Corollary: Highest representative degree and degree projections

In the convention of Lemma lem:null, the same nonzero reconstructed kernel has representatives

of different highest polynomial degrees. Consequently, highest degree of a chosen presentation is not an invariant of the reconstructed kernel.

Likewise, projection to the degree-one summand does not descend: it sends the null presentation in (eq:null) to a summand with nonzero reconstruction.

Proof. Both displayed representatives reconstruct to $\delta (q)\otimes R$. The degree-one projection of the null presentation reconstructs to $-\delta (q)\otimes R\ne 0$. Apply Proposition prop:descent.