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The Stacks project

Remark 31.22.5. The codimension of a relative quasi-regular immersion, if it is constant, does not change after a base change. In fact, if we have a ring map A \to B and a quasi-regular sequence f_1, \ldots , f_ r \in B such that B/(f_1, \ldots , f_ r) is flat over A, then for any ring map A \to A' we have a quasi-regular sequence f_1 \otimes 1, \ldots , f_ r \otimes 1 in B' = B \otimes _ A A' by More on Algebra, Lemma 15.31.4 (which was used in the proof of Lemma 31.22.1 above). Now the proof of Lemma 31.22.4 shows that if A \to B is flat and locally of finite presentation, then for every prime ideal \mathfrak q' \subset B' the sequence f_1 \otimes 1, \ldots , f_ r \otimes 1 is even a regular sequence in the local ring B'_{\mathfrak q'}.


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