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The Stacks project

Lemma 16.11.2. Let k be a field of characteristic p > 0. Let \Lambda be a Noetherian geometrically regular k-algebra. Let \mathfrak q \subset \Lambda be a prime ideal. Let n \geq 1 be an integer and let E \subset \Lambda _\mathfrak q/\mathfrak q^ n\Lambda _\mathfrak q be a finite subset. Then we can find m \geq 0 and \varphi : k[y_1, \ldots , y_ m] \to \Lambda with the following properties

  1. setting \mathfrak p = \varphi ^{-1}(\mathfrak q) we have \mathfrak q\Lambda _\mathfrak q = \mathfrak p \Lambda _\mathfrak q and k[y_1, \ldots , y_ m]_\mathfrak p \to \Lambda _\mathfrak q is flat,

  2. there is a factorization by homomorphisms of local Artinian rings

    k[y_1, \ldots , y_ m]_\mathfrak p/\mathfrak p^ n k[y_1, \ldots , y_ m]_\mathfrak p \to D \to \Lambda _\mathfrak q/\mathfrak q^ n\Lambda _\mathfrak q

    where the first arrow is essentially smooth and the second is flat,

  3. E is contained in D modulo \mathfrak q^ n\Lambda _\mathfrak q.

Proof. Set \bar\Lambda = \Lambda _\mathfrak q/\mathfrak q^ n\Lambda _\mathfrak q. Note that \dim H_1(L_{\kappa (\mathfrak q)/k}) < \infty by More on Algebra, Proposition 15.35.1. Pick A \subset \bar\Lambda containing E such that A is local Artinian, essentially of finite type over k, the map A \to \bar\Lambda is flat, and \mathfrak m_ A generates the maximal ideal of \bar\Lambda , see Lemma 16.11.1. Denote F = A/\mathfrak m_ A the residue field so that k \subset F \subset K. Pick \lambda _1, \ldots , \lambda _ t \in \Lambda which map to elements of A in \bar\Lambda such that moreover the images of \text{d}\lambda _1, \ldots , \text{d}\lambda _ t form a basis of \Omega _{F/k}. Consider the map \varphi ' : k[y_1, \ldots , y_ t] \to \Lambda sending y_ j to \lambda _ j. Set \mathfrak p' = (\varphi ')^{-1}(\mathfrak q). By More on Algebra, Lemma 15.35.2 the ring map k[y_1, \ldots , y_ t]_{\mathfrak p'} \to \Lambda _\mathfrak q is flat and \Lambda _\mathfrak q/\mathfrak p' \Lambda _\mathfrak q is regular. Thus we can choose further elements \lambda _{t + 1}, \ldots , \lambda _ m \in \Lambda which map into A \subset \bar\Lambda and which map to a regular system of parameters of \Lambda _\mathfrak q/\mathfrak p' \Lambda _\mathfrak q. We obtain \varphi : k[y_1, \ldots , y_ m] \to \Lambda having property (1) such that k[y_1, \ldots , y_ m]_\mathfrak p/\mathfrak p^ n k[y_1, \ldots , y_ m]_\mathfrak p \to \bar\Lambda factors through A. Thus k[y_1, \ldots , y_ m]_\mathfrak p/\mathfrak p^ n k[y_1, \ldots , y_ m]_\mathfrak p \to A is flat by Algebra, Lemma 10.39.9. By construction the residue field extension F/\kappa (\mathfrak p) is finitely generated and \Omega _{F/\kappa (\mathfrak p)} = 0. Hence it is finite separable by More on Algebra, Lemma 15.34.1. Thus k[y_1, \ldots , y_ m]_\mathfrak p/\mathfrak p^ n k[y_1, \ldots , y_ m]_\mathfrak p \to A is finite by Algebra, Lemma 10.54.4. Finally, we conclude that it is étale by Algebra, Lemma 10.143.7. Since an étale ring map is certainly essentially smooth we win. \square


Comments (2)

Comment #8618 by Nick Addington on

"Essentially smooth" is used here and in the next two lemmas. I guess it means formally smooth and essentially of finite type, or maybe essentially of finite presentation, or (equivalently?) it's a localization of a smooth map; but the definition doesn't seem to be in the stacks project.


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