Degree-zero correction to radial homotopy ($dh+hd=\mathrm{id}-F_0^{\ast }$)

For radial contraction on a star-shaped domain, the homotopy formula is the identity in positive form degrees because pullback by the constant map vanishes there. On functions it is $h_{1}df=f-f(0)$. A nonzero constant is a counterexample to the uncorrected degree-zero identity. The Poincaré lemma is therefore an exactness assertion in positive degrees, while closed degree-zero forms are locally constant.