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. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (3)
Comment #355 by Nuno on
Comment #356 by Johan on
Comment #358 by Nuno on