# stacks/stacks-project

Fix indices in limits

Thanks to Minseon Shin
 @@ -1061,8 +1061,8 @@ \section{Absolute Noetherian Approximation} \begin{enumerate} \item the transition morphisms $g_{jj'} : S_j \to S_{j'}$ are affine, \item each $S_j$ is of finite type over $\mathbf{Z}$, \item $g_{jj'}^{-1}(V_{j'}) = V_j$, \item $S = \lim S_j$ and $V = \lim V_j$, and \item $g_{jj'}^{-1}(V'_{j'}) = V'_j$, \item $S = \lim S_j$ and $V = \lim V'_j$, and \item the diagrams $$\vcenter{ @@ -1074,8 +1074,8 @@ \section{Absolute Noetherian Approximation} \quad\text{and}\quad \vcenter{ \xymatrix{ V_j \ar[r] \ar[d] & V_{\alpha(j)} \ar[d] \\ V_{j'} \ar[r] & V_{\alpha(j')} V'_j \ar[r] \ar[d] & V_{\alpha(j)} \ar[d] \\ V'_{j'} \ar[r] & V_{\alpha(j')} } }$$ @@ -1106,10 +1106,11 @@ \section{Absolute Noetherian Approximation} an inverse system over a directed set $I$, each $f_{ii'}$ affine and each $V_i$ of finite type over $\mathbf{Z}$. \end{enumerate} Denote $f_i : V \to V_i$ the projections. Set $W = U \cap V$. As $S$ is quasi-separated, this is a quasi-compact open of $V$. By Lemma \ref{lemma-descend-opens} (and after shrinking $I$) we may assume that there exist opens $W_i \subset V_i$ such that $f_{ij}^{-1}(W_j) = W_i$ opens $W_i \subset V_i$ such that $f_{ii'}^{-1}(W_{i'}) = W_i$ and such that $f_i^{-1}(W_i) = W$. Since $W$ is a quasi-compact open of $U$ it is quasi-affine. Hence we may assume (after shrinking $I$ again) that $W_i$ is quasi-affine for all $i$, see @@ -1165,7 +1166,7 @@ \section{Absolute Noetherian Approximation} assume the corresponding opens $D(g_{l, i}) \subset \Spec(R_i)$ are contained in $W_i$ for $l = 1, \ldots, m$ and cover $W_i$. We conclude that the morphism $W_i \to \Spec(R_i) \to \Spec(B_i)$ is an open immersion, see Lemma \ref{lemma-diagram} is an open immersion, see Lemma \ref{lemma-diagram}. \medskip\noindent By Lemma \ref{lemma-quasi-affine-finite-type-over-Z}