An elementary proof of the homological properties of constant-rank operators

We give a simple and constructive extension of Raiţă's result that every constant-rank operator possesses an exact potential and an exact annihilator. Our construction is completely self-contained and provides an improvement on the order of the operators constructed by Raiţă, as well as the order of the explicit annihilators for elliptic operators due to Van Schaftingen. We also give an abstract construction of an optimal annihilator for constant-rank operators, which extends the optimal construction of Van Schaftingen for elliptic operators. Lastly, we establish a generalized Poincaré lemma for constant-rank operators and homogeneous spaces on $\mathbb{R}^d$, and we prove that the existence of potentials on spaces of periodic maps requires a strictly weaker condition than the constant-rank property.