The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

15.29 Koszul regular sequences

Please take a look at Algebra, Sections 10.67, 10.68, and 10.71 before looking at this one.

Definition 15.29.1. Let $R$ be a ring. Let $r \geq 0$ and let $f_1, \ldots , f_ r \in R$ be a sequence of elements. Let $M$ be an $R$-module. The sequence $f_1, \ldots , f_ r$ is called

  1. $M$-Koszul-regular if $H_ i(K_\bullet (f_1, \ldots , f_ r) \otimes _ R M) = 0$ for all $i \not= 0$,

  2. $M$-$H_1$-regular if $H_1(K_\bullet (f_1, \ldots , f_ r) \otimes _ R M) = 0$,

  3. Koszul-regular if $H_ i(K_\bullet (f_1, \ldots , f_ r)) = 0$ for all $i \not= 0$, and

  4. $H_1$-regular if $H_1(K_\bullet (f_1, \ldots , f_ r)) = 0$.

We will see in Lemmas 15.29.2, 15.29.3, and 15.29.6 that for elements $f_1, \ldots , f_ r$ of a ring $R$ we have the following implications

\begin{align*} f_1, \ldots , f_ r\text{ is a regular sequence} & \Rightarrow f_1, \ldots , f_ r\text{ is a Koszul-regular sequence} \\ & \Rightarrow f_1, \ldots , f_ r\text{ is an }H_1\text{-regular sequence} \\ & \Rightarrow f_1, \ldots , f_ r\text{ is a quasi-regular sequence.} \end{align*}

In general none of these implications can be reversed, but if $R$ is a Noetherian local ring and $f_1, \ldots , f_ r \in \mathfrak m_ R$, then the four conditions are all equivalent (Lemma 15.29.7). If $f = f_1 \in R$ is a length $1$ sequence and $f$ is not a unit of $R$ then it is clear that the following are all equivalent

  1. $f$ is a regular sequence of length one,

  2. $f$ is a Koszul-regular sequence of length one, and

  3. $f$ is a $H_1$-regular sequence of length one.

It is also clear that these imply that $f$ is a quasi-regular sequence of length one. But there do exist quasi-regular sequences of length $1$ which are not regular sequences. Namely, let

\[ R = k[x, y_0, y_1, \ldots ]/(xy_0, xy_1 - y_0, xy_2 - y_1, \ldots ) \]

and let $f$ be the image of $x$ in $R$. Then $f$ is a zerodivisor, but $\bigoplus _{n \geq 0} (f^ n)/(f^{n + 1}) \cong k[x]$ is a polynomial ring.

Lemma 15.29.2. An $M$-regular sequence is $M$-Koszul-regular. A regular sequence is Koszul-regular.

Proof. Let $R$ be a ring and let $M$ be an $R$-module. It is immediate that an $M$-regular sequence of length $1$ is $M$-Koszul-regular. Let $f_1, \ldots , f_ r$ be an $M$-regular sequence. Then $f_1$ is a nonzerodivisor on $M$. Hence

\[ 0 \to K_\bullet (f_2, \ldots , f_ r) \otimes M \xrightarrow {f_1} K_\bullet (f_2, \ldots , f_ r) \otimes M \to K_\bullet (\overline{f}_2, \ldots , \overline{f}_ r) \otimes M/f_1M \to 0 \]

is a short exact sequence of complexes where $\overline{f}_ i$ is the image of $f_ i$ in $R/(f_1)$. By Lemma 15.28.8 the complex $K_\bullet (R, f_1, \ldots , f_ r)$ is isomorphic to the cone of multiplication by $f_1$ on $K_\bullet (f_2, \ldots , f_ r)$. Thus $K_\bullet (R, f_1, \ldots , f_ r) \otimes M$ is isomorphic to the cone on the first map. Hence $K_\bullet (\overline{f}_2, \ldots , \overline{f}_ r) \otimes M/f_1M$ is quasi-isomorphic to $K_\bullet (f_1, \ldots , f_ r) \otimes M$. As $\overline{f}_2, \ldots , \overline{f}_ r$ is an $M/f_1M$-regular sequence in $R/(f_1)$ the result follows from the case $r = 1$ and induction. $\square$

Lemma 15.29.3. A $M$-Koszul-regular sequence is $M$-$H_1$-regular. A Koszul-regular sequence is $H_1$-regular.

