Lemma 10.133.6. Let $R \to S$ be a ring map. Let $M$ be an $S$-module. There is a canonical short exact sequence
\[ 0 \to \Omega _{S/R} \otimes _ S M \to P^1_{S/R}(M) \to M \to 0 \]
functorial in $M$ called the sequence of principal parts.
Proof. The map $P^1_{S/R}(M) \to M$ is given above. Let $N$ be an $S$-module and let $D : M \to N$ be a differential operator of order $1$. For $m \in M$ the map
\[ g \longmapsto D(gm) - gD(m) \]
is an $R$-derivation $S \to N$ by the axioms for differential operators of order $1$. Thus it corresponds to a linear map $D_ m : \Omega _{S/R} \to N$ determined by the rule $a\text{d}b \mapsto aD(bm) - abD(m)$ (see Lemma 10.131.3). The map
\[ \Omega _{S/R} \times M \longrightarrow N,\quad (\eta , m) \longmapsto D_ m(\eta ) \]
is $S$-bilinear (details omitted) and hence determines an $S$-linear map
\[ \sigma _ D : \Omega _{S/R} \otimes _ S M \to N \]
In this way we obtain a map $\text{Diff}^1(M, N) \to \mathop{\mathrm{Hom}}\nolimits _ S(\Omega _{S/R} \otimes _ S M, N)$, $D \mapsto \sigma _ D$ functorial in $N$. By the Yoneda lemma this corresponds a map $\Omega _{S/R} \otimes _ S M \to P^1_{S/R}(M)$. It is immediate from the construction that this map is functorial in $M$. The sequence
\[ \Omega _{S/R} \otimes _ S M \to P^1_{S/R}(M) \to M \to 0 \]
is exact because for every module $N$ the sequence
\[ 0 \to \mathop{\mathrm{Hom}}\nolimits _ S(M, N) \to \text{Diff}^1(M, N) \to \mathop{\mathrm{Hom}}\nolimits _ S(\Omega _{S/R} \otimes _ S M, N) \]
is exact by inspection.
To see that $\Omega _{S/R} \otimes _ S M \to P^1_{S/R}(M)$ is injective we argue as follows. Choose an exact sequence
\[ 0 \to M' \to F \to M \to 0 \]
with $F$ a free $S$-module. This induces an exact sequence
\[ 0 \to \text{Diff}^1(M, N) \to \text{Diff}^1(F, N) \to \text{Diff}^1(M', N) \]
for all $N$. This proves that in the commutative diagram
\[ \xymatrix{ 0 \ar[r] & \Omega _{S/R} \otimes _ S M' \ar[r] \ar[d] & P^1_{S/R}(M') \ar[r] \ar[d] & M' \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \Omega _{S/R} \otimes _ S F \ar[r] \ar[d] & P^1_{S/R}(F) \ar[r] \ar[d] & F \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \Omega _{S/R} \otimes _ S M \ar[r] \ar[d] & P^1_{S/R}(M) \ar[r] \ar[d] & M \ar[r] \ar[d] & 0 \\ & 0 & 0 & 0 } \]
the middle column is exact. The left column is exact by right exactness of $\Omega _{S/R} \otimes _ S -$. By the snake lemma (see Section 10.4) it suffices to prove exactness on the left for the free module $F$. Using that $P^1_{S/R}(-)$ commutes with direct sums we reduce to the case $M = S$. This case is a consequence of the discussion in Example 10.133.5. $\square$
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