Question about the exact sequences of sheaves of relative differentials

Question

Hartshorne gave the exact sequences in Chapter II.8 and just say they followed from the affine case. But this view should consider the compkex glueing construction. I am disturbed completely. How should I thunk this? Thanks!

Answer 1

I think the usual arguments "by gluing" can be replaced by more conceptual, global arguments. Namely, the sheaf of differentials has itsself a universal property, classifying global derivations. Actually this works for every morphism of (locally) ringed spaces. Then you can just write down the proof for rings also for ringed sapces, nothing changes at all.

In detail, if $f : X \to S$ is a morphism of ringed spaces and $M$ is a $\mathcal{O}_X$ module, then a $\mathcal{O}_S$-derivation $d : \mathcal{O}_X \to M$ is a morphism of abelian sheaves on $X$ such that for all $U \subseteq S$ open and $W \subseteq f^{-1}(U)$ open, the induced map $\mathcal{O}_X(W) \to M(W)$ is a $\mathcal{O}_S(U)$-derivation. Here, $\mathcal{O}_X(W)$ is considered as a $\mathcal{O}_S(U)$-algebra via $f$. We get the set of derivations $\text{Der}_{\mathcal{O}_S}(\mathcal{O}_X,M)$.

Now it is easy to see that the functor $\text{Der}_{\mathcal{O}_S}(\mathcal{O}_X,-)$ is representable by an object ${\Omega}^1_{X/S}$, by some colimit construction of the ${\Omega}^1_{\mathcal{O}_X(W) / \mathcal{O}_S(U)}$. For the details, see here. There you can also find a simple argument why $\Omega^1_{X/S}$ is quasi-coherent for a morphism of schemes $X/S$ and coincides with the usual gluing ad hoc definition.

Now let us consider for example the following exact sequence:

Let $X \to S$, $Y \to S$ be morphisms of ringed spaces and $f : Y \to X$ a morphism of ringed spaces over $S$. Then there is an exact sequence of $\mathcal{O}_Y$-modules $f^* \Omega^1_{X/S} \to \Omega^1_{Y/S} \to \Omega^1_{Y/X} \to 0$.

Proof: By the Yoneda-Lemma and the definition of a cokernel, it suffices to construct for ${\mathcal{O}}_Y$-modules $N$ a natural exact sequence

$0 \to Hom(\Omega^1_{Y/X},N) \to Hom(\Omega^1_{Y/S},N) \to Hom(f^* \Omega^1_{X/S},N)$

of abelian groups. This corresponds to an exact sequence

$0 \to \text{Der}_{\mathcal{O}_X}(\mathcal{O}_Y,N) \to \text{Der}_{\mathcal{O}_S}(\mathcal{O}_Y,N) \to \text{Der}_{\mathcal{O}_S}(\mathcal{O}_X,f_* N)$.

But now the left map is defined to be the obvious inclusion, and the right one is given by composing with ${\mathcal{O}}_X \to f_{*} {\mathcal{O}}_Y$. The exactness is now clear. If not, use the definition of derivations given above to reduce to the case of rings. QED

Answer 2

Actually, the point is, once you have a map, exactness is local. In other words, the maps exist in general (this does follow by gluing if you want, but it's easy, because the maps are natural and I think Hartshorne does this earlier). Then to check that the sequence is actually exact you can restrict to any open (in particular affine) subset.