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| Then $\NL(\alpha) \otimes_R R' = \NL(\alpha) \otimes_S S' = \NL(\alpha')$. | ||
| In particular, the canonical map | ||
| $$ | ||
| \NL_{S/R} \otimes_R R' \longrightarrow \NL_{S \otimes_R R'/R'} | ||
| \NL_{S/R} \otimes_S S' \longrightarrow \NL_{S \otimes_R R'/R'} | ||
| $$ | ||
| is a homotopy equivalence if $R \to R'$ is flat. | ||
| \end{lemma} |
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| \section{The naive cotangent complex} | ||
| \label{section-netherlander} | ||
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| \noindent | ||
| In this section we continue the discussion started in | ||
| Algebra, Section \ref{algebra-section-netherlander}. | ||
| We begin with a discussion of base change. | ||
| The first lemma shows that taking the naive tensor product | ||
| of the naive cotangent complex with a ring extension | ||
| isn't quite as naive as one might think. | ||
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| \begin{lemma} | ||
| \label{lemma-tensor-NL} | ||
| Let $R \to S$ and $S \to S'$ be ring maps. The canonical map | ||
| $\NL_{S/R} \otimes_S^\mathbf{L} S' \to \NL_{S/R} \otimes_S S'$ | ||
| induces an isomorphism | ||
| $\tau_{\geq -1}(\NL_{S/R} \otimes_S^\mathbf{L} S') \to \NL_{S/R} \otimes_S S'$ | ||
| in $D(S')$. Similarly, given a presentation $\alpha$ of $S$ over $R$ | ||
| the canonical map | ||
| $\NL(\alpha) \otimes_S^\mathbf{L} S' \to \NL(\alpha) \otimes_S S'$ | ||
| induces an isomorphism $\tau_{\geq -1}(\NL(\alpha) \otimes_S^\mathbf{L} S') \to | ||
| \NL(\alpha) \otimes_S S'$ in $D(S')$. | ||
| \end{lemma} | ||
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| \begin{proof} | ||
| Let $K^\bullet$ be a complex of $S$-modules with $K^n = 0$ | ||
| for $n \not \in [-1, 0]$ and $K^0$ flat, for example | ||
| $K^\bullet = \NL_{S/R}$ or $K^\bullet = \NL(\alpha)$. | ||
| Then we have a distinguished triangle | ||
| $$ | ||
| K^0 \to K^\bullet \to K^{-1}[1] \to K^0[1] | ||
| $$ | ||
| in $D(S)$. This determines a map of distinguished triangles | ||
| $$ | ||
| \xymatrix{ | ||
| K^0 \otimes_S^\mathbf{L} S' \ar[d] \ar[r] & | ||
| K^\bullet \otimes_S^\mathbf{L} S' \ar[r] \ar[d] & | ||
| K^{-1} \otimes_S^\mathbf{L} S'[1] \ar[r] \ar[d] & | ||
| K^0 \otimes_S^\mathbf{L} S'[1] \ar[d] \\ | ||
| K^0 \otimes_S S' \ar[r] & | ||
| K^\bullet \otimes_S S' \ar[r] & | ||
| K^{-1} \otimes_S S'[1] \ar[r] & | ||
| K^0 \otimes_S S'[1] | ||
| } | ||
| $$ | ||
| The left and right vertical arrows are isomorphisms as $K^0$ is flat. | ||
| Since $K^{-1} \otimes_S^\mathbf{L} S' \to K^{-1} \otimes_S S'$ | ||
| is an isomorphism on cohomology in degree $0$ we conclude. | ||
| \end{proof} | ||
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| \begin{lemma} | ||
| \label{lemma-base-change-NL} | ||
| Let $R \to S$ and $R \to R'$ be ring maps. | ||
| Let $\alpha : P \to S$ be a presentation of $S$ over $R$. | ||
| Then $\alpha' : P \otimes_R R' \to S \otimes_R R'$ is a | ||
| presentation of $S' = S \otimes_R R'$ over $R'$. | ||
| The canonical map | ||
| $$ | ||
| NL(\alpha) \otimes_S S' \to \NL(\alpha') | ||
| $$ | ||
| is an isomorphism on $H^0$ and surjective on $H^{-1}$. In particular, | ||
| the canonical map | ||
| $$ | ||
| \NL_{S/R} \otimes_S S' \to \NL_{S'/R'} | ||
| $$ | ||
| is an isomorphism on $H^0$ and surjective on $H^{-1}$. | ||
| \end{lemma} | ||
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| \begin{proof} | ||
| Denote $I = \Ker(P \to S)$. Denote $P' = P \otimes_R R'$ and | ||
| $I' = \Ker(P' \to S')$. Suppose $P$ is a polynomial algebra | ||
| on $x_j$ for $j \in J$. The map displayed in the lemma becomes | ||
| $$ | ||
| \xymatrix{ | ||
| \bigoplus_{j \in J} S' \text{d}x_j \ar[r] & | ||
| \bigoplus_{j \in J} S' \text{d}x_j \\ | ||
| I/I^2 \otimes_S S' \ar[r] \ar[u] & | ||
| I'/(I')^2 \ar[u] | ||
| } | ||
