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*}

16.9 Local tricks

Situation 16.9.1. We are given a Noetherian ring $R$ and an $R$-algebra map $A \to \Lambda $ and a prime $\mathfrak q \subset \Lambda $. We assume $A$ is of finite presentation over $R$. In this situation we denote $\mathfrak h_ A = \sqrt{H_{A/R} \Lambda }$.

Let $R \to A \to \Lambda \supset \mathfrak q$ be as in Situation 16.9.1. We say $R \to A \to \Lambda \supset \mathfrak q$ can be resolved if there exists a factorization $A \to B \to \Lambda $ with $B$ of finite presentation and $\mathfrak h_ A \subset \mathfrak h_ B \not\subset \mathfrak q$. In this case we will call the factorization $A \to B \to \Lambda $ a resolution of $R \to A \to \Lambda \supset \mathfrak q$.

Lemma 16.9.2. Let $R \to A \to \Lambda \supset \mathfrak q$ be as in Situation 16.9.1. Let $r \geq 1$ and $\pi _1, \ldots , \pi _ r \in R$ map to elements of $\mathfrak q$. Assume

  1. for $i = 1, \ldots , r$ we have

    \[ \text{Ann}_{R/(\pi _1^8, \ldots , \pi _{i - 1}^8)R}(\pi _ i) = \text{Ann}_{R/(\pi _1^8, \ldots , \pi _{i - 1}^8)R}(\pi _ i^2) \]

    and

    \[ \text{Ann}_{\Lambda /(\pi _1^8, \ldots , \pi _{i - 1}^8)\Lambda }(\pi _ i) = \text{Ann}_{\Lambda /(\pi _1^8, \ldots , \pi _{i - 1}^8)\Lambda }(\pi _ i^2) \]
  2. for $i = 1, \ldots , r$ the element $\pi _ i$ maps to a strictly standard element in $A$ over $R$.

Then, if

\[ R/(\pi _1^8, \ldots , \pi _ r^8)R \to A/(\pi _1^8, \ldots , \pi _ r^8)A \to \Lambda /(\pi _1^8, \ldots , \pi _ r^8)\Lambda \supset \mathfrak q/(\pi _1^8, \ldots , \pi _ r^8)\Lambda \]

can be resolved, so can $R \to A \to \Lambda \supset \mathfrak q$.

Proof. We are going to prove this by induction on $r$.

The case $r = 1$. Here the assumption is that there exists a factorization $A/\pi _1^8 \to \bar C \to \Lambda /\pi _1^8$ which resolves the situation modulo $\pi _1^8$. Conditions (1) and (2) are the assumptions needed to apply Lemma 16.7.3. Thus we can “lift” the resolution $\bar C$ to a resolution of $R \to A \to \Lambda \supset \mathfrak q$.

The case $r > 1$. In this case we apply the induction hypothesis for $r - 1$ to the situation $R/\pi _1^8 \to A/\pi _1^8 \to \Lambda /\pi _1^8 \supset \mathfrak q/\pi _1^8\Lambda $. Note that property (2) is preserved by Lemma 16.2.7. $\square$

Lemma 16.9.3. Let $R \to A \to \Lambda \supset \mathfrak q$ be as in Situation 16.9.1. Let $\mathfrak p = R \cap \mathfrak q$. Assume that $\mathfrak q$ is minimal over $\mathfrak h_ A$ and that $R_\mathfrak p \to A_\mathfrak p \to \Lambda _\mathfrak q \supset \mathfrak q\Lambda _\mathfrak q$ can be resolved. Then there exists a factorization $A \to C \to \Lambda $ with $C$ of finite presentation such that $H_{C/R} \Lambda \not\subset \mathfrak q$.

Proof. Let $A_\mathfrak p \to C \to \Lambda _\mathfrak q$ be a resolution of $R_\mathfrak p \to A_\mathfrak p \to \Lambda _\mathfrak q \supset \mathfrak q\Lambda _\mathfrak q$. By our assumption that $\mathfrak q$ is minimal over $\mathfrak h_ A$ this means that $H_{C/R_\mathfrak p} \Lambda _\mathfrak q = \Lambda _\mathfrak q$. By Lemma 16.2.8 we may assume that $C$ is smooth over $\Lambda _\mathfrak p$. By Lemma 16.3.4 we may assume that $C$ is standard smooth over $R_\mathfrak p$. Write $A = R[x_1, \ldots , x_ n]/(g_1, \ldots , g_ t)$ and say $A \to \Lambda $ is given by $x_ i \mapsto \lambda _ i$. Write $C = R_\mathfrak p[x_1, \ldots , x_{n + m}]/(f_1, \ldots , f_ c)$ for some $c \geq n$ such that $A \to C$ maps $x_ i$ to $x_ i$ and such that $\det (\partial f_ j/\partial x_ i)_{i, j = 1, \ldots , c}$ is invertible in $C$, see Lemma 16.3.6. After clearing denominators we may assume $f_1, \ldots , f_ c$ are elements of $R[x_1, \ldots , x_{n + m}]$. Of course $\det (\partial f_ j/\partial x_ i)_{i, j = 1, \ldots , c}$ is not invertible in $R[x_1, \ldots , x_{n + m}]/(f_1, \ldots , f_ c)$ but it becomes invertible after inverting some element $s_0 \in R$, $s_0 \not\in \mathfrak p$. As $g_ j$ maps to zero under $R[x_1, \ldots , x_ n] \to A \to C$ we can find $s_ j \in R$, $s_ j \not\in \mathfrak p$ such that $s_ j g_ j$ is zero in $R[x_1, \ldots , x_{n + m}]/(f_1, \ldots , f_ c)$. Write $f_ j = F_ j(x_1, \ldots , x_{n + m}, 1)$ for some polynomial $F_ j \in R[x_1, \ldots , x_ n, X_{n + 1}, \ldots , X_{n + m + 1}]$ homogeneous in $X_{n + 1}, \ldots , X_{n + m + 1}$. Pick $\lambda _{n + i} \in \Lambda $, $i = 1, \ldots , m + 1$ with $\lambda _{n + m + 1} \not\in \mathfrak q$ such that $x_{n + i}$ maps to $\lambda _{n + i}/\lambda _{n + m + 1}$ in $\Lambda _\mathfrak q$. Then

