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\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Lemma 10.132.4 (Jacobi-Zariski sequence). Let $A \to B \to C$ be ring maps. Choose a presentation $\alpha : A[x_ s, s \in S] \to B$ with kernel $I$. Choose a presentation $\beta : B[y_ t, t \in T] \to C$ with kernel $J$. Let $\gamma : A[x_ s, y_ t] \to C$ be the induced presentation of $C$ with kernel $K$. Then we get a canonical commutative diagram

\[ \xymatrix{ 0 \ar[r] & \Omega _{A[x_ s]/A} \otimes C \ar[r] & \Omega _{A[x_ s, y_ t]/A} \otimes C \ar[r] & \Omega _{B[y_ t]/B} \otimes C \ar[r] & 0 \\ & I/I^2 \otimes C \ar[r] \ar[u] & K/K^2 \ar[r] \ar[u] & J/J^2 \ar[r] \ar[u] & 0 } \]

with exact rows. We get the following exact sequence of homology groups

\[ H_1(\mathop{N\! L}\nolimits _{B/A} \otimes _ B C) \to H_1(L_{C/A}) \to H_1(L_{C/B}) \to C \otimes _ B \Omega _{B/A} \to \Omega _{C/A} \to \Omega _{C/B} \to 0 \]

of $C$-modules extending the sequence of Lemma 10.130.7. If $\text{Tor}_1^ B(\Omega _{B/A}, C) = 0$, then $H_1(\mathop{N\! L}\nolimits _{B/A} \otimes _ B C) = H_1(L_{B/A}) \otimes _ B C$.

Proof. The precise definition of the maps is omitted. The exactness of the top row follows as the $\text{d}x_ s$, $\text{d}y_ t$ form a basis for the middle module. The map $\gamma $ factors

\[ A[x_ s, y_ t] \to B[y_ t] \to C \]

with surjective first arrow and second arrow equal to $\beta $. Thus we see that $K \to J$ is surjective. Moreover, the kernel of the first displayed arrow is $IA[x_ s, y_ t]$. Hence $I/I^2 \otimes C$ surjects onto the kernel of $K/K^2 \to J/J^2$. Finally, we can use Lemma 10.132.2 to identify the terms as homology groups of the naive cotangent complexes. The final assertion follows as the degree $0$ term of the complex $\mathop{N\! L}\nolimits _{B/A}$ is a free $B$-module. $\square$


Comments (2)

Comment #694 by Keenan Kidwell on

In the third line up from the bottom, the map should go in the opposite direction.

Comment #1718 by Yogesh More on

I couldn't immediately see why being the kernel of the first map implies that surjects onto the kernel of , and in case anyone else is wondering, here is one explanation why: is the therefore the kernel of , and hence tensoring the exact sequence by gives an exact sequence . Since , we have , and similarly so our exact sequence is .

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  • 7 comment(s) on Section 10.132: The naive cotangent complex

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