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    The sheaf of differentials, and its quasicoherence

    D-4GCH6

    (Kähler differentials)

    Let \(A\to B\) be a ring map. The module of Kähler differentials \(\Omega_{B/A}\) is the \(B\)-module generated by symbols \(db\) for \(b\in B\), subject to additivity, \(da=0\) for \(a\) in the image of \(A\), and the Leibniz rule \(d(bb')=b\,db'+b'\,db\). For every \(B\)-module \(M\), composition with \(d\) gives a natural bijection \[ \Hom_B(\Omega_{B/A},M)\cong\Der_A(B,M). \] Here \(\Der_A(B,M)\) denotes the additive maps \(D:B\to M\) annihilating \(A\) and satisfying \(D(bb')=bD(b')+b'D(b)\) [@Har10a, Chapter II, §8].

    (Sheaf of differentials)

    For a morphism of schemes \(f:X\to Y\), the sheaf of relative differentials \(\Omega_{X/Y}\) is obtained by gluing the sheaves \(\widetilde{\Omega_{B/A}}\) on affine opens \(\Spec B\subseteq X\) mapping into affine opens \(\Spec A\subseteq Y\) [@Har10a, Chapter II, §8].

    \(\Omega_{X/Y}\) is quasi-coherent, and it is coherent when \(Y\) is noetherian and \(f\) is of finite type. Differentials commute with localization: \(\Omega_{B_s/A}\cong(\Omega_{B/A})_s\). For the affine diagonal with ideal \(J=\ker(B\otimes_A B\to B)\), there is a canonical isomorphism \(\Omega_{B/A}\cong J/J^2\), sending \(db\) to \(1\otimes b-b\otimes1\). Thus differentials are also the conormal sheaf of the diagonal; this description is computed on affine neighborhoods, and for a separated morphism the diagonal itself is a closed immersion [@Har10a, Chapter II, §8].

    (Differential exact sequences)

    For morphisms \(X\xrightarrow{f}Y\to S\), there is a natural exact sequence \[ f^*\Omega_{Y/S} \to \Omega_{X/S} \to \Omega_{X/Y} \to 0 . \] For an \(S\)-scheme \(X\) and a closed immersion \(i:Z\hookrightarrow X\) with ideal sheaf \(\mci\), there is a natural exact sequence \[ \mci/\mci^2\longrightarrow i^*\Omega_{X/S}\longrightarrow\Omega_{Z/S}\longrightarrow0. \] Its first map sends the class of a local section \(g\) of \(\mci\) to \(dg\otimes1\). Affinely, for \(B\to A\) and an ideal \(I\subseteq A\), it is the map \(I/I^2\to\Omega_{A/B}\otimes_A A/I\) [@Har10a, Propositions II.8.4A, II.8.11 and II.8.12].

    (Regularity and differential rank)

    Let \(X\) be an irreducible scheme of finite type over a perfect field \(k\), of dimension \(n\). For every point \(x\in X\), the local ring \(\OO_{X,x}\) is regular if and only if \((\Omega_{X/k})_x\) is free of rank \(n\), as proved in P-AGH281DIFFNONCLOSED, part (c). Consequently \(X\) is smooth over \(k\) if and only if \(\Omega_{X/k}\) is locally free of rank \(n\). Here regularity and geometric regularity agree over a perfect field, and geometric regularity is equivalent to smoothness for schemes locally of finite type over a field.

    (Zero differentials without finite type)

    Let \(K\) be an algebraic closure of \(\QQ\). The scheme \(\Spec K\) has dimension zero and \(\Omega_{K/\QQ}=0\): for \(\alpha\in K\), differentiation of its separable minimal polynomial gives \(p'(\alpha)d\alpha=0\) with \(p'(\alpha)\ne0\), so \(d\alpha=0\). Nevertheless \(\Spec K\to\Spec\QQ\) is not smooth, because it is not of finite type. A finite list of algebraic generators would generate a finite extension of \(\QQ\), whereas \(K\) contains elements of arbitrarily large degree, for example roots of the Eisenstein polynomials \(T^m-2\).

    Define differentials.
    Are differentials quasicoherent?
    What is $\Omega_{X/k}$ for $X$ smooth?

    Dependencies

    • D-QNTZYQuasicoherent and coherent sheavesuses

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