Proof. This is immediate from the definition. $\square$

Lemma 15.29.4. Let $f_1, \ldots , f_{r - 1} \in R$ be a sequence and $f, g \in R$. Let $M$ be an $R$-module.

  1. If $f_1, \ldots , f_{r - 1}, f$ and $f_1, \ldots , f_{r - 1}, g$ are $M$-$H_1$-regular then $f_1, \ldots , f_{r - 1}, fg$ is $M$-$H_1$-regular too.

  2. If $f_1, \ldots , f_{r - 1}, f$ and $f_1, \ldots , f_{r - 1}, f$ are $M$-Koszul-regular then $f_1, \ldots , f_{r - 1}, fg$ is $M$-Koszul-regular too.

Proof. By Lemma 15.28.11 we have exact sequences

\[ H_ i(K_\bullet (f_1, \ldots , f_{r - 1}, f) \otimes M) \to H_ i(K_\bullet (f_1, \ldots , f_{r - 1}, fg) \otimes M) \to H_ i(K_\bullet (f_1, \ldots , f_{r - 1}, g) \otimes M) \]

for all $i$. $\square$

Lemma 15.29.5. Let $\varphi : R \to S$ be a flat ring map. Let $f_1, \ldots , f_ r \in R$. Let $M$ be an $R$-module and set $N = M \otimes _ R S$.

  1. If $f_1, \ldots , f_ r$ in $R$ is an $M$-$H_1$-regular sequence, then $\varphi (f_1), \ldots , \varphi (f_ r)$ is an $N$-$H_1$-regular sequence in $S$.

  2. If $f_1, \ldots , f_ r$ is an $M$-Koszul-regular sequence in $R$, then $\varphi (f_1), \ldots , \varphi (f_ r)$ is an $N$-Koszul-regular sequence in $S$.

Proof. This is true because $K_\bullet (f_1, \ldots , f_ r) \otimes _ R S = K_\bullet (\varphi (f_1), \ldots , \varphi (f_ r))$ and therefore $(K_\bullet (f_1, \ldots , f_ r) \otimes _ R M) \otimes _ R S = K_\bullet (\varphi (f_1), \ldots , \varphi (f_ r)) \otimes _ S N$. $\square$

Proof. Let $R$ be a ring and let $M$ be an $R$-module. Let $f_1, \ldots , f_ r$ be an $M$-$H_1$-regular sequence. Denote $J = (f_1, \ldots , f_ r)$. The assumption means that we have an exact sequence

\[ \wedge ^2(R^ r) \otimes M \to R^{\oplus r} \otimes M \to JM \to 0 \]

where the first arrow is given by $e_ i \wedge e_ j \otimes m \mapsto (f_ ie_ j - f_ je_ i) \otimes m$. In particular this implies that

\[ JM/J^2M = JM \otimes _ R R/J = (M/JM)^{\oplus r} \]

is a finite free module. To finish the proof we have to prove for every $n \geq 2$ the following: if

\[ \xi = \sum \nolimits _{|I| = n, I = (i_1, \ldots , i_ r)} m_ I f_1^{i_1} \ldots f_ r^{i_ r} \in J^{n + 1}M \]

then $m_ I \in JM$ for all $I$. Note that $f_1, \ldots , f_{r - 1}, f_ r^ n$ is an $M$-$H_1$-regular sequence by Lemma 15.29.4. Hence we see that the required result holds for the multi-index $I = (0, \ldots , 0, n)$. It turns out that we can reduce the general case to this case as follows.

Let $S = R[x_1, x_2, \ldots , x_ r, 1/x_ r]$. The ring map $R \to S$ is faithfully flat, hence $f_1, \ldots , f_ r$ is an $M$-$H_1$-regular sequence in $S$, see Lemma 15.29.5. By Lemma 15.28.4 we see that

\[ g_1 = f_1 - x_1/x_ r f_ r, \ldots g_{r - 1} = f_{r - 1} - x_{r - 1}/x_ r f_ r, g_ r = (1/x_ r)f_ r \]

is an $M$-$H_1$-regular sequence in $S$. Finally, note that our element $\xi $ can be rewritten

\[ \xi = \sum \nolimits _{|I| = n, I = (i_1, \ldots , i_ r)} m_ I (g_1 + x_ r g_ r)^{i_1} \ldots (g_{r - 1} + x_ r g_ r)^{i_{r - 1}} (x_ rg_ r)^{i_ r} \]

and the coefficient of $g_ r^ n$ in this expression is

\[ \sum m_ I x_1^{i_1} \ldots x_ r^{i_ r} \in J(M \otimes _ R S). \]

Since the monomials $x_1^{i_1} \ldots x_ r^{i_ r}$ form part of an $R$-basis of $S$ over $R$ we conclude that $m_ I \in J$ for all $I$ as desired. $\square$

For nonzero finite modules over Noetherian local rings all of the types of regular sequences introduced so far are equivalent.

