# stacks/stacks-project

Another characterization of lci map

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},