Lemma 15.127.2. In Situation 15.127.1 let $x \in \Omega $. There exists a canonical short exact sequence

\[ 0 \to B(x) \to M(x) \to V(x) \to 0 \]

of $\kappa (x)$-vector spaces which the following property: for $s_1, \ldots , s_ r \in M$ the following are equivalent

there exists an $f \in R$, $f \not\in x$ such that the map $s_1, \ldots , s_ r : R^{\oplus r} \to M$ becomes the inclusion of a direct summand after inverting $f$, and

$s_1(x), \ldots , s_ r(x)$ map to linearly independent elements of $V(x)$.

**Proof.**
Define $B(x) \subset M(x)$ as the perpendicular of the image of the map

\[ \mathop{\mathrm{Hom}}\nolimits _ R(M, R) \to \mathop{\mathrm{Hom}}\nolimits _{\kappa (x)}(M(x), \kappa (x)) \]

and set $V(x) = M(x)/B(x)$. Then any $R$-linear map $\varphi : M \to R$ induces a map $\overline{\varphi } : V(x) \to \kappa (x)$ and conversely any $\kappa (x)$-linear map $\lambda : V(x) \to \kappa (x)$ is equal to $\overline{\varphi }$ for some $\varphi $. Let $s_1, \ldots , s_ r \in M$.

Suppose $s_1, \ldots , s_ r$ map to linearly independent elements of $V(x)$. Then we can find $\varphi _1, \ldots , \varphi _ r \in \mathop{\mathrm{Hom}}\nolimits _ R(M, R)$ such that $\varphi _ i(s_ j)$ maps to $\delta _{ij}$^{1} in $\kappa (x)$. Hence the matrix of the composition

\[ R^{\oplus r} \xrightarrow {s_1, \ldots , s_ r} M \xrightarrow {\varphi _1, \ldots , \varphi _ r} R^{\oplus r} \]

has a determinant $f \in R$ which maps to $1$ in $\kappa (x)$ Clearly, this implies that $s_1, \ldots , s_ r : R^{\oplus r} \to M$ is the inclusion of a direct summand after inverting $f$.

Conversely, suppose that we have an $f \in R$, $f \not\in x$ such that $s_1, \ldots , s_ r : R^{\oplus r} \to M$ is the inclusion of a direct summand after inverting $f$. Hence we can find $R_ f$-linear maps $\varphi _ i : M_ f \to R_ f$ such that $\varphi _ i(s_ j) = \delta _{ij} \in R_ f$. Since $\mathop{\mathrm{Hom}}\nolimits _ R(M, R)_ f = \mathop{\mathrm{Hom}}\nolimits _{R_ f}(M_ f, R_ f)$ by Algebra, Lemma 10.10.2 we conclude that we can find $n \geq 0$ and $\varphi '_ i \in \mathop{\mathrm{Hom}}\nolimits _ R(M, R)$ such that $\varphi '_ i(s_ j) = f^ n\delta _{ij} \in R$. It follows that $s_1, \ldots , s_ r$ map to linearly independent elements of $V(x)$ as $\overline{\varphi }'_ i(s_ j) = f^ n\delta _{ij}$.
$\square$

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