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The Stacks project

Lemma 15.32.4. Let A \to B be a faithfully flat ring map. Let I \subset A be an ideal. If IB is a Koszul-regular (resp. H_1-regular, resp. quasi-regular) ideal in B, then I is a Koszul-regular (resp. H_1-regular, resp. quasi-regular) ideal in A.

Proof. We fix the prime \mathfrak p \supset I throughout the proof. Assume IB is quasi-regular. By Lemma 15.32.2 IB is a finite module, hence I is a finite A-module by Algebra, Lemma 10.83.2. As A \to B is flat we see that

I/I^2 \otimes _{A/I} B/IB = I/I^2 \otimes _ A B = IB/(IB)^2.

As IB is quasi-regular, the B/IB-module IB/(IB)^2 is finite locally free. Hence I/I^2 is finite projective, see Algebra, Proposition 10.83.3. In particular, after replacing A by A_ f for some f \in A, f \not\in \mathfrak p we may assume that I/I^2 is free of rank r. Pick f_1, \ldots , f_ r \in I which give a basis of I/I^2. By Nakayama's lemma (see Algebra, Lemma 10.20.1) we see that, after another replacement A \leadsto A_ f as above, I is generated by f_1, \ldots , f_ r.

Proof of the “quasi-regular” case. Above we have seen that I/I^2 is free on the r-generators f_1, \ldots , f_ r. To finish the proof in this case we have to show that the maps \text{Sym}^ d(I/I^2) \to I^ d/I^{d + 1} are isomorphisms for each d \geq 2. This is clear as the faithfully flat base changes \text{Sym}^ d(IB/(IB)^2) \to (IB)^ d/(IB)^{d + 1} are isomorphisms locally on B by assumption. Details omitted.

Proof of the “H_1-regular” and “Koszul-regular” case. Consider the sequence of elements f_1, \ldots , f_ r generating I we constructed above. By Lemma 15.30.15 we see that f_1, \ldots , f_ r map to a H_1-regular or Koszul-regular sequence in B_ g for any g \in B such that IB is generated by an H_1-regular or Koszul-regular sequence. Hence K_\bullet (A, f_1, \ldots , f_ r) \otimes _ A B_ g has vanishing H_1 or H_ i, i > 0. Since the homology of K_\bullet (B, f_1, \ldots , f_ r) = K_\bullet (A, f_1, \ldots , f_ r) \otimes _ A B is annihilated by IB (see Lemma 15.28.6) and since V(IB) \subset \bigcup _{g\text{ as above}} D(g) we conclude that K_\bullet (A, f_1, \ldots , f_ r) \otimes _ A B has vanishing homology in degree 1 or all positive degrees. Using that A \to B is faithfully flat we conclude that the same is true for K_\bullet (A, f_1, \ldots , f_ r). \square


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