The Stacks project

Lemma 96.10.1. Let $S$ be a locally Noetherian scheme. Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids. Let $x$ be an object of $\mathcal{X}$ lying over $\mathop{\mathrm{Spec}}(R)$ where $R$ is a Noetherian complete local ring with residue field $k$ of finite type over $S$. Let $s \in S$ be the image of $\mathop{\mathrm{Spec}}(k) \to S$. Assume that (a) $\mathcal{O}_{S, s}$ is a G-ring and (b) $p$ is limit preserving on objects. Then for every integer $N \geq 1$ there exist

  1. a finite type $S$-algebra $A$,

  2. a maximal ideal $\mathfrak m_ A \subset A$,

  3. an object $x_ A$ of $\mathcal{X}$ over $\mathop{\mathrm{Spec}}(A)$,

  4. an $S$-isomorphism $R/\mathfrak m_ R^ N \cong A/\mathfrak m_ A^ N$,

  5. an isomorphism $x|_{\mathop{\mathrm{Spec}}(R/\mathfrak m_ R^ N)} \cong x_ A|_{\mathop{\mathrm{Spec}}(A/\mathfrak m_ A^ N)}$ compatible with (4), and

  6. an isomorphism $\text{Gr}_{\mathfrak m_ R}(R) \cong \text{Gr}_{\mathfrak m_ A}(A)$ of graded $k$-algebras.

Proof. Choose an affine open $\mathop{\mathrm{Spec}}(\Lambda ) \subset S$ such that $k$ is a finite $\Lambda $-algebra, see Morphisms, Lemma 29.16.1. We may and do replace $S$ by $\mathop{\mathrm{Spec}}(\Lambda )$.

We may write $R$ as a directed colimit $R = \mathop{\mathrm{colim}}\nolimits C_ j$ where each $C_ j$ is a finite type $\Lambda $-algebra (see Algebra, Lemma 10.126.2). By assumption (b) the object $x$ is isomorphic to the restriction of an object over one of the $C_ j$. Hence we may choose a finite type $\Lambda $-algebra $C$, a $\Lambda $-algebra map $C \to R$, and an object $x_ C$ of $\mathcal{X}$ over $\mathop{\mathrm{Spec}}(C)$ such that $x = x_ C|_{\mathop{\mathrm{Spec}}(R)}$. The choice of $C$ is a bookkeeping device and could be avoided. For later use, let us write $C = \Lambda [y_1, \ldots , y_ u]/(f_1, \ldots , f_ v)$ and we denote $\overline{a}_ i \in R$ the image of $y_ i$ under the map $C \to R$. Set $\mathfrak m_ C = C \cap \mathfrak m_ R$.

Choose a $\Lambda $-algebra surjection $\Lambda [x_1, \ldots , x_ s] \to k$ and denote $\mathfrak m'$ the kernel. By the universal property of polynomial rings we may lift this to a $\Lambda $-algebra map $\Lambda [x_1, \ldots , x_ s] \to R$. We add some variables (i.e., we increase $s$ a bit) mapping to generators of $\mathfrak m_ R$. Having done this we see that $\Lambda [x_1, \ldots , x_ s] \to R/\mathfrak m_ R^2$ is surjective. Then we see that
\begin{equation} \label{artin-equation-surjection} P = \Lambda [x_1, \ldots , x_ s]_{\mathfrak m'}^\wedge \longrightarrow R \end{equation}

is a surjective map of Noetherian complete local rings, see for example Formal Deformation Theory, Lemma 88.4.2.

Choose lifts $a_ i \in P$ of $\overline{a}_ i$ we found above. Choose generators $b_1, \ldots , b_ r \in P$ for the kernel of ( Choose $c_{ji} \in P$ such that

\[ f_ j(a_1, \ldots , a_ u) = \sum c_{ji} b_ i \]

in $P$ which is possible by the choices made so far. Choose generators

\[ k_1, \ldots , k_ t \in \mathop{\mathrm{Ker}}(P^{\oplus r} \xrightarrow {(b_1, \ldots , b_ r)} P) \]

and write $k_ i = (k_{i1}, \ldots , k_{ir})$ and $K = (k_{ij})$ so that

\[ P^{\oplus t} \xrightarrow {K} P^{\oplus r} \xrightarrow {(b_1, \ldots , b_ r)} P \to R \to 0 \]

is an exact sequence of $P$-modules. In particular we have $\sum k_{ij} b_ j = 0$. After possibly increasing $N$ we may assume $N - 1$ works in the Artin-Rees lemma for the first two maps of this exact sequence (see More on Algebra, Section 15.4 for terminology).

