Lemma 37.39.5. 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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