Lemma 15.30.10. Let $A$ be a ring. Let $I$ be an ideal generated by a quasi-regular sequence $f_1, \ldots , f_ n$ in $A$. Let $g_1, \ldots , g_ m \in A$ be elements whose images $\overline{g}_1, \ldots , \overline{g}_ m$ form an $H_1$-regular sequence in $A/I$. Then $f_1, \ldots , f_ n, g_1, \ldots , g_ m$ is a quasi-regular sequence in $A$.

Proof. We claim that $g_1, \ldots , g_ m$ forms an $H_1$-regular sequence in $A/I^ d$ for every $d$. By induction assume that this holds in $A/I^{d - 1}$. We have a short exact sequence of complexes

$0 \to K_\bullet (A, g_\bullet ) \otimes _ A I^{d - 1}/I^ d \to K_\bullet (A/I^ d, g_\bullet ) \to K_\bullet (A/I^{d - 1}, g_\bullet ) \to 0$

Since $f_1, \ldots , f_ n$ is quasi-regular we see that the first complex is a direct sum of copies of $K_\bullet (A/I, g_1, \ldots , g_ m)$ hence acyclic in degree $1$. By induction hypothesis the last complex is acyclic in degree $1$. Hence also the middle complex is. In particular, the sequence $g_1, \ldots , g_ m$ forms a quasi-regular sequence in $A/I^ d$ for every $d \geq 1$, see Lemma 15.30.6. Now we are ready to prove that $f_1, \ldots , f_ n, g_1, \ldots , g_ m$ is a quasi-regular sequence in $A$. Namely, set $J = (f_1, \ldots , f_ n, g_1, \ldots , g_ m)$ and suppose that (with multinomial notation)

$\sum \nolimits _{|N| + |M| = d} a_{N, M} f^ N g^ M \in J^{d + 1}$

for some $a_{N, M} \in A$. We have to show that $a_{N, M} \in J$ for all $N, M$. Let $e \in \{ 0, 1, \ldots , d\}$. Then

$\sum \nolimits _{|N| = d - e, \ |M| = e} a_{N, M} f^ N g^ M \in (g_1, \ldots , g_ m)^{e + 1} + I^{d - e + 1}$

Because $g_1, \ldots , g_ m$ is a quasi-regular sequence in $A/I^{d - e + 1}$ we deduce

$\sum \nolimits _{|N| = d - e} a_{N, M} f^ N \in (g_1, \ldots , g_ m) + I^{d - e + 1}$

for each $M$ with $|M| = e$. By Lemma 15.30.8 applied to $I^{d - e}/I^{d - e + 1}$ in the ring $A/I^{d - e + 1}$ this implies $\sum _{|N| = d - e} a_{N, M} f^ N \in I^{d - e}(g_1, \ldots , g_ m)$. Since $f_1, \ldots , f_ n$ is quasi-regular in $A$ this implies that $a_{N, M} \in J$ for each $N, M$ with $|N| = d - e$ and $|M| = e$. This proves the lemma. $\square$

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