Check we're using all assumptions

Thanks to Et
https://stacks.math.columbia.edu/tag/00N1#comment-8330

algebra.tex

@@ -24541,9 +24541,14 @@ \section{What makes a complex exact?}
 contains a regular sequence of length $i$.
 
 \medskip\noindent
-Assume (2)(a) and (2)(b) hold. We claim that for any prime
-$\mathfrak p \subset R$ conditions (2)(a) and (2)(b)
-hold for the complex
+Assume (2)(a) and (2)(b) hold. We will prove that (1) holds by
+induction on $\dim(R)$. If $\dim(R) = 0$, then we must have
+$I(\varphi_i) = R$ for $1 \leq i \leq e$ by (2)(b). Since the
+coefficients of $\varphi_i$ are contained in the maximal ideal
+this can happen only if $r_i = 0$ for all $i$. By (2)(a) we
+conclude that $e = 0$ and (1) holds. Assume $\dim(R) > 0$.
+We claim that for any prime $\mathfrak p \subset R$ conditions
+(2)(a) and (2)(b) hold for the complex
 $0 \to R_\mathfrak p^{n_e} \to R_\mathfrak p^{n_{e - 1}} \to \ldots \to
 R_\mathfrak p^{n_0}$ with maps $\varphi_{i, \mathfrak p}$
 over $R_\mathfrak p$. Namely, since $I(\varphi_i)$ contains a