The Stacks project

Lemma 86.10.2. Let $A$ be a Noetherian ring. Let $I \subset A$ be an ideal. Let $B$ be an object of ( which is rig-étale over $(A, I)$. Then there exists a finite type $A$-algebra $C$ and an isomorphism $B \cong C^\wedge $.

Proof. We prove this lemma by induction on the number of generators of $I$. Say $I = (a_1, \ldots , a_ t)$. If $t = 0$, then $I = 0$ and there is nothing to prove. If $t = 1$, then the lemma follows from Lemma 86.10.1. Assume $t > 1$.

For any $m \geq 1$ set $\bar A_ m = A/(a_ t^ m)$. Consider the ideal $\bar I_ m = (\bar a_1, \ldots , \bar a_{t - 1})$ in $\bar A_ m$. Observe that $V(I \bar A_ m) = V(\bar I_ m)$. Let $B_ m = B/(a_ t^ m)$ be the base change of $B$ for the map $(A, I) \to (\bar A_ m, \bar I_ m)$, see Remark 86.2.4. By Lemma 86.8.6 we find that $B_ m$ is rig-étale over $(\bar A_ m, \bar I_ m)$.

By induction hypothesis (on $t$) we can find a finite type $\bar A_ m$-algebra $C_ m$ and a map $C_ m \to B_ m$ which induces an isomorphism $C_ m^\wedge \cong B_ m$ where the completion is with respect to $\bar I_ m$. By Lemma 86.8.4 we may assume that $\mathop{\mathrm{Spec}}(C_ m) \to \mathop{\mathrm{Spec}}(\bar A_ m)$ is étale over $\mathop{\mathrm{Spec}}(\bar A_ m) \setminus V(\bar I_ m)$.

We claim that we may choose $A_ m \to C_ m \to B_ m$ as in the previous paragraph such that moreover there are isomorphisms $C_ m/(a_ t^{m - 1}) \to C_{m - 1}$ compatible with the given $A$-algebra structure and the maps to $B_{m - 1} = B_ m/(a_ t^{m - 1})$. Namely, first fix a choice of $A_1 \to C_1 \to B_1$. Suppose we have found $C_{m - 1} \to C_{m - 2} \to \ldots \to C_1$ with the desired properties. Note that $C_ m/(a_ t^{m - 1})$ is étale over $\mathop{\mathrm{Spec}}(\bar A_{m - 1}) \setminus V(\bar I_{m - 1})$. Hence by Lemma 86.8.7 there exists an étale extension $C_{m - 1} \to C'_{m - 1}$ which induces an isomorphism modulo $\bar I_{m - 1}$ and an $\bar A_{m - 1}$-algebra map $C_ m/(a_ t^{m - 1}) \to C'_{m - 1}$ inducing the isomorphism $B_ m/(a_ t^{m - 1}) \to B_{m - 1}$ on completions. Note that $C_ m/(a_ t^{m - 1}) \to C'_{m - 1}$ is étale over the complement of $V(\bar I_{m - 1})$ by Morphisms, Lemma 29.36.18 and over $V(\bar I_{m - 1})$ induces an isomorphism on completions hence is étale there too (for example by More on Morphisms, Lemma 37.12.3). Thus $C_ m/(a_ t^{m - 1}) \to C'_{m - 1}$ is étale. By the topological invariance of étale morphisms (Étale Morphisms, Theorem 41.15.2) there exists an étale ring map $C_ m \to C'_ m$ such that $C_ m/(a_ t^{m - 1}) \to C'_{m - 1}$ is isomorphic to $C_ m/(a_ t^{m - 1}) \to C'_ m/(a_ t^{m - 1})$. Observe that the $\bar I_ m$-adic completion of $C'_ m$ is equal to the $\bar I_ m$-adic completion of $C_ m$, i.e., to $B_ m$ (details omitted). We apply Lemma 86.9.1 to the diagram

\[ \xymatrix{ & C'_ m \ar[r] & C'_ m/(a_ t^{m - 1}) \\ C''_ m \ar@{..>}[ru] \ar@{..>}[rr] & & C_{m - 1} \ar[u] \\ & \bar A_ m \ar[r] \ar[uu] \ar@{..>}[lu] & \bar A_{m - 1} \ar[u] } \]

to see that there exists a “lift” of $C''_ m$ of $C_{m - 1}$ to an algebra over $\bar A_ m$ with all the desired properties.

