The Stacks project

Proof. Write $S = R[x_1, \ldots , x_ n]/J$ and denote $N = S \oplus M$ viewed as a module over $P = R[x_1, \ldots , x_ n]$. If we can prove the lemma in case $S$ is a polynomial algebra over $R$, then we can find $I, a$ satisfying (1), (2), (3) such that the strict transform $N_{I, a}$ of $N$ along $R \to R[\frac{I}{a}]$ is flat over $R$ and finitely presented as a module over the strict transform $P_{I, a}]$ of $P$. Since $P_{I, a} = R[\frac{I}{a}][x_1, \ldots , x_ n]$ (small detail omitted) we find that the summand $S_{I, a} \subset N_{I, a}$ is flat over $R$ and finitely presented as a module over $R[\frac{I}{a}][x_1, \ldots , x_ n]$. Hence $S_{I, a}$ is finitely presented as an $R[\frac{I}{a}]$-algebra. Moreover, the summand $M_{I, a} \subset N_{I, a}$ is flat over $R$ and finitely presented as a module over $P_{I, a}$ hence also finitely presented as a module over $S_{I, a}$, see Algebra, Lemma 10.6.4. This reduces us to the case discussed in the next paragraph.

Assume $S = R[x_1, \ldots , x_ n]$. Choose a presentation

Let $M_ A$ be the quotient of $M \otimes _ R A$ by its torsion submodule, see Lemma 15.22.2. Then $M_ A$ is a finite module over $S_ A = A[x_1, \ldots , x_ n]$. By Lemma 15.22.10 we see that $M_ A$ is flat over $A$. By Lemma 15.25.6 we see that $M_ A$ is finitely presented. Hence there exist finitely many elements $k_1, \ldots , k_ t \in S_ A^{\oplus r}$ which generate the kernel of the presentation $S_ A^{\oplus r} \to M_ A$ as an $S_ A$-module. For any choice of $a \in I \subset \mathfrak m$ satisfying (1), (2), and (3) we denote $M_{I, a}$ the strict transform of $M$ along $R \to R[\frac{I}{a}]$. It is a finite module over $S_{I, a} = R[\frac{I}{a}][x_1, \ldots , x_ n]$. By Algebra, Lemma 10.70.12 we have $A = \mathop{\mathrm{colim}}\nolimits _{I, a} R[\frac{I}{a}]$. This implies that $S_ A = \mathop{\mathrm{colim}}\nolimits S_{I, a}$ and

Choose $I, a$ and lifts $k_1, \ldots , k_ t \in S_{I, a}^{\oplus r}$. Since $M_ A$ is the quotient of $M \otimes _ R A$ by torsion, we see that the images of $k_1, \ldots , k_ t$ in $M \otimes _ R A$ are annihilated by a nonzero element $\alpha \in A$. After replacing $I, a$ by a different pair (recall that the colimit is filtered), we may assume $\alpha = x/a^ n$ for some $x \in I^ n$ nonzero. Then we find that $x k_1, \ldots , x k_ t$ map to zero in $M \otimes _ R A$. Hence after replacing $I, a$ by a different pair we may assume $x k_1, \ldots , x k_ t$ map to zero in $M \otimes _ R R[\frac{I}{a}]$ for some nonzero $x \in R$. Then finally replacing $I, a$ by $xI, xa$ we find that we may assume $k_1, \ldots , k_ t$ map to $a$-power torsion elements of $M \otimes _ R R[\frac{I}{a}]$. For any such pair $(I, a)$ we set

Since $M_ A = S_ A^{\oplus r}/ \sum S_ Ak_ j$ we see that $M_ A = \mathop{\mathrm{colim}}\nolimits _{I, a} M'_{I, a}$. At this point we finally apply Algebra, Lemma 10.168.1 (3) to conclude that $M'_{I, a}$ is flat for some pair $(I, a)$ as above. This lemma does not apply a priori to the system of strict transforms

as the transition maps may not satisfy the assumptions of the lemma. But now, since flatness implies torsion free (Lemma 15.22.9) and since $M_{I, a}$ is the quotient of $M'_{I, a}$ (because we arranged it so the elements $k_1, \ldots , k_ t$ map to zero in $M_{I, a}$) by the $a$-power torsion submodule we also conclude that $M'_{I, a} = M_{I, a}$ for such a pair and we win. $\square$

