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Applications of 30836b4 and 4ad9d51

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aisejohan committed Sep 11, 2019
1 parent 4ad9d51 commit 9a720816ac17dfe8267971913c44a603c17c9831
Showing with 144 additions and 3 deletions.
  1. +38 −3 discriminant.tex
  2. +106 −0 more-morphisms.tex
@@ -1685,7 +1685,7 @@ \section{Quasi-finite syntomic morphisms}
\item $f$ is locally quasi-finite, flat, and a local complete intersection
morphism,
\item $f$ is locally quasi-finite, flat, locally of finite presentation,
and the fibres of $f$ are local complete intersections.
and the fibres of $f$ are local complete intersections,
\item $f$ is locally quasi-finite and for every $y \in Y$ there are
affine opens $y \in V = \Spec(B) \subset Y$, $U = \Spec(A) \subset X$
with $f(V) \subset U$ an integer $n$ and
@@ -1694,7 +1694,11 @@ \section{Quasi-finite syntomic morphisms}
\item for every $y \in Y$ there are affine opens
$y \in V = \Spec(B) \subset Y$, $U = \Spec(A) \subset X$
with $f(V) \subset U$ such that $A \to B$ is a relative global complete
intersection of the form $B = A[x_1, \ldots, x_n]/(f_1, \ldots, f_n)$.
intersection of the form $B = A[x_1, \ldots, x_n]/(f_1, \ldots, f_n)$,
\item $f$ is locally quasi-finite, flat, locally of finite presentation,
and $\NL_{X/Y}$ has tor-amplitude in $[-1, 0]$, and
\item $f$ is flat, locally of finite presentation,
$\NL_{X/Y}$ is perfect of rank $0$ with tor-amplitude in $[-1, 0]$,
\end{enumerate}
\end{lemma}

@@ -1724,9 +1728,40 @@ \section{Quasi-finite syntomic morphisms}
Lemma \ref{algebra-lemma-isolated-point-fibre}.

\medskip\noindent
Finally, either Algebra, Lemma \ref{algebra-lemma-syntomic} or
Either Algebra, Lemma \ref{algebra-lemma-syntomic} or
Morphisms, Lemma \ref{morphisms-lemma-syntomic-locally-standard-syntomic}
shows that (1) implies (5).

\medskip\noindent
More on Morphisms, Lemma \ref{more-morphisms-lemma-flat-fp-NL-lci} shows that
(6) is equivalent to (1). If the equivalent conditions (1) -- (6) hold,
then we see that affine locally $Y \to X$ is given by a relative global
complete intersection $B = A[x_1, \ldots, x_n]/(f_1, \ldots, f_n)$
with the same number of variables as the number of
equations. Using this presentation we see that
$$
\NL_{B/A} =\left(
(f_1, \ldots, f_n)/(f_1, \ldots, f_n)^2
\longrightarrow
\bigoplus\nolimits_{i = 1, \ldots, n} B \text{d} x_i\right)
$$
By Algebra, Lemma
\ref{algebra-lemma-relative-global-complete-intersection-conormal}
the module $(f_1, \ldots, f_n)/(f_1, \ldots, f_n)^2$
is free with generators the congruence classes of the elements
$f_1, \ldots, f_n$. Thus $\NL_{B/A}$ has rank $0$ and so does $\NL_{Y/X}$.
In this way we see that (1) -- (6) imply (7).

\medskip\noindent
Finally, assume (7). By
More on Morphisms, Lemma \ref{more-morphisms-lemma-flat-fp-NL-lci}
we see that $f$ is syntomic. Thus on suitable affine opens
$f$ is given by a relative global complete intersection
$A \to B = A[x_1, \ldots, x_n]/(f_1, \ldots, f_m)$, see
Morphisms, Lemma \ref{morphisms-lemma-syntomic-locally-standard-syntomic}.
Exactly as above we see that $\NL_{B/A}$ is a perfect complex
of rank $n - m$. Thus $n = m$ and we see that (5) holds.
This finishes the proof.
\end{proof}

\begin{lemma}
@@ -3824,6 +3824,49 @@ \section{The naive cotangent complex}
Special case of Modules, Lemma \ref{modules-lemma-exact-sequence-NL}.
\end{proof}

