The de Rham and smooth singular Mayer–Vietoris diagram commutes away from connectors

For any form $\omega$ on $M$, [F1] applied to $j_U$ gives $I_U(\omega {\mathrm{\mid }}_U)=I_M(\omega ){\mathrm{\mid }}_U$ as cochains, and application to $j_V$ gives the corresponding equality on $V$. Taking the ordered pair yields the restriction square. For a closed $\omega$, passing to its classes gives the first asserted equality using [F2] and [F3].

For forms $\alpha$ on $U$ and $\beta$ on $V$, naturality for ${\mathrm{\ell }}_U,{\mathrm{\ell }}_V$ and linearity give $I_W(\beta {\mathrm{\mid }}_W-\alpha {\mathrm{\mid }}_W)=I_V(\beta ){\mathrm{\mid }}_W-I_U(\alpha ){\mathrm{\mid }}_W\mathrm{.}$ This is exactly $V-U$ on both rows, not its negative. For closed representatives it passes to the difference square on cohomology. Replacing either representative by an exact form changes its integration cochain by a coboundary by [F1], so the computed square is independent of representatives.

In [F3] the identification between the first term and the small-complex cohomology is restriction along the actual inclusion of small chains. Restricting a global integration cochain to a simplex lying in $U$ or $V$ gives exactly its integral in that open set. Hence step 1.1 computes the stated Mayer–Vietoris arrows even under that identification; no auxiliary small-chain inverse or change of sign enters either square.

If an open or the overlap is empty, the corresponding cochain group is zero and the formulas still hold. If $U=V=M$, step 1.1 is the diagonal square and step 1.2 is ordinary subtraction. In degree zero they are the two pointwise restriction/subtraction identities; degree one and top degree use the same coefficient equalities. Negative degrees are zero. Degenerate simplices are evaluated by the same integration rule. Assumption [F4] is needed only for [F2]'s partition existence, not for any computation above; with a supplied partition both exact rows and these compatibilities are choice-free.