By construction $(C_ m)$ is an object of the category ( for the principal ideal $(a_ t)$. Thus the inverse limit $B' = \mathop{\mathrm{lim}}\nolimits C_ m$ is an $(a_ t)$-adically complete $A$-algebra such that $B'/a_ t B'$ is of finite type over $A/(a_ t)$, see Lemma 86.2.1. By construction the $I$-adic completion of $B'$ is isomorphic to $B$ (details omitted). Consider the complex $\mathop{N\! L}\nolimits _{B'/A}^\wedge $ constructed using the $(a_ t)$-adic topology. Choosing a presentation for $B'$ (which induces a similar presentation for $B$) the reader immediately sees that $\mathop{N\! L}\nolimits _{B'/A}^\wedge \otimes _{B'} B = \mathop{N\! L}\nolimits _{B/A}^\wedge $. Since $a_ t \in I$ and since the cohomology modules of $\mathop{N\! L}\nolimits _{B'/A}^\wedge $ are finite $B'$-modules (hence complete for the $a_ t$-adic topology), we conclude that $a_ t^ c$ acts as zero on these cohomologies as the same thing is true by assumption for $\mathop{N\! L}\nolimits _{B/A}^\wedge $. Thus $B'$ is rig-étale over $(A, (a_ t))$ by Lemma 86.8.2. Hence finally, we may apply Lemma 86.10.1 to $B'$ over $(A, (a_ t))$ to finish the proof. $\square$

Proof of Lemma 86.7.3 in case $A$ is a G-ring. This proof is easier in that it does not depend on the somewhat delicate deformation theory argument given in the proof of Lemma 86.7.2, but of course it requires a very strong assumption on the Noetherian ring $A$.

Choose a presentation $B = A[x_1, \ldots , x_ r]^\wedge /J$. Choose generators $g_1, \ldots , g_ m \in J$. Choose generators $k_1, \ldots , k_ t$ of the module of relations between $g_1, \ldots , g_ m$, i.e., such that

\[ (A[x_1, \ldots , x_ r]^\wedge )^{\oplus t} \xrightarrow {k_1, \ldots , k_ t} (A[x_1, \ldots , x_ r]^\wedge )^{\oplus m} \xrightarrow {g_1, \ldots , g_ m} A[x_1, \ldots , x_ r]^\wedge \]

is exact in the middle. Write $k_ i = (k_{i1}, \ldots , k_{im})$ so that we have
\begin{equation} \label{restricted-equation-relations-straight-up} \sum k_{ij}g_ j = 0 \end{equation}

for $i = 1, \ldots , t$. Let $I^ c = (a_1, \ldots , a_ s)$. For each $l \in \{ 1, \ldots , s\} $ we know that multiplication by $a_ l$ on $\mathop{N\! L}\nolimits ^\wedge _{B/A}$ is zero in $D(B)$. By Lemma 86.3.4 we can find a map $\alpha _ l : \bigoplus B\text{d}x_ i \to J/J^2$ such that $\text{d} \circ \alpha _ l$ and $\alpha _ l \circ \text{d}$ are both multiplication by $a_ l$. Pick an element $f_{l, i} \in J$ whose class modulo $J^2$ is equal to $\alpha _ l(\text{d}x_ i)$. Then we have for all $l = 1, \ldots , s$ and $i = 1, \ldots , r$ that
\begin{equation} \label{restricted-equation-derivatives} \sum \nolimits _{i'} (\partial f_{l, i}/ \partial x_{i'}) \text{d}x_{i'} = a_ l \text{d}x_ i + \sum h_{l, i}^{j', i'} g_{j'} \text{d}x_{i'} \end{equation}

