Sheaf of relative Kähler differentials

Let $f\text{\hspace{0.17em}⁣}:X\to S$ be a morphism of schemes (Morphisms of schemes), so that $f$ is in particular a morphism of ringed spaces and comes with a map of sheaves of rings $f^{\mathrm{♯}}\text{\hspace{0.17em}⁣}:f^{-1}{\mathcal{O}}_S\to {\mathcal{O}}_X$ (Inverse image presheaf and inverse image sheaf); the pair $(X,f)$ is an $S$-scheme (Schemes and morphisms over a base). Let $\mathcal{F}$ be a sheaf of ${\mathcal{O}}_X$-modules (Modules on a ringed space).

$S$-derivations. An $S$-derivation of ${\mathcal{O}}_X$ into $\mathcal{F}$ is a morphism of sheaves of abelian groups $D\text{\hspace{0.17em}⁣}:{\mathcal{O}}_X\to \mathcal{F}$ such that for all local sections $a,a^{\mathrm{\prime }}$ of ${\mathcal{O}}_X$ over a common open set $W\subseteq X$ one has

$D(a+a^{\mathrm{\prime }})=D(a)+D(a^{\mathrm{\prime }}),D(a{a}^{\mathrm{\prime }})=aD(a^{\mathrm{\prime }})+a^{\mathrm{\prime }}D(a),$

and such that $D$ annihilates the image of $f^{\mathrm{♯}}\text{\hspace{0.17em}⁣}:f^{-1}{\mathcal{O}}_S\to {\mathcal{O}}_X$: the composite $f^{-1}{\mathcal{O}}_S\stackrel{f^{\mathrm{♯}}}{\to }{\mathcal{O}}_X\stackrel{D}{\to }\mathcal{F}$ is the zero map. Here the products and sums are taken in the rings ${\mathcal{O}}_X(W)$. The set of all such $D$ is written ${\mathrm{Der}}_S({\mathcal{O}}_X,\mathcal{F})$; it is a $\mathrm{\Gamma }(X,{\mathcal{O}}_X)$-module under the pointwise operations, and for a morphism $\mathcal{F}\to \mathcal{G}$ of ${\mathcal{O}}_X$-modules, postcomposition $D\mapsto \alpha \circ D$ maps ${\mathrm{Der}}_S({\mathcal{O}}_X,\mathcal{F})$ to ${\mathrm{Der}}_S({\mathcal{O}}_X,\mathcal{G})$. For $S$-schemes and $S$-morphisms the condition is that $D$ kills $f^{-1}{\mathcal{O}}_S$; when $S$ is fixed one simply says derivation.

Construction of ${\mathrm{\Omega }}_{X\mathrm{/}S}$. Consider the presheaf of ${\mathcal{O}}_X$-modules

$P\text{\hspace{0.17em}⁣}:W⟼{\mathrm{\Omega }}_{{\mathcal{O}}_X(W)\mathrm{/}(f^{-1}{\mathcal{O}}_S)(W)},$

where $W\subseteq X$ is open, the ring $(f^{-1}{\mathcal{O}}_S)(W)$ maps to ${\mathcal{O}}_X(W)$ by $f_W^{\mathrm{♯}}$, and ${\mathrm{\Omega }}_{{\mathcal{O}}_X(W)\mathrm{/}(f^{-1}{\mathcal{O}}_S)(W)}$ is the Kähler differential module of that ring map (Universal Kähler differential module), which exists by Existence and generators of Kähler differentials. For $W^{\mathrm{\prime }}\subseteq W$ the restriction $P(W)\to P(W^{\mathrm{\prime }})$ is the unique ${\mathcal{O}}_X(W)$-linear map induced, via the universal property of $P(W)$, by the derivation ${\mathcal{O}}_X(W)\to {\mathcal{O}}_X(W^{\mathrm{\prime }})\to P(W^{\mathrm{\prime }})$, where $P(W^{\mathrm{\prime }})$ is viewed as an ${\mathcal{O}}_X(W)$-module by restriction of scalars; the restriction maps compose, so $P$ is a presheaf of ${\mathcal{O}}_X$-modules. Define

${\mathrm{\Omega }}_{X\mathrm{/}S}:=aP,$

the sheafification of $P$ (Sheafification of a presheaf); this is a sheaf of ${\mathcal{O}}_X$-modules by defining scalar multiplication on local representatives in the double-plus construction, with equality on germs making the operation well defined. The universal derivations ${\mathrm{d}}_W\text{\hspace{0.17em}⁣}:{\mathcal{O}}_X(W)\to P(W)$ are compatible with the restriction maps by construction, so they define a morphism of presheaves and hence a morphism of sheaves

${\mathrm{d}}_{X\mathrm{/}S}\text{\hspace{0.17em}⁣}:{\mathcal{O}}_X⟶{\mathrm{\Omega }}_{X\mathrm{/}S},$

the universal $S$-derivation of $X$ over $S$, and ${\mathrm{d}}_{X\mathrm{/}S}$ is an $S$-derivation because each ${\mathrm{d}}_W$ is an $A_W$-derivation for $A_W=(f^{-1}{\mathcal{O}}_S)(W)$ and the maps $A_W\to {\mathcal{O}}_X(W)$ are the structure maps $f_W^{\mathrm{♯}}$.

Local descriptions. Two descriptions are used constantly and are recorded here for orientation; both are proved from the universal property in Universal property of relative differential sheaves and Affine charts recover the algebraic module of differentials.

1. Functor of points form. For every ${\mathcal{O}}_X$-module $\mathcal{F}$ there is a natural bijection ${\mathrm{Hom}}_{{\mathcal{O}}_X}({\mathrm{\Omega }}_{X\mathrm{/}S},\mathcal{F})\cong {\mathrm{Der}}_S({\mathcal{O}}_X,\mathcal{F})$, $g\mapsto g\circ {\mathrm{d}}_{X\mathrm{/}S}$; this is the universal property that characterizes ${\mathrm{\Omega }}_{X\mathrm{/}S}$.
2. Affine charts. If $U=\mathrm{Spec}B\subseteq X$ is an affine open subscheme and $f(U)\subseteq V=\mathrm{Spec}A$ for an affine open $V\subseteq S$, then the map $B\to {\mathrm{\Omega }}_{X\mathrm{/}S}(U)$, $b\mapsto {\mathrm{d}}_{X\mathrm{/}S}(b)$ exhibits ${\mathrm{\Omega }}_{X\mathrm{/}S}(U)$ as ${\mathrm{\Omega }}_{B\mathrm{/}A}$, compatibly with the universal derivations, and restriction to a basic open $D(g)\subseteq U$ corresponds to the localization ${\mathrm{\Omega }}_{B\mathrm{/}A}\to {\mathrm{\Omega }}_{B_g\mathrm{/}A}$.

Affine module convention. The affine description (2) identifies ${\mathrm{\Omega }}_{X\mathrm{/}S}$ on each affine chart with the sheaf attached to ${\mathrm{\Omega }}_{B\mathrm{/}A}$, with localization as restriction. This local description is the part used below; it needs no finiteness, flatness or separatedness hypothesis on $f$ and also applies to the identity $X\to X$.