The equivalence of (1), (2), and (3) is immediate from More on Algebra, Lemma \ref{more-algebra-lemma-ext-1-annihilated}. \medskip\noindent Assume $b_1, \ldots, b_s$ are as in (4). Since $B$ is Noetherian the inclusion $V(b_1, \ldots, b_s) \subset V(IB)$ implies $I^cB \subset (b_1, \ldots, b_s)$ for some $c \geq 0$ (for example by Algebra, Lemma \ref{algebra-lemma-Noetherian-power-ideal-kills-module}). Pick $1 \leq l \leq s$ and $m \geq 0$ and $f_1, \ldots, f_m \in J$ and $T \subset \{1, \ldots, n\}$ with $|T| = m$ satisfying (4)(a) and (b). Then if we invert $b_l$ we see that $$ \NL_{B/A}^\wedge \otimes_B B_{b_l} = \left( \bigoplus\nolimits_{j \leq m} B_{b_l} f_j \longrightarrow \bigoplus\nolimits_{i = 1, \ldots, n} B_{b_l} \text{d}x_i \right) $$ and moreover the arrow is isomorphic to the inclusion of the direct summand $\bigoplus_{i \in T} B_{b_l} \text{d}x_i$. We conclude that $H^{-1}(\NL_{B/A}^\wedge)$ is $b_l$-power torsion and that $H^0(\NL_{B/A}^\wedge)$ becomes finite free after inverting $b_l$. Combined with the inclusion $I^cB \subset (b_1, \ldots, b_s)$ we see that $H^{-1}(\NL_{B/A}^\wedge)$ is $IB$-power torsion. Hence we see that condition (4) of More on Algebra, Lemma \ref{more-algebra-lemma-ext-1-annihilated} holds. In this way we see that (4) implies (2). \medskip\noindent Assume the equivalent conditions (1), (2), and (3) hold. We will prove that (4) holds, but we strongly urge the reader to convince themselves of this. The complex $\NL_{B/A}^\wedge$ determines an object of $D^b_{\textit{Coh}}(\Spec(B))$ whose restriction to the Zariski open $U = \Spec(B) \setminus V(IB)$ is a finite locally free module $\mathcal{E}$ placed in degree $0$ (this follows for example from the the fourth equivalent condition in More on Algebra, Lemma \ref{more-algebra-lemma-ext-1-annihilated}). Choose generators $f_1, \ldots, f_M$ for $J$. This determines an exact sequence $$ \bigoplus\nolimits_{j = 1, \ldots, M} \mathcal{O}_U \cdot f_j \to \bigoplus\nolimits_{i = 1, \ldots, n} \mathcal{O}_U \cdot \text{d}x_i \to \mathcal{E} \to 0 $$ Let $U = \bigcup_{l = 1, \ldots, s} U_l$ be a finite affine open covering such that $\mathcal{E}|_{U_l}$ is free of rank $r_l = n - m_l$ for some integer $n \geq m_l \geq 0$. After replacing each $U_l$ by an affine open covering we may assume there exists a subset $T_l \subset \{1, \ldots, n\}$ such that the elements $\text{d}x_i$, $i \in \{1, \ldots, n\} \setminus T_l$ map to a basis for $\mathcal{E}|_{U_l}$. Repeating the argument, we may assume there exists a subset $T'_l \subset \{1, \ldots, M\}$ of cardinality $m_l$ such that $f_j$, $j \in T'_l$ map to a basis of the kernel of $\mathcal{O}_{U_l} \cdot \text{d}x_i \to \mathcal{E}|_{U_l}$. Finally, since the open covering $U = \bigcup U_l$ may be refined by a open covering by standard opens (Algebra, Lemma \ref{algebra-lemma-Zariski-topology}) we may assume $U_l = D(g_l)$ for some $g_l \in B$. In particular we have $V(g_1, \ldots, g_s) = V(IB)$. A linear algebra argument using our choices above shows that $\det_{i \in T_l, j \in T'_l}(\partial f_j/ \partial x_i)$ maps to an invertible element of $B_{b_l}$. Similarly, the vanishing of cohomology of $\NL_{B/A}^\wedge$ in degree $-1$ over $U_l$ shows that $J/J^2 + (f_j; j \in T')$ is annihilated by a power of $b_l$. After replacing each $g_l$ by a suitable power we obtain conditions (4)(a) and (4)(b) of the lemma. Some details omitted.