\begin{lemma}
\label{lemma-flat-base-change-NL}
Consider a cartesian diagram of schemes
$$
\xymatrix{
X' \ar[r] \ar[d] & X \ar[d] \\
Y' \ar[r] & Y
}
$$
If $Y' \to Y$ is flat, then the canonical map
$f^*\NL_{X/Y} \to \NL_{X'/Y'}$ is a quasi-isomorphism.
\end{lemma}

\begin{proof}
By Lemma \ref{lemma-NL-affine} this follows from
Algebra, Lemma \ref{algebra-lemma-change-base-NL}.
\end{proof}

\begin{lemma}
\label{lemma-base-change-NL}
Consider a cartesian diagram of schemes
$$
\xymatrix{
X' \ar[r] \ar[d] & X \ar[d] \\
Y' \ar[r] & Y
}
$$
If $X \to Y$ is flat, then the canonical map
$f^*\NL_{X/Y} \to \NL_{X'/Y'}$ is a quasi-isomorphism.
If in addition $\NL_{X/Y}$ has tor-amplitude in $[-1, 0]$
then $Lf^*\NL_{X/Y} \to \NL_{X'/Y'}$ is a quasi-isomorphism too.
\end{lemma}

\begin{proof}
Translated into algebra this is
Divided Power Algebra, Lemma \ref{dpa-lemma-base-change-NL}.
To do the translation use Lemma \ref{lemma-NL-affine}
and Derived Categories of Schemes, Lemmas
\ref{perfect-lemma-affine-compare-bounded} and
\ref{perfect-lemma-tor-dimension-affine}.
\end{proof}





(and Morphisms, Lemma \ref{morphisms-lemma-permanence-finite-type}).
\end{proof}

\begin{lemma}
\label{lemma-perfect-conormal-free-lci}
Let $i : X \to Y$ be an immersion. If
\begin{enumerate}
\item $i$ is perfect,
\item $Y$ is locally Noetherian, and
\item the conormal sheaf $\mathcal{C}_{Z/X}$ is finite locally free,
\end{enumerate}
then $i$ is a regular immersion.
\end{lemma}

\begin{proof}
Translated into algebra, this is
Divided Power Algebra, Proposition \ref{dpa-proposition-regular-ideal}.
\end{proof}

\begin{lemma}
\label{lemma-perfect-NL-lci}
Let $f : X \to Y$ be a perfect morphism of locally Noetherian schemes.
The following are equivalent
\begin{enumerate}
\item $f$ is a local complete intersection morphism,
\item $\NL_{X/Y}$ has tor-amplitude in $[-1, 0]$, and
\item $\NL_{X/Y}$ is perfect with tor-amplitude in $[-1, 0]$.
\end{enumerate}
\end{lemma}

\begin{proof}
Translated into algebra this is
Divided Power Algebra, Lemma \ref{dpa-lemma-perfect-NL-lci}.
To do the translation use
Lemmas \ref{lemma-affine-lci} and
\ref{lemma-NL-affine} as well as
Derived Categories of Schemes, Lemmas
\ref{perfect-lemma-affine-compare-bounded},
\ref{perfect-lemma-tor-dimension-affine} and
\ref{perfect-lemma-perfect-affine}.
\end{proof}

\begin{lemma}
\label{lemma-flat-fp-NL-lci}
Let $f : X \to Y$ be a flat morphism of finite presentation.
The following are equivalent
\begin{enumerate}
\item $f$ is a local complete intersection morphism,
\item $f$ is syntomic,
\item $\NL_{X/Y}$ has tor-amplitude in $[-1, 0]$, and
\item $\NL_{X/Y}$ is perfect with tor-amplitude in $[-1, 0]$.
\end{enumerate}
\end{lemma}

\begin{proof}
Translated into algebra this is
Divided Power Algebra, Lemma \ref{dpa-lemma-flat-fp-NL-lci}.
To do the translation use
Lemmas \ref{lemma-affine-lci} and
\ref{lemma-NL-affine} as well as
Derived Categories of Schemes, Lemmas
\ref{perfect-lemma-affine-compare-bounded},
\ref{perfect-lemma-tor-dimension-affine} and
\ref{perfect-lemma-perfect-affine}.
\end{proof}

\noindent
The following lemma gives a characterization of smooth morphisms as flat
morphisms whose diagonal is perfect.

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