for some $h_{l, i}^{j', i'} \in A[x_1, \ldots , x_ r]^\wedge $. We also have for $j = 1, \ldots , m$ and $l = 1, \ldots , s$ that
\begin{equation} \label{restricted-equation-ci} a_ l g_ j = \sum h_{l, j}^ if_{l, i} + \sum h_{l, j}^{j', j''}g_{j'} g_{j''} \end{equation}

for some $h_{l, j}^ i$ and $h_{l, j}^{j', j''}$ in $A[x_1, \ldots , x_ r]^\wedge $. Of course, since $f_{l, i} \in J$ we can write for $l = 1, \ldots , s$ and $i = 1, \ldots , r$
\begin{equation} \label{restricted-equation-in-ideal} f_{l, i} = \sum h_{l, i}^ jg_ j \end{equation}

for some $h_{l, i}^ j$ in $A[x_1, \ldots , x_ r]^\wedge $.

Let $A[x_1, \ldots , x_ r]^ h$ be the henselization of the pair $(A[x_1, \ldots , x_ r], IA[x_1, \ldots , x_ r])$, see More on Algebra, Lemma 15.12.1. Since $A$ is a Noetherian G-ring, so is $A[x_1, \ldots , x_ r]$, see More on Algebra, Proposition 15.50.10. Hence we have approximation for the map $A[x_1, \ldots , x_ r]^ h \to A[x_1, \ldots , x_ r]^\wedge $ with respect to the ideal generated by $I$, see Smoothing Ring Maps, Lemma 16.14.1. Choose a large integer $M$. Choose

\[ G_ j, K_{ij}, F_{l, i}, H_{l, j}^ i, H_{l, j}^{j', j''}, H_{l, i}^ j \in A[x_1, \ldots , x_ r]^ h \]

such that analogues of equations (, (, and ( hold for these elements in $A[x_1, \ldots , x_ r]^ h$, i.e.,

\[ \sum K_{ij}G_ j = 0,\quad a_ l G_ j = \sum H_{l, j}^ iF_{l, i} + \sum H_{l, j}^{j', j''} G_{j'} G_{j''},\quad F_{l, i} = \sum H_{l, i}^ j G_ j \]

and such that we have

\[ G_ j - g_ j, K_{ij} - k_{ij}, F_{l, i} - f_{l, i}, H_{l, j}^ i - h_{l, j}^ i, H_{l, j}^{j', j''} - h_{l, j}^{j', j''}, H_{l, i}^ j - h_{l, i}^ j \in I^ M A[x_1, \ldots , x_ r]^ h \]

where we take liberty of thinking of $A[x_1, \ldots , x_ r]^ h$ as a subring of $A[x_1, \ldots , x_ r]^\wedge $. Note that we cannot guarantee that the analogue of ( holds in $A[x_1, \ldots , x_ r]^ h$, because it is not a polynomial equation. But since taking partial derivatives is $A$-linear, we do get the analogue modulo $I^ M$. More precisely, we see that
\begin{equation} \label{restricted-equation-derivatives-analogue} \sum \nolimits _{i'} (\partial F_{l, i}/ \partial x_{i'}) \text{d}x_{i'} - a_ l \text{d}x_ i - \sum h_{l, i}^{j', i'} G_{j'} \text{d}x_{i'} \in I^ MA[x_1, \ldots , x_ r]^\wedge \end{equation}

for $l = 1, \ldots , s$ and $i = 1, \ldots , r$.

With these choices, consider the ring

\[ C^ h = A[x_1, \ldots , x_ r]^ h/(G_1, \ldots , G_ r) \]

and denote $C^\wedge $ its $I$-adic completion, namely

\[ C^\wedge = A[x_1, \ldots , x_ r]^\wedge /J',\quad J' = (G_1, \ldots , G_ r)A[x_1, \ldots , x_ r]^\wedge \]

In the following paragraphs we establish the fact that $C^\wedge $ is isomorphic to $B$. Then in the final paragraph we deal with show that $C^ h$ comes from a finite type algebra over $A$ as in the statement of the lemma.

