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*}

Lemma 12.5.11. Let $\mathcal{A}$ be an abelian category. Let

\[ \xymatrix{ w\ar[r]^ f\ar[d]_ g & y\ar[d]^ h\\ x\ar[r]^ k & z } \]

be a commutative diagram.

  1. The diagram is cartesian if and only if

    \[ 0 \to w \xrightarrow {(g, f)} x \oplus y \xrightarrow {(k, -h)} z \]

    is exact.

  2. The diagram is cocartesian if and only if

    \[ w \xrightarrow {(g, -f)} x \oplus y \xrightarrow {(k, h)} z \to 0 \]

    is exact.

Proof. Let $u = (g, f) : w \to x \oplus y$ and $v = (k, -h) : x \oplus y \to z$. Let $p : x \oplus y \to x$ and $q : x \oplus y \to y$ be the canonical projections. Let $i : \mathop{\mathrm{Ker}}(v) \to x \oplus y$ be the canonical injection. By Example 12.5.6, the diagram is cartesian if and only if there exists an isomorphism $r : \mathop{\mathrm{Ker}}(v) \to w$ with $f \circ r = q \circ i$ and $g \circ r = p \circ i$. The sequence $0 \to w \overset {u} \to x \oplus y \overset {v} \to z$ is exact if and only if there exists an isomorphism $r : \mathop{\mathrm{Ker}}(v) \to w$ with $u \circ r = i$. But given $r : \mathop{\mathrm{Ker}}(v) \to w$, we have $f \circ r = q \circ i$ and $g \circ r = p \circ i$ if and only if $q \circ u \circ r= f \circ r = q \circ i$ and $p \circ u \circ r = g \circ r = p \circ i$, hence if and only if $u \circ r = i$. This proves (1), and then (2) follows by duality. $\square$


Comments (4)

Comment #380 by Fan on

Typo: the canonical injection is , not to .

I'm afraid that there is a sign error somewhere, as the sequence is not even a chain if I take and .

Comment #386 by on

@#380: Fixed this and most of your other comments. See here. Thanks very much!

Comment #3843 by Junho Won on

I think the second exact sequence is missing a 0 on the left.

Comment #3844 by Laurent Moret-Bailly on

@Junho Won: no. For instance, if , the diagram is cocartesian for any .

There are also:

  • 3 comment(s) on Section 12.5: Abelian categories

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