The Stacks project

Lemma 37.38.5. Let $T \to S$ be finite type morphisms of Noetherian schemes. Let $t \in T$ map to $s \in S$ and let $\sigma : \mathcal{O}_{T, t} \to \mathcal{O}_{S, s}^\wedge $ be a local $\mathcal{O}_{S, s}$-algebra map. For every $N \geq 1$ there exists a finite type morphism $(T', t') \to (T, t)$ such that $\sigma $ factors through $\mathcal{O}_{T, t} \to \mathcal{O}_{T', t'}$ and such that for every local $\mathcal{O}_{S, s}$-algebra map $\sigma ' : \mathcal{O}_{T, t} \to \mathcal{O}_{S, s}^\wedge $ which factors through $\mathcal{O}_{T, t} \to \mathcal{O}_{T', t'}$ the maps $\sigma $ and $\sigma '$ agree modulo $\mathfrak m_ s^ N$.

Proof. We may assume $S$ and $T$ are affine. Say $S = \mathop{\mathrm{Spec}}(R)$ and $T = \mathop{\mathrm{Spec}}(C)$. Let $c_1, \ldots , c_ n \in C$ be generators of $C$ as an $R$-algebra. Let $\mathfrak p \subset R$ be the prime ideal corresponding to $s$. Say $\mathfrak p = (f_1, \ldots , f_ m)$. After replacing $R$ by a principal localization (to clear denominators in $R_\mathfrak p$) we may assume there exist $r_1, \ldots , r_ n \in R$ and $a_{i, I} \in \mathcal{O}_{S, s}^\wedge $ where $I = (i_1, \ldots , i_ m)$ with $\sum i_ j = N$ such that

\[ \sigma (c_ i) = r_ i + \sum \nolimits _ I a_{i, I} f_1^{i_1} \ldots f_ m^{i_ m} \]

in $\mathcal{O}_{S, s}^\wedge $. Then we consider

\[ C' = C[t_{i, I}]/ \left(c_ i - r_ i - \sum \nolimits _ I t_{i, I} f_1^{i_1} \ldots f_ m^{i_ m}\right) \]

with $\mathfrak p' = \mathfrak pC' + (t_{i, I})$ and factorization of $\sigma : C \to \mathcal{O}_{S, s}^\wedge $ through $C'$ given by sending $t_{i, I}$ to $a_{i, I}$. Taking $T' = \mathop{\mathrm{Spec}}(C')$ works because any $\sigma '$ as in the statement of the lemma will send $c_ i$ to $r_ i$ modulo the maximal ideal to the power $N$. $\square$

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