| $$ | ||
| where the left column is $\NL(\alpha) \otimes_S S'$ and the right | ||
| column is $\NL(\alpha')$. By right exactness of tensor product we | ||
| see that $I \otimes_R R' \to I'$ is surjective. Hence the bottom arrow | ||
| is a surjection. This proves the first statement of the lemma. | ||
| The statement for $\NL_{S/R} \otimes_S S' \to \NL_{S'/R'}$ | ||
| follows as these complexes are homotopic to $\NL(\alpha) \otimes_S S'$ and | ||
| $\NL(\alpha')$. | ||
| \end{proof} | ||
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| \begin{lemma} | ||
| \label{lemma-base-change-NL-flat} | ||
| Consider a cocartesian diagram of rings | ||
| $$ | ||
| \xymatrix{ | ||
| B \ar[r] & B' \\ | ||
| A \ar[r] \ar[u] & A' \ar[u] | ||
| } | ||
| $$ | ||
| If $B$ is flat over $A$, then the canonical map | ||
| $\NL_{B/A} \otimes_B B' \to \NL_{B'/A'}$ is a quasi-isomorphism. | ||
| If in addition $\NL_{B/A}$ has tor-amplitude in $[-1, 0]$ | ||
| then $\NL_{B/A} \otimes_B^\mathbf{L} B' \to \NL_{B'/A'}$ | ||
| is a quasi-isomorphism too. | ||
| \end{lemma} | ||
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| \begin{proof} | ||
| Choose a presentation $\alpha : P \to B$ as in | ||
| Algebra, Section \ref{algebra-section-netherlander}. | ||
| Let $I = \Ker(\alpha)$. Set $P' = P \otimes_A A'$ and denote | ||
| $\alpha' : P' \to B'$ the corresponding presentation of $B'$ over $A'$. | ||
| As $B$ is flat over $A$ we see that $I' = \Ker(\alpha')$ is equal | ||
| to $I \otimes_A A'$. Hence | ||
| $$ | ||
| I'/(I')^2 = \Coker(I^2 \otimes_A A' \to I \otimes_A A') = | ||
| I/I^2 \otimes_A A' = I/I^2 \otimes_B B' | ||
| $$ | ||
| We have $\Omega_{P'/A'} = \Omega_{P/A} \otimes_A A'$ | ||
| because both sides have the same basis. It follows that | ||
| $\Omega_{P'/A'} \otimes_{P'} B' = \Omega_{P/A} \otimes_P B \otimes_B B'$. | ||
| This proves that $\NL(\alpha) \otimes_B B' \to \NL(\alpha')$ | ||
| is an isomorphism of complexes and hence the first statement holds. | ||
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| \medskip\noindent | ||
| We have | ||
| $$ | ||
| \NL(\alpha) = I/I^2 \longrightarrow \Omega_{P/A} \otimes_P B | ||
| $$ | ||
| as a complex of $B$-modules with $I/I^2$ placed in degree $-1$. | ||
| Since the term in degree $0$ is free, this complex has tor-amplitude | ||
| in $[-1, 0]$ if and only if $I/I^2$ is a flat $B$-module, see | ||
| Lemma \ref{lemma-last-one-flat}. | ||
| If this holds, then $\NL(\alpha) \otimes_B^\mathbf{L} B' = | ||
| \NL(\alpha) \otimes_B B'$ and we get the second statement. | ||
| \end{proof} | ||
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| \begin{lemma} | ||
| \label{lemma-lci-NL} | ||
| Let $A \to B$ be a local complete intersection as in | ||
| Definition \ref{definition-local-complete-intersection}. | ||
| Then $\NL_{B/A}$ is a perfect object of | ||
| $D(B)$ with tor amplitude in $[-1, 0]$. | ||
| \end{lemma} | ||
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| \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 | ||
| Lemma \ref{lemma-last-one-flat}. By definition $I$ is a Koszul regular | ||
| ideal and hence a quasi-regular ideal, see Section \ref{section-ideals}. | ||
| Thus $I/I^2$ is a finite projective $B$-module | ||
| (Lemma \ref{lemma-quasi-regular-ideal-finite-projective}) | ||
| and we conclude both that $\NL_{B/A}$ is perfect and that it has tor amplitude | ||
| in $[-1, 0]$. | ||
| \end{proof} | ||
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| 0FJR,dpa-lemma-vasconcelos | ||
| 0FJS,dpa-proposition-regular-ideal | ||
| 0FJT,dpa-lemma-perfect-NL-lci | ||
| 0FJU,dpa-lemma-base-change-NL | ||
| 0FJU,more-algebra-lemma-base-change-NL-flat | ||
| 0FJV,dpa-lemma-flat-fp-NL-lci | ||
| 0FJW,perfect-section-det-two-terms | ||
| 0FJX,perfect-lemma-determinant-two-term-complexes |
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