The sheaf of differentials, and its quasicoherence (Tag D-4GCH6)
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 \(T^m-2\).
Quasicoherence follows because the affine construction commutes with localization. For \(X\) of finite type over a field \(k\) and equidimensional of dimension \(n\), \(X\) is smooth over \(k\) exactly when \(\Omega_{X/k}\) is locally free of rank \(n\); on an affine chart this is the Jacobian criterion.
The two sequences
The relative sequence, for \(X \to Y \to S\): \[ f^*\Omega_{Y/S} \to \Omega_{X/S} \to \Omega_{X/Y} \to 0 . \] The conormal sequence, for a closed immersion \(Z \subseteq X\) with ideal \(\mci\): \[ \mci/\mci^2 \to \Omega_{X/S}\ro{}{Z} \to \Omega_{Z/S} \to 0 . \]
Neither sequence is left exact in general. For the Frobenius \(F\colon\AA^1_k\to\AA^1_k\) in characteristic \(p\), the map \(F^*\Omega_{\AA^1}\to\Omega_{\AA^1}\) is zero. For \(Z=V(x^2)\subset\AA^1_k\) with \(\operatorname{char}k\neq3\), the class of \(x^3\) in \(\mci/\mci^2=(x^2)/(x^4)\) is nonzero and maps to \(3x^2\,dx=0\) in \(\Omega_{\AA^1}|_Z\).
The relative sequence is exact on the left when \(X\to Y\) is smooth. For a finite separable morphism \(f\colon X\to Y\) of smooth curves, \(0\to f^*\Omega_Y\to\Omega_X\to\Omega_{X/Y}\to0\) is exact, \(\Omega_{X/Y}\) is a torsion sheaf whose lengths give the ramification divisor \(R\), and \(\Omega_X\cong f^*\Omega_Y(R)\). The conormal sequence is exact on the left when \(X\) and \(Z\) are smooth over a field, and taking determinants then gives adjunction, \[ \omega_Z = \left( \omega_X \tensor \dualof{\det(\mci/\mci^2)} \right)\ro{}{Z} . \] For a smooth curve \(C\) on a smooth surface \(X\), \(\det(\mci/\mci^2)^\vee=\OO_X(C)|_C\), so \(\omega_C=(\omega_X\otimes\OO_X(C))|_C\); see curves and surfaces.
The conormal and normal sheaves, and adjunction (Tag D-MODCONORM)
Let \(i:Z\hookrightarrow X\) be a closed immersion of schemes with ideal sheaf \(\mci\). The conormal sheaf is \(\mci/\mci^2\), regarded as an \(\OO_Z\)-module. Its dual is the normal sheaf \[ \mcn_{Z/X}\coloneqq\sheafhom_{\OO_Z}(\mci/\mci^2,\OO_Z). \]
(Determinant)
For a locally free sheaf \(\mcf\) of finite rank \(r\) on a scheme \(X\), its determinant is the invertible sheaf \(\det\mcf\coloneqq\bigwedge^r\mcf\). For a smooth variety \(X\) over a field \(k\), the canonical invertible sheaf is \(\omega_X=\det\Omega_{X/k}\) [@Har10a, Chapter II, §8].
(Determinants of exact sequences)
Let \(X\) be a scheme and let \(0\to\mcf_1\to\cdots\to\mcf_n\to0\) be an exact sequence of locally free sheaves of finite rank. With exponent \(-1\) denoting the dual of an invertible sheaf, there is a canonical isomorphism \[ \bigotimes_{i=1}^n(\det\mcf_i)^{\otimes(-1)^{i+1}}\cong\OO_X. \] For \(n=3\) it is \(\det\mcf_2\cong\det\mcf_1\otimes\det\mcf_3\). The short-exact-sequence map is proved in P-AGH2516TENSOROPS, part (d); applying it to the successive image sheaves gives the displayed cancellation for a longer exact sequence [@Har10a, Exercise II.5.16].
(Adjunction)
Let \(i:Z\hookrightarrow X\) be a closed immersion of smooth varieties over a field \(k\), of dimensions \(m\) and \(n\), with ideal sheaf \(\mci\). Then the conormal sequence is short exact, \[ 0\longrightarrow\mci/\mci^2\longrightarrow i^*\Omega_{X/k}\longrightarrow\Omega_{Z/k}\longrightarrow0, \] and \(\mci/\mci^2\) is locally free of rank \(n-m\). Taking determinants gives \[ \omega_Z\cong i^*\omega_X\otimes_{\OO_Z}\bigwedge^{n-m}\mcn_{Z/X}. \] [@Har10a, Theorem II.8.17 and Proposition II.8.20]
(Adjunction for a divisor)
Let \(X\) be a smooth variety over a field \(k\), and let \(i:Z\hookrightarrow X\) be a smooth effective Cartier divisor. Its ideal sheaf is \(\OO_X(-Z)\), and \[ \omega_Z\cong i^*\bigl(\omega_X\otimes_{\OO_X}\OO_X(Z)\bigr). \] [@Har10a, Proposition II.8.20]
(A free conormal module on a nonsmooth subscheme)
Let \(k\) be a field, let \(X=\Spec k[x]\), and let \(Z\) be the closed subscheme defined by \(I=(x^2)\). Put \(B=k[x]/(x^2)\). The map \[ B\longrightarrow I/I^2,\qquad a\bmod x^2\longmapsto ax^2\bmod x^4 \] is an isomorphism: it is surjective, and \(ax^2\in(x^4)\) holds exactly when \(a\in(x^2)\). Thus the conormal sheaf is free of rank one on \(Z\). The element \(x\bmod x^2\) is nonzero and nilpotent, so \(Z\) is not reduced and is not smooth over \(k\) [@Har10a, Chapter III, §10]. Consequently local freeness of the conormal sheaf does not imply smoothness of the closed subscheme.