First consider the cokernel

\[ \Omega = \mathop{\mathrm{Coker}}(J'/(J')^2 \longrightarrow \bigoplus C^\wedge \text{d}x_ i) \]

This $C^\wedge $ module is generated by the images of the elements $\text{d}x_ i$. Since $F_{l, i} \in J'$ by the analogue of ( we see from ( we see that $a_ l \text{d}x_ i \in I^ M\Omega $. As $I^ c = (a_ l)$ we see that $I^ c \Omega \subset I^ M \Omega $. Since $M > c$ we conclude that $I^ c \Omega = 0$ by Algebra, Lemma 10.19.1.

Next, consider the kernel

\[ H_1 = \mathop{\mathrm{Ker}}(J'/(J')^2 \longrightarrow \bigoplus C^\wedge \text{d}x_ i) \]

By the analogue of ( we see that $a_ l J' \subset (F_{l, i}) + (J')^2$. On the other hand, the determinant $\Delta _ l$ of the matrix $(\partial F_{l, i}/ \partial x_{i'})$ satisfies $\Delta _ l = a_ l^ r \bmod I^ M C^\wedge $ by ( It follows that $a_ l^{r + 1} H_1 \subset I^ M H_1$ (some details omitted; use Algebra, Lemma 10.14.5). Now $(a_1^{r + 1}, \ldots , a_ s^{r + 1}) \supset I^{(sr + 1)c}$. Hence $I^{(sr + 1)c}H_1 \subset I^ M H_1$ and since $M > (sr + 1)c$ we conclude that $I^{(sr + 1)c}H_1 = 0$.

By Lemma 86.3.5 we conclude that multiplication by an element of $I^{2(sr + 1)c}$ on $\mathop{N\! L}\nolimits ^\wedge _{C^\wedge /A}$ is zero (note that the bound does not depend on $M$ or the choice of the approximation, as long as $M$ is large enough). Since $G_ j - g_ j$ is in the ideal generated by $I^ M$ we see that there is an isomorphism

\[ \psi _ M : C^\wedge /I^ MC^\wedge \to B/I^ MB \]

As $M$ is large enough we can use Lemma 86.7.1 with $d = d(I \subset A \to B)$, with $C^\wedge $ playing the role of $B$, with $2(rs + 1)c$ instead of $c$, to find a morphism

\[ \psi : C^\wedge \longrightarrow B \]

which agrees with $\psi _ M$ modulo $I^{q - 2(rs + 1)c}$ where $q$ is the quotient of $M$ by the number of generators of $I$. We claim $\psi $ is an isomorphism. Since $C^\wedge $ and $B$ are $I$-adically complete the map $\psi $ is surjective because it is surjective modulo $I$ (see Algebra, Lemma 10.95.1). On the other hand, as $M$ is large enough we see that

\[ \text{Gr}_ I(C^\wedge ) \cong \text{Gr}_ I(B) \]

as graded $\text{Gr}_ I(A[x_1, \ldots , x_ r]^\wedge )$-modules by More on Algebra, Lemma 15.4.2. Since $\psi $ is compatible with this isomorphism as it agrees with $\psi _ M$ modulo $I$, this means that $\text{Gr}_ I(\psi )$ is an isomorphism. As $C^\wedge $ and $B$ are $I$-adically complete, it follows that $\psi $ is an isomorphism.

This paragraph serves to deal with the issue that $C^ h$ is not of finite type over $A$. Namely, the ring $A[x_1, \ldots , x_ r]^ h$ is a filtered colimit of étale $A[x_1, \ldots , x_ r]$ algebras $A'$ such that $A/I[x_1, \ldots , x_ r] \to A'/IA'$ is an isomorphism (see proof of More on Algebra, Lemma 15.12.1). Pick an $A'$ such that $G_1, \ldots , G_ m$ are the images of $G'_1, \ldots , G'_ m \in A'$. Setting $C = A'/(G'_1, \ldots , G'_ m)$ we get the finite type algebra we were looking for. $\square$

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