The equivalence of (1) and (\ref{item-zero-on-NL}) is a restatement of Definition \ref{definition-rig-etale-homomorphism}. \medskip\noindent The equivalence of (\ref{item-zero-on-NL}) and (\ref{item-zero-on-cohomology-NL}) follows from More on Algebra, Lemma \ref{more-algebra-lemma-zero-in-derived}. \medskip\noindent The equivalence of (\ref{item-zero-on-cohomology-NL}) and (\ref{item-zero-on-cohomology-NL-truncations}) follows from the fact that the systems $\{\NL_{B_n/A_n}\}$ and $\NL_{B/A}^\wedge \otimes_B B_n$ are strictly isomorphic, see Lemma \ref{lemma-NL-is-limit}. Some details omitted. \medskip\noindent Assume (\ref{item-zero-on-NL}). Let $a \in I$. Let $c$ be such that multiplication by $a^c$ is zero on $\NL_{B/A}^\wedge$. By More on Algebra, Lemma \ref{more-algebra-lemma-map-out-of-almost-free} part (1) there exists a map $\alpha : \bigoplus B\text{d}x_i \to J/J^2$ such that $\text{d} \circ \alpha$ and $\alpha \circ \text{d}$ are both multiplication by $a^c$. Let $f_i \in J$ be an element whose class modulo $J^2$ is equal to $\alpha(\text{d}x_i)$. A simple calculation gives that (\ref{item-condition-artin-pre})(a), (b) hold. \medskip\noindent We omit the verification that (\ref{item-condition-artin-pre}) implies (\ref{item-condition-artin-pre-pre}); it is just a statement on two term complexes over $B$ of the form $M \to B^{\oplus r}$. \medskip\noindent Assume (\ref{item-condition-artin-pre-pre}) holds. Say $I = (a_1, \ldots, a_t)$. Let $c_i \geq 0$ be the integer such that (\ref{item-condition-artin-pre-pre})(a), (b) hold for $a_i^{c_i}$. Then we see that $I^{\sum c_i}$ annihilates $H^0(\NL_{B/A}^\wedge)$. Let $f_{i, 1}, \ldots, f_{i, r} \in J$ be as in (\ref{item-condition-artin-pre-pre})(b) for $a_i$. Consider the composition $$ B^{\oplus r} \to J/J^2 \to \bigoplus B\text{d}x_i $$ where the $j$th basis vector is mapped to the class of $f_{i, j}$ in $J/J^2$. By (\ref{item-condition-artin-pre-pre})(a) and (b) the cokernel of the composition is annihilated by $a_i^{2c_i}$. Thus this map is surjective after inverting $a_i^{c_i}$, and hence an isomorphism (Algebra, Lemma \ref{algebra-lemma-fun}). Thus the kernel of $B^{\oplus r} \to \bigoplus B\text{d}x_i$ is $a_i$-power torsion, and hence $H^{-1}(\NL_{B/A}^\wedge) = \Ker(J/J^2 \to \bigoplus B\text{d}x_i)$ is $a_i$-power torsion. Since $B$ is Noetherian (Lemma \ref{lemma-topologically-finite-type-Noetherian}), all modules including $H^{-1}(\NL_{B/A}^\wedge)$ are finite. Thus $a_i^{d_i}$ annihilates $H^{-1}(\NL_{B/A}^\wedge)$ for some $d_i \geq 0$. It follows that $I^{\sum d_i}$ annihilates $H^{-1}(\NL_{B/A}^\wedge)$ and we see that (\ref{item-zero-on-cohomology-NL}) holds. \medskip\noindent Thus conditions (\ref{item-zero-on-NL}), (\ref{item-zero-on-cohomology-NL}), (\ref{item-zero-on-cohomology-NL-truncations}), (\ref{item-condition-artin-pre-pre}), and (\ref{item-condition-artin-pre}) are equivalent. Thus it remains to show that these conditions are equivalent with (\ref{item-condition-artin}). Observe that the Cramer ideal $\text{Fit}_r(J) B$ is equal to $\text{Fit}_r(J/J^2)$ as $J/J^2 = J \otimes_{A[x_1, \ldots, x_r]^\wedge} B$, see More on Algebra, Lemma \ref{more-algebra-lemma-fitting-ideal-basics} part (3). Also, observe that the Jacobian ideal is just $\text{Fit}_0(H^0(\NL_{B/A}^\wedge))$. Thus we see that the equivalence of (\ref{item-zero-on-cohomology-NL}) and (\ref{item-condition-artin}) is a purely algebraic question which we discuss in the next paragraph. \medskip\noindent Let $R$ be a Noetherian ring and let $I \subset R$ be an ideal. Let $M \xrightarrow{d} R^{\oplus r}$ be a two term complex. We have to show that the following are equivalent \begin{enumerate} \item[(A)] the cohomology of $M \to R^{\oplus r}$ is annihilated by a power of $I$, and \item[(B)] the ideals $\text{Fit}_r(M)$ and $\text{Fit}_0(\text{Coker}(d))$ contain a power of $I$. \end{enumerate} Since $R$ is Noetherian, we can reformulate part (2) as an inclusion of the corresponding closed subschemes, see Algebra, Lemmas \ref{algebra-lemma-Zariski-topology} and \ref{algebra-lemma-Noetherian-power}. On the other hand, over the complement of $V(\text{Fit}_0(\Coker(d)))$ the cokernel of $d$ vanishes and over the complement of $V(\text{Fit}_r(M))$ the module $M$ is locally generated by $r$ elements, see More on Algebra, Lemma \ref{more-algebra-lemma-fitting-ideal-generate-locally}. Thus (B) is equivalent to \begin{enumerate} \item[(C)] away from $V(I)$ the cokernel of $d$ vanishes and the module $M$ is locally generated by $\leq r$ elements. \end{enumerate} Of course this is equivalent to the condition that $M \to R^{\oplus r}$ has vanishing cohomology over $\Spec(R) \setminus V(I)$ which in turn is equivalent to (A). This finishes the proof.