(Canonical sheaf and genus of a smooth plane curve)
Let \(k\) be algebraically closed and let \(C\subseteq\PP_k^2\) be a smooth integral plane curve of degree \(d\ge1\). The identities \(\omega_{\PP_k^2}\cong\OO(-3)\) and \(\OO(C)\cong\OO(d)\) give \[ \omega_C\cong\OO_C(d-3),\qquad\deg\omega_C=d(d-3). \] Since \(\deg\OO_C(1)=d\) and \(\deg\omega_C=2g(C)-2\), it follows that \[ g(C)=\frac{(d-1)(d-2)}2. \] [@Har10a, Example II.8.20.3 and Chapter IV, §1]
The canonical sheaf
For \(X\) smooth of dimension \(n\), \(\omega_X = \det \Omega_{X/k} = \bigwedge^n \Omega_{X/k}\). On \(\PP^n\) it is \(\OO(-n-1)\). Hence \(H^0(\PP^1,\Omega^1)=H^0(\PP^1,\OO(-2))=0\); \(\omega_{\PP^n}^\vee=\OO(n+1)\) is ample, so \(\PP^n\) is Fano; and adjunction gives \(\omega_C\cong\OO_C(d-3)\) for a smooth plane curve \(C\) of degree \(d\), so \(2g-2=d(d-3)\).
Projective space
The Euler sequence on projective space (Tag T-MODEULER)
For \(X = \PP^n_A\) there is a short exact sequence of \(\OO_X\)-modules \[ 0 \to \Omega_{X/A} \to \OO_X(-1)\sumpower{n+1} \to \OO_X \to 0 . \]
The sequence is the sheaf-level record of \(\PP^n = (\AA^{n+1} \setminus 0)/\GG_m\): the middle term is the pullback of the cotangent bundle of the affine space, twisted, and the surjection onto \(\OO_X\) is contraction with the Euler vector field, whose kernel is the forms that descend. Dually, \(0 \to \OO_X \to \OO_X(1)\sumpower{n+1} \to T_X \to 0\), and the \(\OO_X\) is the Euler field itself.
Its determinant computes the canonical bundle. Taking top exterior powers in a short exact sequence multiplies them, so \[ \omega_{\PP^n} = \Extpower^n \Omega = \Extpower^{n+1}\qty{\OO(-1)\sumpower{n+1}} \tensor \dualof{\OO} = \OO(-n-1) , \] which is the one computation that the genus formulas, the Fano property of \(\PP^n\), and every application of adjunction in projective space come out of.
The sequence does not split for \(n \geq 1\), and for \(n \geq 2\) the sheaf \(\Omega_{\PP^n}\) is locally free but not a direct sum of line bundles: the Euler sequence is the standard witness that locally free does not mean split.
On \(\PP^1\) with coordinates \(x, y\), the dual Euler sequence is \(0 \to \OO \xrightarrow{(x,\,y)} \OO(1)^{\oplus 2} \to T_{\PP^1} \to 0\). So the cokernel of \((x, y) \colon \OO \to \OO(1)^{\oplus 2}\) is \(T_{\PP^1} \cong \OO(2)\), the determinant of \(\OO(1)^{\oplus 2}\); explicitly \((s, t) \mapsto y s - x t\) identifies it with \(\OO(2)\).
The Euler sequence and its top exterior power give \(\omega_{\PP^n}\cong\OO(-n-1)\).
The de Rham complex
The de Rham differential is not $\OO_X$-linear (Tag FE-DERHAMNONLIN)
The de Rham differential \(d \colon \OO_X \to \Omega^1_{X/k}\) satisfies the Leibniz rule \(d(fg) = f\, dg + g\, df\), which differs from the \(\OO_X\)-linearity condition \(d(fg) = f\, dg\) by the term \(g\, df\). On \(X = \AA^1_k = \Spec k[x]\), take \(f = x\) and \(g = 1\): \(d(x \cdot 1) = dx\), while \(x\, d(1) = 0\), and \(dx \neq 0\) since \(\Omega^1_{k[x]/k}\) is free on \(dx\). So \(d\) is \(k\)-linear but not \(\OO_X\)-linear, and the de Rham complex \(\Omega^\bullet_{X/k}\) is a complex of sheaves of \(k\)-vector spaces whose terms are \(\OO_X\)-modules, not a complex of \(\OO_X\)-modules.
This is why algebraic de Rham cohomology is the hypercohomology \(\mathbb{H}^\bullet(X, \Omega^\bullet_{X/k})\) of a complex of \(k\)-sheaves, and why its Hodge-to-de Rham spectral sequence starts from the \(\OO_X\)-module cohomology groups \(H^q(X, \Omega^p_{X/k})\) of the individual terms.