stacks/stacks-project

 @@ -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.