Reading Hartshorne Chapter II.8 I was looking up some commutative algebra, derivations and differentials, in “Commutative Algebra” by Matsumura and am stuck in the proof of Theorem 57. (The first fundamental exact sequence) on page 186.
It states that given ring-homomorphisms $\phi \colon k\to A$ and $\psi \colon A\to B$ there is an exact sequence of $B$-modules
$$\Omega_{A/k}\otimes_A B \to \Omega_{B/k}\to \Omega_{B/A}\to 0 \,.$$
With the first map $v$ defined by $v(d_{A/k}(a)\otimes b) = b \cdot d_{B/k}(\psi(a))$ and the second map $u$ defined by $u(b \cdot d_{B/k}(b'))=b \cdot d_{B/A}(b')$ ($\Omega_{R/S}$ is the Kähler differential of $R$ an $S$-algebra.)
In the proof of the theorem it now states that it is clear that $u$ is surjective. I think this is the case because the map by which we construct the Kähler differential for $B$ over $k$ factors through $A$, am I right?
And then it goes on that because $d_{B/A}\psi(a) = 0$, $uv = 0$ and it remains to show that $\mathrm{Ker}(u) = \mathrm{Im}(v)$ and it is enough to show that
$$\mathrm{Hom}_B(\Omega_{A/k}\otimes_A B,T) \gets \mathrm{Hom}_B(\Omega_{B/k},T) \gets \mathrm{Hom}_B(\Omega_{B/A},T)$$
is exact for any $B$-module $T$ and take $T = \mathrm{Coker}(v)$ and this is the point I don’t understand. How does that prove $\mathrm{Ker}(u) = \mathrm{Im}(v)$?
Best Answer
The map $u$ is surjective, since the differentials are constructed by taking a quotient of a free module over $B$ on symbols $db$ for $b\in B$, modulo certain relations. And the map $u$ is just quotienting by more relations, namely asking that $d(\Psi(a))=0$ for all $a\in A$, instead of just $d(\Psi(\Phi(x))=0$ for all $x\in k$.
For your second question. Since $uv=0$, the map $u$ factors as $\Omega_{B/k}\to coker(v)\to \Omega_{B/k}$. Now consider the natural projection map $\Omega_{B/k}\to coker(v)$. It maps to the zero map in $Hom(\Omega_{A/k}\otimes B, coker(v))$ and so (using the exactness of the sequence of Hom's) is induced by a map $\Omega_{B/A}\to coker(v)$, which is easily seen to be inverse to the map $coker(v)\to \Omega_{B/k}$.