Lemma: Multidegree, corrected

Let $P_d$ project onto field multidegree $d$ and let $E_u$ be the Euler operator of field $u$. Then $E_u{P}_d=P_{d-e_u}{E}_u$. ($E_u$ lowers the $u$-degree by one and preserves the others; total derivatives preserve multidegree.) Consequently, if $E_{u}F=0$ for all $u$, then $E_u(P_{d}F)=P_{d-e_u}(E_{u}F)=0$ for all $d$ and $u$, so every homogeneous component $P_{d}F$ ($|d|\ge 1$) is a divergence; conversely a sum of divergences is a divergence.