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Another characterization of lci map

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aisejohan committed Sep 10, 2019
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@@ -2407,6 +2407,158 @@ \section{Smooth ring maps and diagonals}




\section{Freeness of the conormal module}
\label{section-freeness-conormal}

\noindent
Tate resolutions and derivations on them can be used to prove
(stronger) versions of the results in this section, see \cite{Iyengar}.
Two more elementary references are
\cite{Vasconcelos} and \cite{Ferrand-lci}.

\begin{lemma}
\label{lemma-free-summand-in-ideal-finite-proj-dim}
\begin{reference}
\cite{Vasconcelos}
\end{reference}
Let $R$ be a Noetherian local ring. Let $I \subset R$ be an ideal
of finite projective dimension over $R$. If $F \subset I/I^2$ is a
direct summand isomorphic to $R/I$, then there exists a nonzerodivisor
$x \in I$ such that the image of $x$ in $I/I^2$ generates $F$.
\end{lemma}

\begin{proof}
By assumption we may choose a finite free resolution
$$
0 \to R^{\oplus n_e} \to R^{\oplus n_{e-1}} \to \ldots \to
R^{\oplus n_1} \to R \to R/I \to 0
$$
Then $\varphi_1 : R^{\oplus n_1} \to R$ has rank $1$ and
we see that $I$ contains a nonzerodivisor $y$ by
Algebra, Proposition \ref{algebra-proposition-what-exact}.
Let $\mathfrak p_1, \ldots, \mathfrak p_n$ be the associated
primes of $R$, see Algebra, Lemma \ref{algebra-lemma-finite-ass}.
Let $I^2 \subset J \subset I$ be an ideal such that $J/I^2 = F$.
Then $J \not \subset \mathfrak p_i$ for all $i$
as $y^2 \in J$ and $y^2 \not \in \mathfrak p_i$, see
Algebra, Lemma \ref{algebra-lemma-ass-zero-divisors}.
By Nakayama's lemma (Algebra, Lemma \ref{algebra-lemma-NAK})
we have $J \not \subset \mathfrak m J + I^2$.
By Algebra, Lemma \ref{algebra-lemma-silly}
we can pick $x \in J$, $x \not \in \mathfrak m J + I^2$ and
$x \not \in \mathfrak p_i$ for $i = 1, \ldots, n$.
Then $x$ is a nonzerodivisor and the image
of $x$ in $I/I^2$ generates (by Nakayama's lemma)
the summand $J/I^2 \cong R/I$.
\end{proof}

\begin{lemma}
\label{lemma-vasconcelos}
\begin{reference}
Local version of \cite[Theorem 1.1]{Vasconcelos}
\end{reference}
Let $R$ be a Noetherian local ring. Let $I \subset R$ be an ideal
of finite projective dimension over $R$. If $F \subset I/I^2$
is a direct summand free of rank $r$, then there exists a regular sequence
$x_1, \ldots, x_r \in I$ such that $x_1 \bmod I^2, \ldots, x_r \bmod I^2$
generate $F$.
\end{lemma}

\begin{proof}
If $r = 0$ there is nothing to prove. Assume $r > 0$. We may pick
$x \in I$ such that $x$ is a nonzerodivisor and $x \bmod I^2$
generates a summand of $F$ isomorphic to $R/I$, see
Lemma \ref{lemma-free-summand-in-ideal-finite-proj-dim}.
Consider the ring $R' = R/(x)$ and the ideal $I' = I/(x)$.
Of course $R'/I' = R/I$. The short exact sequence
$$
0 \to R/I \xrightarrow{x} I/xI \to I' \to 0
$$
splits because the map $I/xI \to I/I^2$ sends $xR/xI$
to a direct summand. Now $I/xI = I \otimes_R^\mathbf{L} R'$ has
finite projective dimension over $R'$, see
More on Algebra, Lemmas \ref{more-algebra-lemma-perfect-module} and
\ref{more-algebra-lemma-pull-perfect}.
Hence the summand $I'$ has finite projective dimension over $R'$.
On the other hand, we have the short exact sequence
$0 \to xR/xI \to I/I^2 \to I'/(I')^2 \to 0$ and we conclude
$I'/(I')^2$ has the free direct summand $F' = F/(R/I \cdot x)$
of rank $r - 1$. By induction on $r$ we may
we pick a regular sequence $x'_2, \ldots, x'_r \in I'$
such that there congruence classes freely generate $F'$.
If $x_1 = x$ and $x_2, \ldots, x_r$ are any elements lifting
$x'_1, \ldots, x'_r$ in $R$, then we see that the lemma holds.
\end{proof}

\begin{proposition}
\label{proposition-regular-ideal}
\begin{reference}
Variant of \cite[Corollary 1]{Vasconcelos}. See also
\cite{Iyengar} and \cite{Ferrand-lci}.
\end{reference}
Let $R$ be a Noetherian ring. Let $I \subset R$ be an ideal
which has finite projective dimension and such that $I/I^2$ is
finite locally free over $R/I$. Then $I$ is a regular ideal
(More on Algebra, Definition \ref{more-algebra-definition-regular-ideal}).
\end{proposition}

