Corollary: Sheet-compatible contacts are zero classes

Let $N_h:\mathcal{D}\to \mathcal{D}$ be any proposed modification, and set

Then the following are equivalent:

1. ${\mathcal{R}}_N$ satisfies the same normalization sheet;

2. $N_{h}c\in P_m\mathcal{D}$ for every $c$;

3. $\mathbf{\varphi }(N_{h}c)=0$ for every $c$.

In that case ${\mathcal{R}}_N=R_1(V_{1,g}[h];\cdot )$. The same statement applies coefficientwise to formal modifications.

Proof. Subtract (eq:sheet) for the two maps and apply Lemmas lem:detect and lem:graded.