By assumption $\mathcal{O}_{S, s} = \Lambda _{\Lambda \cap \mathfrak m'}$ is a G-ring. Hence by More on Algebra, Proposition 15.49.10 the ring $\Lambda [x_1, \ldots , x_ s]_{\mathfrak m'}$ is a $G$-ring. Hence by Smoothing Ring Maps, Theorem 16.13.2 there exist an étale ring map

\[ \Lambda [x_1, \ldots , x_ s]_{\mathfrak m'} \to B, \]

a maximal ideal $\mathfrak m_ B$ of $B$ lying over $\mathfrak m'$, and elements $a'_ i, b'_ i, c'_{ij}, k'_{ij} \in B'$ such that

  1. $\kappa (\mathfrak m') = \kappa (\mathfrak m_ B)$ which implies that $\Lambda [x_1, \ldots , x_ s]_{\mathfrak m'} \subset B_{\mathfrak m_ B} \subset P$ and $P$ is identified with the completion of $B$ at $\mathfrak m_ B$, see remark preceding Smoothing Ring Maps, Theorem 16.13.2,

  2. $a_ i - a'_ i, b_ i - b'_ i, c_{ij} - c'_{ij}, k_{ij} - k'_{ij} \in (\mathfrak m')^ N P$, and

  3. $f_ j(a'_1, \ldots , a'_ u) = \sum c'_{ji} b'_ i$ and $\sum k'_{ij}b'_ j = 0$.

Set $A = B/(b'_1, \ldots , b'_ r)$ and denote $\mathfrak m_ A$ the image of $\mathfrak m_ B$ in $A$. (Note that $A$ is essentially of finite type over $\Lambda $; at the end of the proof we will show how to obtain an $A$ which is of finite type over $\Lambda $.) There is a ring map $C \to A$ sending $y_ i \mapsto a'_ i$ because the $a'_ i$ satisfy the desired equations modulo $(b'_1, \ldots , b'_ r)$. Note that $A/\mathfrak m_ A^ N = R/\mathfrak m_ R^ N$ as quotients of $P = B^\wedge $ by property (2) above. Set $x_ A = x_ C|_{\mathop{\mathrm{Spec}}(A)}$. Since the maps

\[ C \to A \to A/\mathfrak m_ A^ N \cong R/\mathfrak m_ R^ N \quad \text{and}\quad C \to R \to R/\mathfrak m_ R^ N \]

are equal we see that $x_ A$ and $x$ agree modulo $\mathfrak m_ R^ N$ via the isomorphism $A/\mathfrak m_ A^ N = R/\mathfrak m_ R^ N$. At this point we have shown properties (1) – (5) of the statement of the lemma. To see (6) note that

\[ P^{\oplus t} \xrightarrow {K} P^{\oplus r} \xrightarrow {(b_1, \ldots , b_ r)} P \quad \text{and}\quad P^{\oplus t} \xrightarrow {K'} P^{\oplus r} \xrightarrow {(b'_1, \ldots , b'_ r)} P \]

are two complexes of $P$-modules which are congruent modulo $(\mathfrak m')^ N$ with the first one being exact. By our choice of $N$ above we see from More on Algebra, Lemma 15.4.2 that $R = P/(b_1, \ldots , b_ r)$ and $P/(b'_1, \ldots , b'_ r) = B^\wedge /(b'_1, \ldots , b'_ r) = A^\wedge $ have isomorphic associated graded algebras, which is what we wanted to show.

This last paragraph of the proof serves to clean up the issue that $A$ is essentially of finite type over $S$ and not yet of finite type. The construction above gives $A = B/(b'_1, \ldots , b'_ r)$ and $\mathfrak m_ A \subset A$ with $B$ étale over $\Lambda [x_1, \ldots , x_ s]_{\mathfrak m'}$. Hence $A$ is of finite type over the Noetherian ring $\Lambda [x_1, \ldots , x_ s]_{\mathfrak m'}$. Thus we can write $A = (A_0)_{\mathfrak m'}$ for some finite type $\Lambda [x_1, \ldots , x_ n]$ algebra $A_0$. Then $A = \mathop{\mathrm{colim}}\nolimits (A_0)_ f$ where $f \in \Lambda [x_1, \ldots , x_ n] \setminus \mathfrak m'$, see Algebra, Lemma 10.9.9. Because $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ is limit preserving on objects, we see that $x_ A$ comes from some object $x_{(A_0)_ f}$ over $\mathop{\mathrm{Spec}}((A_0)_ f)$ for an $f$ as above. After replacing $A$ by $(A_0)_ f$ and $x_ A$ by $x_{(A_0)_ f}$ and $\mathfrak m_ A$ by $(A_0)_ f \cap \mathfrak m_ A$ the proof is finished. $\square$

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