\begin{align*} F_ j(\lambda _1, \ldots , \lambda _{n + m + 1}) & = (\lambda _{n + m + 1})^{\deg (F_ j)} F_ j(\lambda _1, \ldots , \lambda _ n, \frac{\lambda _{n + 1}}{\lambda _{n + m + 1}}, \ldots , \frac{\lambda _{n + m}}{\lambda _{n + m + 1}}, 1) \\ & = (\lambda _{n + m + 1})^{\deg (F_ j)} f_ j(\lambda _1, \ldots , \lambda _ n, \frac{\lambda _{n + 1}}{\lambda _{n + m + 1}}, \ldots , \frac{\lambda _{n + m}}{\lambda _{n + m + 1}}) \\ & = 0 \end{align*}

in $\Lambda _\mathfrak q$. Thus we can find $\lambda _0 \in \Lambda $, $\lambda _0 \not\in \mathfrak q$ such that $\lambda _0 F_ j(\lambda _1, \ldots , \lambda _{n + m + 1}) = 0$ in $\Lambda $. Now we set $B$ equal to

\[ R[x_0, \ldots , x_{n + m + 1}]/ (g_1, \ldots , g_ t, x_0F_1(x_1, \ldots , x_{n + m + 1}), \ldots , x_0F_ c(x_1, \ldots , x_{n + m + 1})) \]

which we map to $\Lambda $ by mapping $x_ i$ to $\lambda _ i$. Let $b$ be the image of $x_0 x_1 s_0 s_1 \ldots s_ t$ in $B$. Then $B_ b$ is isomorphic to

\[ R_{s_0s_1}[x_0, x_1, \ldots , x_{n + m + 1}, 1/x_0x_{n + m + 1}]/ (f_1, \ldots , f_ c) \]

which is smooth over $R$ by construction. Since $b$ does not map to an element of $\mathfrak q$, we win. $\square$

Lemma 16.9.4. Let $R \to A \to \Lambda \supset \mathfrak q$ be as in Situation 16.9.1. Let $\mathfrak p = R \cap \mathfrak q$. Assume

  1. $\mathfrak q$ is minimal over $\mathfrak h_ A$,

  2. $R_\mathfrak p \to A_\mathfrak p \to \Lambda _\mathfrak q \supset \mathfrak q\Lambda _\mathfrak q$ can be resolved, and

  3. $\dim (\Lambda _\mathfrak q) = 0$.

Then $R \to A \to \Lambda \supset \mathfrak q$ can be resolved.

Proof. By (3) the ring $\Lambda _\mathfrak q$ is Artinian local hence $\mathfrak q\Lambda _\mathfrak q$ is nilpotent. Thus $(\mathfrak h_ A)^ N \Lambda _\mathfrak q = 0$ for some $N > 0$. Thus there exists a $\lambda \in \Lambda $, $\lambda \not\in \mathfrak q$ such that $\lambda (\mathfrak h_ A)^ N = 0$ in $\Lambda $. Say $H_{A/R} = (a_1, \ldots , a_ r)$ so that $\lambda a_ i^ N = 0$ in $\Lambda $. By Lemma 16.9.3 we can find a factorization $A \to C \to \Lambda $ with $C$ of finite presentation such that $\mathfrak h_ C \not\subset \mathfrak q$. Write $C = A[x_1, \ldots , x_ n]/(f_1, \ldots , f_ m)$. Set

\[ B = A[x_1, \ldots , x_ n, y_1, \ldots , y_ r, z, t_{ij}]/ (f_ j - \sum y_ i t_{ij}, zy_ i) \]

where $t_{ij}$ is a set of $rm$ variables. Note that there is a map $B \to C[y_ i, z]/(y_ iz)$ given by setting $t_{ij}$ equal to zero. The map $B \to \Lambda $ is the composition $B \to C[y_ i, z]/(y_ iz) \to \Lambda $ where $C[y_ i, z]/(y_ iz) \to \Lambda $ is the given map $C \to \Lambda $, maps $z$ to $\lambda $, and maps $y_ i$ to the image of $a_ i^ N$ in $\Lambda $.

We claim that $B$ is a solution for $R \to A \to \Lambda \supset \mathfrak q$. First note that $B_ z$ is isomorphic to $C[y_1, \ldots , y_ r, z, z^{-1}]$ and hence is smooth. On the other hand, $B_{y_\ell } \cong A[x_ i, y_ i, y_\ell ^{-1}, t_{ij}, i \not= \ell ]$ which is smooth over $A$. Thus we see that $z$ and $a_\ell y_\ell $ (compositions of smooth maps are smooth) are all elements of $H_{B/R}$. This proves the lemma. $\square$


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