\begin{proof}
By Algebra, Lemma \ref{algebra-lemma-regular-sequence-in-neighbourhood}
it suffices to show that $I_\mathfrak p \subset R_\mathfrak p$ is generated
by a regular sequence for every $\mathfrak p \supset I$. Thus we may
assume $R$ is local. If $I/I^2$ has rank $r$, then by
Lemma \ref{lemma-vasconcelos} we find a regular sequence
$x_1, \ldots, x_r \in I$ generating $I/I^2$. By
Nakayama (Algebra, Lemma \ref{algebra-lemma-NAK})
we conclude that $I$ is generated by $x_1, \ldots, x_r$.
\end{proof}

\begin{lemma}
\label{lemma-perfect-NL-lci}
Let $A \to B$ be a perfect (More on Algebra, Definition
\ref{more-algebra-definition-pseudo-coherent-perfect})
ring homomorphism of Noetherian rings. Then the following are equivalent
\begin{enumerate}
\item $\NL_{B/A}$ has tor-amplitude in $[-1, 0]$,
\item $\NL_{B/A}$ is a perfect object of $D(B)$
with tor-amplitude in $[-1, 0]$, and
\item $A \to B$ is a local complete intersection
(More on Algebra, Definition
\ref{more-algebra-definition-local-complete-intersection}).
\end{enumerate}
\end{lemma}

\begin{proof}
Write $B = A[x_1, \ldots, x_n]/I$. Then $\NL_{B/A}$ is represented by
the complex
$$
I/I^2 \longrightarrow \bigoplus B \text{d}x_i
$$
of $B$-modules with $I/I^2$ placed in degree $-1$. Since the term in
degree $0$ is finite free, this complex has tor-amplitude in $[-1, 0]$ if and
only if $I/I^2$ is a flat $B$-module, see
More on Algebra, Lemma \ref{more-algebra-lemma-last-one-flat}.
Since $I/I^2$ is a finite $B$-module and $B$ is Noetherian, this is true
if and only if $I/I^2$ is a finite locally free $B$-module
(Algebra, Lemma \ref{algebra-lemma-finite-projective}).
Thus the equivalence of (1) and (2) is clear. Moreover, the equivalence
of (1) and (3) also follows if we apply
Proposition \ref{proposition-regular-ideal}
(and the observation that a regular ideal is a Koszul regular
ideal as well as a quasi-regular ideal, see
More on Algebra, Section \ref{more-algebra-section-ideals}).
\end{proof}










\input{chapters}

\bibliography{my}
35 my.bib
@@ -1854,6 +1854,16 @@ @ARTICLE{Ferrand-Conducteur
ISSN = {0037-9484},
}

@ARTICLE{Ferrand-lci,
AUTHOR = {Ferrand, Daniel},
TITLE = {Suite r\'{e}guli\`ere et intersection compl\`ete},
JOURNAL = {C. R. Acad. Sci. Paris S\'{e}r. A-B},
VOLUME = {264},
YEAR = {1967},
PAGES = {A427--A428},
ISSN = {0151-0509},
}

@ARTICLE{Flenner,
AUTHOR = {Flenner, Hubert},
TITLE = {Ein {K}riterium f\"ur die {O}ffenheit der {V}ersalit\"at},
@@ -3015,6 +3025,20 @@ @ARTICLE{iwanari_chow
YEAR = {2007}
}

@ARTICLE{Iyengar,
AUTHOR = {Iyengar, Srikanth},
TITLE = {Free summands of conormal modules and central elements in homotopy {L}ie algebras of local rings},
JOURNAL = {Proc. Amer. Math. Soc.},
VOLUME = {129},
YEAR = {2001},
NUMBER = {6},
PAGES = {1563--1572},
ISSN = {0002-9939},
URL = {https://doi.org/10.1090/S0002-9939-01-05565-4},
}




%%%%%%%%%%%%%%%%%%%%%% J %%%%%%%%%%%%%%%%%%%%%%
@@ -6120,6 +6144,17 @@ @ARTICLE{Varbaro
URL = {https://doi.org/10.1016/j.jalgebra.2009.01.018},
}

@ARTICLE{Vasconcelos,
AUTHOR = {Vasconcelos, Wolmer V.},
TITLE = {Ideals generated by {$R$}-sequences},
JOURNAL = {J. Algebra},
VOLUME = {6},
YEAR = {1967},
PAGES = {309--316},
ISSN = {0021-8693},
URL = {https://doi.org/10.1016/0021-8693(67)90086-5},
}

@ARTICLE{Verdier,
AUTHOR = {Verdier, Jean-Louis},
TITLE = {Des cat\'egories d\'eriv\'ees des cat\'egories ab\'eliennes},

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