Lemma 15.29.7. Let $(R, \mathfrak m)$ be a Noetherian local ring. Let $M$ be a nonzero finite $R$-module. Let $f_1, \ldots , f_ r \in \mathfrak m$. The following are equivalent

  1. $f_1, \ldots , f_ r$ is an $M$-regular sequence,

  2. $f_1, \ldots , f_ r$ is a $M$-Koszul-regular sequence,

  3. $f_1, \ldots , f_ r$ is an $M$-$H_1$-regular sequence,

  4. $f_1, \ldots , f_ r$ is an $M$-quasi-regular sequence.

In particular the sequence $f_1, \ldots , f_ r$ is a regular sequence in $R$ if and only if it is a Koszul regular sequence, if and only if it is a $H_1$-regular sequence, if and only if it is a quasi-regular sequence.

Proof. The implication (1) $\Rightarrow $ (2) is Lemma 15.29.2. The implication (2) $\Rightarrow $ (3) is Lemma 15.29.3. The implication (3) $\Rightarrow $ (4) is Lemma 15.29.6. The implication (4) $\Rightarrow $ (1) is Algebra, Lemma 10.68.6. $\square$

Lemma 15.29.8. Let $A$ be a ring. Let $I \subset A$ be an ideal. Let $g_1, \ldots , g_ m$ be a sequence in $A$ whose image in $A/I$ is $H_1$-regular. Then $I \cap (g_1, \ldots , g_ m) = I(g_1, \ldots , g_ m)$.

Proof. Consider the exact sequence of complexes

\[ 0 \to I \otimes _ A K_\bullet (A, g_1, \ldots , g_ m) \to K_\bullet (A, g_1, \ldots , g_ m) \to K_\bullet (A/I, g_1, \ldots , g_ m) \to 0 \]

Since the complex on the right has $H_1 = 0$ by assumption we see that

\[ \mathop{\mathrm{Coker}}(I^{\oplus m} \to I) \longrightarrow \mathop{\mathrm{Coker}}(A^{\oplus m} \to A) \]

is injective. This is equivalent to the assertion of the lemma. $\square$

Lemma 15.29.9. Let $A$ be a ring. Let $I \subset J \subset A$ be ideals. Assume that $J/I \subset A/I$ is generated by an $H_1$-regular sequence. Then $I \cap J^2 = IJ$.

Proof. To prove this choose $g_1, \ldots , g_ m \in J$ whose images in $A/I$ form a $H_1$-regular sequence which generates $J/I$. In particular $J = I + (g_1, \ldots , g_ m)$. Suppose that $x \in I \cap J^2$. Because $x \in J^2$ can write

\[ x = \sum a_{ij} g_ ig_ j + \sum a_ j g_ j + a \]

with $a_{ij} \in A$, $a_ j \in I$ and $a \in I^2$. Then $\sum a_{ij}g_ ig_ j \in I \cap (g_1, \ldots , g_ m)$ hence by Lemma 15.29.8 we see that $\sum a_{ij}g_ ig_ j \in I(g_1, \ldots , g_ m)$. Thus $x \in IJ$ as desired. $\square$

Lemma 15.29.10. 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 15.29.6. 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 15.29.8 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. $\square$

Lemma 15.29.11. Let $A$ be a ring. Let $I$ be an ideal generated by an $H_1$-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 an $H_1$-regular sequence in $A$.

