Lemma 3.12 (0NAH) · ScalingStacks

0NAH

Lemma 3.12. The chain maps $R3_{\pm}\colon\left\llbracket L^{i}_{\pm}\right\rrbracket\to\left\llbracket L^{i+1}_{\pm}\right\rrbracket$ in (3.1) do not decrease the external homological grading.

0NAI

Proof. Since chain maps are of homological degree zero, the statement is equivalent to saying that the Reidemeister III chain maps in (3.1) never increase the internal homological grading. This can be verified by inspecting (3.3) and (3.4). For the reader’s convenience we have highlighted the components of negative internal homological degree in green. All other non-zero components are highlighted blue or yellow and have internal homological degree zero because they map between the yellow and blue layers of the relevant complexes. Thus we only need to worry about components of the chain map which are not highlighted in the diagrams above. With $y=0$ , these components all vanish. ∎