, we can form a simplicial object $X_\bullet$ that is in degree $p$ given by $X_p = \coprod (Y_{i_0} \cap \cdots \cap Y_{i_p})$.
The face operators are induced by the inclusions $Y_{i_0} \cap \cdots \cap Y_{i_p} \hookrightarrow Y_{i_0} \cap \cdots \cap \widehat{Y_{i_\alpha}} \cap \cdots \cap Y_{i_p}$ and the degeneracy operators similarly.
(I hope you understand what I mean.
There is simply no good way of notating this without introducing a lot of auxiliary objects.)
In fact, $X_\bullet$ has an augmentation $X_\bullet \to X$ and we can pull back any open cover $\mathfrak{U}$ of $X$ to obtain an open cover $\mathfrak{U}_p$ on each $X_p$ that is compatible with the simplicial operators.
Thus, applying the Čech cochain complex functor, we obtain a cosimplicial cochain complex $\check{C}{}^q (\mathfrak{U}_p)$, and then as usual we can take the alternating sum of coface operators to obtain a second coboundary operator, and hence a double cochain complex.
Somewhat more concretely, an element $\phi$ of $\check{C}{}^q (\mathfrak{U}_p)$ is an assignment of a scalar function $\phi_{i_0, \ldots, i_p, U_0, \ldots, U_q}$ on $U_0 \cap \cdots \cap U_q \cap Y_{i_0} \cap \cdots \cap Y_{i_p}$ to each tuple $(i_0, \ldots, i_p, U_0, \ldots, U_q)$ where $i_0, \ldots, i_p$ are indices and $U_0, \ldots, U_q$ are elements of the open cover $\mathfrak{U}$.
The coboundary operator $\check{C}{}^q (\mathfrak{U}_p) \to \check{C}{}^q (\mathfrak{U}_{p+1})$ must therefore take assignments of functions to tuples of the form $(i_0, \ldots, i_p, U_0, \ldots, U_q)$ to assignments of functions to tuples of the form $(i_0, \ldots, i_{p+1}, U_0, \ldots, U_q)$.
There is basically only one sensible way of doing this: given $(i_0, \ldots, i_{p+1}, U_0, \ldots, U_q)$, for each $0 \le \alpha \le p+1$, omit $i_\alpha$, take the function assigned to that tuple, multiply it by $(-1)^\alpha$, and then take the sum.
Symbolically, the coboundary of $\phi$ is the assignment
$$(i_0, \ldots, i_{p+1}, U_0, \ldots, U_q) \mapsto \sum_{0 \le \alpha \le p+1} (-1)^\alpha \phi_{i_0, \ldots, \widehat{i_\alpha}, \ldots, i_{p+1}, U_0, \ldots, U_q}$$
which is more or less what appears in the cited article, but with slightly different notation and perhaps some typos.