Proof. We have to show that $H_1(A, f_1, \ldots , f_ n, g_1, \ldots , g_ m) = 0$. To do this consider the commutative diagram

\[ \xymatrix{ \wedge ^2(A^{\oplus n + m}) \ar[r] \ar[d] & A^{\oplus n + m} \ar[r] \ar[d] & A \ar[r] \ar[d] & 0 \\ \wedge ^2(A/I^{\oplus m}) \ar[r] & A/I^{\oplus m} \ar[r] & A/I \ar[r] & 0 } \]

Consider an element $(a_1, \ldots , a_{n + m}) \in A^{\oplus n + m}$ which maps to zero in $A$. Because $\overline{g}_1, \ldots , \overline{g}_ m$ form an $H_1$-regular sequence in $A/I$ we see that $(\overline{a}_{n + 1}, \ldots , \overline{a}_{n + m})$ is the image of some element $\overline{\alpha }$ of $\wedge ^2(A/I^{\oplus m})$. We can lift $\overline{\alpha }$ to an element $\alpha \in \wedge ^2(A^{\oplus n + m})$ and substract the image of it in $A^{\oplus n + m}$ from our element $(a_1, \ldots , a_{n + m})$. Thus we may assume that $a_{n + 1}, \ldots , a_{n + m} \in I$. Since $I = (f_1, \ldots , f_ n)$ we can modify our element $(a_1, \ldots , a_{n + m})$ by linear combinations of the elements

\[ (0, \ldots , g_ j, 0, \ldots , 0, f_ i, 0, \ldots , 0) \]

in the image of the top left horizontal arrow to reduce to the case that $a_{n + 1}, \ldots , a_{n + m}$ are zero. In this case $(a_1, \ldots , a_ n, 0, \ldots , 0)$ defines an element of $H_1(A, f_1, \ldots , f_ n)$ which we assumed to be zero. $\square$

Lemma 15.29.12. Let $A$ be a ring. Let $f_1, \ldots , f_ n, g_1, \ldots , g_ m \in A$ be an $H_1$-regular sequence. Then the images $\overline{g}_1, \ldots , \overline{g}_ m$ in $A/(f_1, \ldots , f_ n)$ form an $H_1$-regular sequence.

Proof. Set $I = (f_1, \ldots , f_ n)$. We have to show that any relation $\sum _{j = 1, \ldots , m} \overline{a}_ j \overline{g}_ j$ in $A/I$ is a linear combination of trivial relations. Because $I = (f_1, \ldots , f_ n)$ we can lift this relation to a relation

\[ \sum \nolimits _{j = 1, \ldots , m} a_ j g_ j + \sum \nolimits _{i = 1, \ldots , n} b_ if_ i = 0 \]

in $A$. By assumption this relation in $A$ is a linear combination of trivial relations. Taking the image in $A/I$ we obtain what we want. $\square$

Lemma 15.29.13. Let $A$ be a ring. Let $I$ be an ideal generated by a Koszul-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 a Koszul-regular sequence in $A/I$. Then $f_1, \ldots , f_ n, g_1, \ldots , g_ m$ is a Koszul-regular sequence in $A$.

Proof. Our assumptions say that $K_\bullet (A, f_1, \ldots , f_ n)$ is a finite free resolution of $A/I$ and $K_\bullet (A/I, \overline{g}_1, \ldots , \overline{g}_ m)$ is a finite free resolution of $A/(f_ i, g_ j)$ over $A/I$. Then

\begin{align*} K_\bullet (A, f_1, \ldots , f_ n, g_1, \ldots , g_ m) & = \text{Tot}(K_\bullet (A, f_1, \ldots , f_ n) \otimes _ A K_\bullet (A, g_1, \ldots , g_ m)) \\ & \cong A/I \otimes _ A K_\bullet (A, g_1, \ldots , g_ m) \\ & = K_\bullet (A/I, \overline{g}_1, \ldots , \overline{g}_ m) \\ & \cong A/(f_ i, g_ j) \end{align*}

The first equality by Lemma 15.28.12. The first quasi-isomorphism $\cong $ by (the dual of) Homology, Lemma 12.22.7 as the $q$th row of the double complex $K_\bullet (A, f_1, \ldots , f_ n) \otimes _ A K_\bullet (A, g_1, \ldots , g_ m)$ is a resolution of $A/I \otimes _ A K_ q(A, g_1, \ldots , g_ m)$. The second equality is clear. The last quasi-isomorphism by assumption. Hence we win. $\square$

To conclude in the following lemma it is necessary to assume that both $f_1, \ldots , f_ n$ and $f_1, \ldots , f_ n, g_1, \ldots , g_ m$ are Koszul-regular. A counter example to dropping the assumption that $f_1, \ldots , f_ n$ is Koszul-regular is Examples, Lemma 102.13.1.

