The Stacks project

\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*}

The formation of the naive cotangent complex commutes with localization at an element.

Lemma 10.132.12. Let $A \to B$ be a ring map. Let $g \in B$. Suppose $\alpha : P \to B$ is a presentation with kernel $I$. Then a presentation of $B_ g$ over $A$ is the map

\[ \beta : P[x] \longrightarrow B_ g \]

extending $\alpha $ and sending $x$ to $1/g$. The kernel $J$ of $\beta $ is generated by $I$ and the element $f x - 1$ where $f \in P$ is an element mapped to $g \in B$ by $\alpha $. In this situation we have

  1. $J/J^2 = (I/I^2)_ g \oplus B_ g (f x - 1)$,

  2. $\Omega _{P[x]/A} \otimes _{P[x]} B_ g = \Omega _{P/A} \otimes _ P B_ g \oplus B_ g \text{d}x$,

  3. $\mathop{N\! L}\nolimits (\beta ) \cong \mathop{N\! L}\nolimits (\alpha ) \otimes _ B B_ g \oplus (B_ g \xrightarrow {g} B_ g)$

Hence the canonical map $\mathop{N\! L}\nolimits _{B/A} \otimes _ B B_ g \to \mathop{N\! L}\nolimits _{B_ g/A}$ is a homotopy equivalence.

Proof. Since $P[x]/(I, fx - 1) = B[x]/(gx - 1) = B_ g$ we get the statement about $I$ and $fx - 1$ generating $J$. Consider the commutative diagram

\[ \xymatrix{ 0 \ar[r] & \Omega _{P/A} \otimes B_ g \ar[r] & \Omega _{P[x]/A} \otimes B_ g \ar[r] & \Omega _{B[x]/B} \otimes B_ g \ar[r] & 0 \\ & (I/I^2)_ g \ar[r] \ar[u] & J/J^2 \ar[r] \ar[u] & (gx - 1)/(gx - 1)^2 \ar[r] \ar[u] & 0 } \]

with exact rows of Lemma 10.132.4. The $B_ g$-module $\Omega _{B[x]/B} \otimes B_ g$ is free of rank $1$ on $\text{d}x$. The element $\text{d}x$ in the $B_ g$-module $\Omega _{P[x]/A} \otimes B_ g$ provides a splitting for the top row. The element $gx - 1 \in (gx - 1)/(gx - 1)^2$ is mapped to $g\text{d}x$ in $\Omega _{B[x]/B} \otimes B_ g$ and hence $(gx - 1)/(gx - 1)^2$ is free of rank $1$ over $B_ g$. (This can also be seen by arguing that $gx - 1$ is a nonzerodivisor in $B[x]$ because it is a polynomial with invertible constant term and any nonzerodivisor gives a quasi-regular sequence of length $1$ by Lemma 10.68.2.)

Let us prove $(I/I^2)_ g \to J/J^2$ injective. Consider the $P$-algebra map

\[ \pi : P[x] \to (P/I^2)_ f = P_ f/I_ f^2 \]

sending $x$ to $1/f$. Since $J$ is generated by $I$ and $fx - 1$ we see that $\pi (J) \subset (I/I^2)_ f = (I/I^2)_ g$. Since this is an ideal of square zero we see that $\pi (J^2) = 0$. If $a \in I$ maps to an element of $J^2$ in $J$, then $\pi (a) = 0$, which implies that $a$ maps to zero in $I_ f/I_ f^2$. This proves the desired injectivity.

Thus we have a short exact sequence of two term complexes

\[ 0 \to \mathop{N\! L}\nolimits (\alpha ) \otimes _ B B_ g \to \mathop{N\! L}\nolimits (\beta ) \to (B_ g \xrightarrow {g} B_ g) \to 0 \]

Such a short exact sequence can always be split in the category of complexes. In our particular case we can take as splittings

\[ J/J^2 = (I/I^2)_ g \oplus B_ g (fx - 1)\quad \text{and}\quad \Omega _{P[x]/A} \otimes B_ g = \Omega _{P/A} \otimes B_ g \oplus B_ g (g^{-2}\text{d}f + \text{d}x) \]

This works because $\text{d}(fx - 1) = x\text{d}f + f \text{d}x = g(g^{-2}\text{d}f + \text{d}x)$ in $\Omega _{P[x]/A} \otimes B_ g$. $\square$


Comments (7)

Comment #701 by Keenan Kidwell on

In the fourth sentence of the statement of the lemma, should be .

Comment #703 by on

Thank you very much for all the comments. As of now, the last batch of fixes can be found in this commit. I intend to update the stacks project later today so then you should be able to see the changes reflected online as well.

Comment #850 by Bhargav Bhatt on

Suggested slogan: The formation of the naive cotangent complex commutes with localization at an element.

Comment #1514 by Rob Roy on

There's an error in the argument for injectivity of , because in the short exact sequence , really represents a non-free quotient of (annihilated by ); so the last sentence of the proof is false. Here's a different argument. It is enough to show that where is the obvious map . So consider the extension of to the P-algebra map given by , and note that , since implies .

Comment #1531 by on

Thanks very much. There is a generalization of this proof (the correct one you give) in the proof of Lemma 15.32. Here is the commit.

Comment #1951 by Brian Conrad on

Perhaps mention that is not a zero-divisor (easy calculation beginning at constant terms), so is a free module (of rank 1) over . This freeness over is implicitly used in the use of the present result in the proof of Tag 07CF.

Comment #2006 by on

OK, I added your remark and I tried to explain the proof better one more time, see here. But I think there is a curse on this lemma which prevents it from being explained clearly.

There are also:

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

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