h : A \to B

, the dual morphism

h : B \to A

is pullback of algebraically trivial line bundles. Translations satisfy

h \circ T_{x} = T_{h (x)} \circ h

, and therefore

ϕ_{h^{*} L} (x) = h^{*} (T_{h (x)}^{*} L \otimes L^{- 1}) = (h \circ ϕ_{L} \circ h) (x) .

Also

ϕ_{L \otimes M} = ϕ_{L} + ϕ_{M}

and

ϕ_{L^{- 1}} = - ϕ_{L}

. Dualizing is additive:

f + g = f + g

, since algebraically trivial line bundles obey the Theorem of the square. Apply these identities to the three factors defining

D_{L}

. The two diagonal terms cancel in the expansion

ϕ_{D_{L} (f, g)} = (f + g) ϕ_{L} (f + g) - f ϕ_{L} f - g ϕ_{L} g = f ϕ_{L} g + g ϕ_{L} f .

Hence $ϕ_{D_{L} (f, g)} = f \circ ϕ_{L} \circ g + g \circ ϕ_{L} \circ f$ .

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