Lemma 15.29.14. Let $A$ be a ring. Let $f_1, \ldots , f_ n, g_1, \ldots , g_ m \in A$. If both $f_1, \ldots , f_ n$ and $f_1, \ldots , f_ n, g_1, \ldots , g_ m$ are Koszul-regular sequences in $A$, then $\overline{g}_1, \ldots , \overline{g}_ m$ in $A/(f_1, \ldots , f_ n)$ form a Koszul-regular sequence.

Proof. Set $I = (f_1, \ldots , f_ n)$. Our assumptions say that $K_\bullet (A, f_1, \ldots , f_ n)$ is a finite free resolution of $A/I$ and $K_\bullet (A, f_1, \ldots , f_ n, g_1, \ldots , g_ m)$ is a finite free resolution of $A/(f_ i, g_ j)$ over $A$. Then

\begin{align*} A/(f_ i, g_ j) & \cong K_\bullet (A, f_1, \ldots , f_ n, g_1, \ldots , g_ m) \\ & = \text{Tot}(K_\bullet (A, f_1, \ldots , f_ n) \otimes _ A K_\bullet (A, g_1, \ldots , g_ m)) \\ & \cong A/I \otimes _ A K_\bullet (A, g_1, \ldots , g_ m) \\ & = K_\bullet (A/I, \overline{g}_1, \ldots , \overline{g}_ m) \end{align*}

The first quasi-isomorphism $\cong $ by assumption. The first equality by Lemma 15.28.12. The second quasi-isomorphism by (the dual of) Homology, Lemma 12.22.7 as the $q$th row of the double complex $K_\bullet (A, f_1, \ldots , f_ n) \otimes _ A K_\bullet (A, g_1, \ldots , g_ m)$ is a resolution of $A/I \otimes _ A K_ q(A, g_1, \ldots , g_ m)$. The second equality is clear. Hence we win. $\square$

Lemma 15.29.15. Let $R$ be a ring. Let $I$ be an ideal generated by $f_1, \ldots , f_ r \in R$.

  1. If $I$ can be generated by a quasi-regular sequence of length $r$, then $f_1, \ldots , f_ r$ is a quasi-regular sequence.

  2. If $I$ can be generated by an $H_1$-regular sequence of length $r$, then $f_1, \ldots , f_ r$ is an $H_1$-regular sequence.

  3. If $I$ can be generated by a Koszul-regular sequence of length $r$, then $f_1, \ldots , f_ r$ is a Koszul-regular sequence.

Proof. If $I$ can be generated by a quasi-regular sequence of length $r$, then $I/I^2$ is free of rank $r$ over $R/I$. Since $f_1, \ldots , f_ r$ generate by assumption we see that the images $\overline{f}_ i$ form a basis of $I/I^2$ over $R/I$. It follows that $f_1, \ldots , f_ r$ is a quasi-regular sequence as all this means, besides the freeness of $I/I^2$, is that the maps $\text{Sym}^ n_{R/I}(I/I^2) \to I^ n/I^{n + 1}$ are isomorphisms.

We continue to assume that $I$ can be generated by a quasi-regular sequence, say $g_1, \ldots , g_ r$. Write $g_ j = \sum a_{ij}f_ i$. As $f_1, \ldots , f_ r$ is quasi-regular according to the previous paragraph, we see that $\det (a_{ij})$ is invertible mod $I$. The matrix $a_{ij}$ gives a map $R^{\oplus r} \to R^{\oplus r}$ which induces a map of Koszul complexes $\alpha : K_\bullet (R, f_1, \ldots , f_ r) \to K_\bullet (R, g_1, \ldots , g_ r)$, see Lemma 15.28.3. This map becomes an isomorphism on inverting $\det (a_{ij})$. Since the cohomology modules of both $K_\bullet (R, f_1, \ldots , f_ r)$ and $K_\bullet (R, g_1, \ldots , g_ r)$ are annihilated by $I$, see Lemma 15.28.6, we see that $\alpha $ is a quasi-isomorphism.