Proof. First observe that $\text{Fit}_ k(M \otimes _ R R') = IR' = aR'$. The first equality by Lemma 15.8.4 part (3) and the second equality by Algebra, Lemma 10.70.2. From Lemma 15.8.9 and exactness of localization we see that $M'_{\mathfrak p'}$ can be generated by $\leq k$ elements for every prime $\mathfrak p'$ of $R'$. Then $\text{Fit}_ k(M') = R'$ for example by Lemma 15.8.7. $\square$

Proof. By Lemma 15.26.3 we have $\text{Fit}_ k(M') = R'$. By Lemma 15.8.8 it suffices to show that $\text{Fit}_{k - 1}(M') = 0$. Recall that $R' \subset R'_ a = R_ a$, see Algebra, Lemma 10.70.2. Hence it suffices to prove that $\text{Fit}_{k - 1}(M')$ maps to zero in $R'_ a = R_ a$. Since clearly $(M')_ a = M_ a$ this reduces us to showing that $\text{Fit}_{k - 1}(M_ a) = 0$ because formation of Fitting ideals commutes with base change according to Lemma 15.8.4 part (3). This is true by our assumption that $M_ a$ is finite locally free of rank $k$ (see Algebra, Lemma 10.78.2) and the already cited Lemma 15.8.8. $\square$

Proof. Choose a surjection $R^{\oplus n} \to M$. Choose a finite submodule $K \subset \mathop{\mathrm{Ker}}(R^{\oplus n} \to M)$ such that $R^{\oplus n}/K \to M$ becomes an isomorphism after inverting $f$. This is possible because $M_ f$ is of finite presentation for example by Algebra, Lemma 10.78.2. Set $M_1 = R^{\oplus n}/K$ and suppose we can prove the lemma for $M_1$. Say $I \subset R$ is the corresponding ideal. Then for $a \in I$ the map

is surjective. It is also an isomorphism after inverting $a$ in $R'$ as $R'_ a = R_ f$, see Algebra, Lemma 10.70.7. But $a$ is a nonzerodivisor on $M'_1$, whence the displayed map is an isomorphism. Thus it suffices to prove the lemma in case $M$ is a finitely presented $R$-module.

Assume $M$ is a finitely presented $R$-module. Then $J = \text{Fit}_ r(M) \subset S$ is a finitely generated ideal. We claim that $I = fJ$ works.

We first check that $V(f) = V(I)$. The inclusion $V(f) \subset V(I)$ is clear. Conversely, if $f \not\in \mathfrak p$, then $\mathfrak p$ is not an element of $V(J)$ by Lemma 15.8.7. Thus $\mathfrak p \not\in V(fJ) = V(I)$.

Let $a \in I$ and set $R' = R[\frac{I}{a}]$. We may write $a = fb$ for some $b \in J$. By Algebra, Lemmas 10.70.2 and 10.70.8 we see that $J R' = b R'$ and $b$ is a nonzerodivisor in $R'$. Let $\mathfrak p' \subset R' = R[\frac{I}{a}]$ be a prime ideal. Then $JR'_{\mathfrak p'}$ is generated by $b$. It follows from Lemma 15.8.9 that $M'_{\mathfrak p'}$ can be generated by $r$ elements. Since $M'$ is finite, there exist $m_1, \ldots , m_ r \in M'$ and $g \in R'$, $g \not\in \mathfrak p'$ such that the corresponding map $(R')^{\oplus r} \to M'$ becomes surjective after inverting $g$.

Finally, consider the ideal $J' = \text{Fit}_{k - 1}(M')$. Note that $J' R'_ g$ is generated by the coefficients of relations between $m_1, \ldots , m_ r$ (compatibility of Fitting ideal with base change). Thus it suffices to show that $J' = 0$, see Lemma 15.8.8. Since $R'_ a = R_ f$ (Algebra, Lemma 10.70.7) and $M'_ a = M_ f$ is free of rank $r$ we see that $J'_ a = 0$. Since $a$ is a nonzerodivisor in $R'$ we conclude that $J' = 0$ and we win. $\square$