Independent generators and off-shell nilpotence

Use independent fields

Their parities are $0,0,1,1,0$, and their ghost numbers are $0,0,1,-1,0$. The odd fields have coefficients in a Grassmann algebra. All fields are ambient smooth configurations before equations are imposed. Compactness belongs to smearing tests and variations, not to a separate antighost configuration space.

Define an odd derivation $s_0$, commuting with spacetime differentiation, by

The graded Leibniz rule is $s_0(FG)=(s_{0}F)G+(-1{)}^{|F|}F(s_{0}G)$.