Now assume that $g_1, \ldots , g_ r$ is a $H_1$-regular sequence generating $I$. Then $g_1, \ldots , g_ r$ is a quasi-regular sequence by Lemma 15.29.6. By the previous paragraph we conclude that $f_1, \ldots , f_ r$ is a $H_1$-regular sequence. Similarly for Koszul-regular sequences. $\square$

reference

Lemma 15.29.16. Let $R$ be a ring. Let $a_1, \ldots , a_ n \in R$ be elements such that $R \to R^{\oplus n}$, $x \mapsto (xa_1, \ldots , xa_ n)$ is injective. Then the element $\sum a_ i t_ i$ of the polynomial ring $R[t_1, \ldots , t_ n]$ is a nonzerodivisor.

Proof. If one of the $a_ i$ is a unit this is just the statement that any element of the form $t_1 + a_2 t_2 + \ldots + a_ n t_ n$ is a nonzerodivisor in the polynomial ring over $R$.

Case I: $R$ is Noetherian. Let $\mathfrak q_ j$, $j = 1, \ldots , m$ be the associated primes of $R$. We have to show that each of the maps

\[ \sum a_ i t_ i : \text{Sym}^ d(R^{\oplus n}) \longrightarrow \text{Sym}^{d + 1}(R^{\oplus n}) \]

is injective. As $\text{Sym}^ d(R^{\oplus n})$ is a free $R$-module its associated primes are $\mathfrak q_ j$, $j = 1, \ldots , m$. For each $j$ there exists an $i = i(j)$ such that $a_ i \not\in \mathfrak q_ j$ because there exists an $x \in R$ with $\mathfrak q_ jx = 0$ but $a_ i x \not= 0$ for some $i$ by assumption. Hence $a_ i$ is a unit in $R_{\mathfrak q_ j}$ and the map is injective after localizing at $\mathfrak q_ j$. Thus the map is injective, see Algebra, Lemma 10.62.19.

Case II: $R$ general. We can write $R$ as the union of Noetherian rings $R_\lambda $ with $a_1, \ldots , a_ n \in R_\lambda $. For each $R_\lambda $ the result holds, hence the result holds for $R$. $\square$

Lemma 15.29.17. Let $R$ be a ring. Let $f_1, \ldots , f_ n$ be a Koszul-regular sequence in $R$ such that $(f_1, \ldots , f_ n) \not= R$. Consider the faithfully flat, smooth ring map

\[ R \longrightarrow S = R[\{ t_{ij}\} _{i \leq j}, t_{11}^{-1}, t_{22}^{-1}, \ldots , t_{nn}^{-1}] \]

For $1 \leq i \leq n$ set

\[ g_ i = \sum \nolimits _{i \leq j} t_{ij} f_ j \in S. \]

Then $g_1, \ldots , g_ n$ is a regular sequence in $S$ and $(f_1, \ldots , f_ n)S = (g_1, \ldots , g_ n)$.

Proof. The equality of ideals is obvious as the matrix

\[ \left( \begin{matrix} t_{11} & t_{12} & t_{13} & \ldots \\ 0 & t_{22} & t_{23} & \ldots \\ 0 & 0 & t_{33} & \ldots \\ \ldots & \ldots & \ldots & \ldots \end{matrix} \right) \]

is invertible in $S$. Because $f_1, \ldots , f_ n$ is a Koszul-regular sequence we see that the kernel of $R \to R^{\oplus n}$, $x \mapsto (xf_1, \ldots , xf_ n)$ is zero (as it computes the $n$the Koszul homology of $R$ w.r.t. $f_1, \ldots , f_ n$). Hence by Lemma 15.29.16 we see that $g_1 = f_1 t_{11} + \ldots + f_ n t_{1n}$ is a nonzerodivisor in $S' = R[t_{11}, t_{12}, \ldots , t_{1n}, t_{11}^{-1}]$. We see that $g_1, f_2, \ldots , f_ n$ is a Koszul-sequence in $S'$ by Lemma 15.29.5 and 15.29.15. We conclude that $\overline{f}_2, \ldots , \overline{f}_ n$ is a Koszul-regular sequence in $S'/(g_1)$ by Lemma 15.29.14. Hence by induction on $n$ we see that the images $\overline{g}_2, \ldots , \overline{g}_ n$ of $g_2, \ldots , g_ n$ in $S'/(g_1)[\{ t_{ij}\} _{2 \leq i \leq j}, t_{22}^{-1}, \ldots , t_{nn}^{-1}]$ form a regular sequence. This in turn means that $g_1, \ldots , g_ n$ forms a regular sequence in $S$. $\square$


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