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 10.106.11. Let $\varphi : R \to S$ be a ring map. Let $g \in S$. The following are equivalent:

  1. $g \otimes 1 = 1 \otimes g$ in $S \otimes _ R S$, and

  2. there exist $n \geq 0$ and elements $y_ i, z_ j \in S$ and $x_{i, j} \in R$ for $1 \leq i, j \leq n$ such that

    1. $g = \sum _{i, j \leq n} x_{i, j} y_ i z_ j$,

    2. for each $j$ we have $\sum x_{i, j}y_ i \in \varphi (R)$, and

    3. for each $i$ we have $\sum x_{i, j}z_ j \in \varphi (R)$.

Proof. It is clear that (2) implies (1). Conversely, suppose that $g \otimes 1 = 1 \otimes g$. Choose generators $\{ s_ i\} _{i \in I}$ of $S$ as an $R$-module with $0, 1 \in I$ and $s_0 = 1$ and $s_1 = g$. Apply Lemma 10.106.10 to the relation $g \otimes s_0 + (-1) \otimes s_1 = 0$. We see that there exist $a_{i, j} \in R$ such that $g = \sum _ i a_{i, 0} s_ i$, $-1 = \sum _ i a_{i, 1} s_ i$, and for $j \not= 0, 1$ we have $0 = \sum _ i a_{i, j} s_ i$, and moreover for all $i$ we have $\sum _ j a_{i, j}s_ j = 0$. Then we have

\[ \sum \nolimits _{i, j \not= 0} a_{i, j} s_ i s_ j = -g + a_{0, 0} \]

and for each $j \not= 0$ we have $\sum _{i \not= 0} a_{i, j}s_ i \in R$. This proves that $-g + a_{0, 0}$ can be written as in (2). It follows that $g$ can be written as in (2). Details omitted. Hint: Show that the set of elements of $S$ which have an expression as in (2) form an $R$-subalgebra of $S$. $\square$

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