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\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*}

Lemma 10.135.7. Let $S = R[x_1, \ldots , x_ n]/(f_1, \ldots , f_ c) = R[x_1, \ldots , x_ n]/I$ be a standard smooth algebra. Then

  1. the ring map $R \to S$ is smooth,

  2. the $S$-module $\Omega _{S/R}$ is free on $\text{d}x_{c + 1}, \ldots , \text{d}x_ n$,

  3. the $S$-module $I/I^2$ is free on the classes of $f_1, \ldots , f_ c$,

  4. for any $g \in S$ the ring map $R \to S_ g$ is standard smooth,

  5. for any ring map $R \to R'$ the base change $R' \to R'\otimes _ R S$ is standard smooth,

  6. if $f \in R$ maps to an invertible element in $S$, then $R_ f \to S$ is standard smooth, and

  7. the ring $S$ is a relative global complete intersection over $R$.

Proof. Consider the naive cotangent complex of the given presentation

\[ (f_1, \ldots , f_ c)/(f_1, \ldots , f_ c)^2 \longrightarrow \bigoplus \nolimits _{i = 1}^ n S \text{d}x_ i \]

Let us compose this map with the projection onto the first $c$ direct summands of the direct sum. According to the definition of a standard smooth algebra the classes $f_ i \bmod (f_1, \ldots , f_ c)^2$ map to a basis of $\bigoplus _{i = 1}^ c S\text{d}x_ i$. We conclude that $(f_1, \ldots , f_ c)/(f_1, \ldots , f_ c)^2$ is free of rank $c$ with a basis given by the elements $f_ i \bmod (f_1, \ldots , f_ c)^2$, and that the homology in degree $0$, i.e., $\Omega _{S/R}$, of the naive cotangent complex is a free $S$-module with basis the images of $\text{d}x_{c + j}$, $j = 1, \ldots , n - c$. In particular, this proves $R \to S$ is smooth.

The proofs of (4) and (6) are omitted. But see the example below and the proof of Lemma 10.134.10.

Let $\varphi : R \to R'$ be any ring map. Denote $S' = R'[x_1, \ldots , x_ n]/(f_1^\varphi , \ldots , f_ c^\varphi )$ where $f^\varphi $ is the polynomial obtained from $f \in R[x_1, \ldots , x_ n]$ by applying $\varphi $ to all the coefficients. Then $S' \cong R' \otimes _ R S$. Moreover, the determinant of Definition 10.135.6 for $S'/R'$ is equal to $g^\varphi $. Its image in $S'$ is therefore the image of $g$ via $R[x_1, \ldots , x_ n] \to S \to S'$ and hence invertible. This proves (5).

To prove (7) it suffices to show that $S \otimes _ R \kappa (\mathfrak p)$ has dimension $n - c$ for every prime $\mathfrak p \subset R$. By (5) it suffices to prove that any standard smooth algebra $k[x_1, \ldots , x_ n]/(f_1, \ldots , f_ c)$ over a field $k$ has dimension $n - c$. We already know that $k[x_1, \ldots , x_ n]/(f_1, \ldots , f_ c)$ is a local complete intersection by Lemma 10.135.5. Hence, since $I/I^2$ is free of rank $c$ we see that $k[x_1, \ldots , x_ n]/(f_1, \ldots , f_ c)$ has dimension $n - c$, by Lemma 10.133.4 for example. $\square$


Comments (2)

Comment #2824 by Dario WeiƟmann on

Typos in the proof of (7):

  • is not introduced as a prime of .

  • "Hence, since is free of rank we see that it dimension ..." Maybe something like "Hence, since is free of rank we see that our standard smooth algebra has dimension ..."?


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