Kähler differential (${\Omega }_{B/A}$)

For a ring map $A\to B$, the module of Kähler differentials represents $A$-derivations: ${\mathrm{Hom}}_B({\Omega }_{B/A},M)\cong {\mathrm{Der}}_A(B,M)$.

For an algebra $B$ over $A$, the module ${\Omega }_{B/A}$ represents $A$-linear derivations: every derivation $B\to N$ factors uniquely through the universal map $d:B\to {\Omega }_{B/A}$.

For finite separable $L/F$, every $k$-derivation on $F$ into an $L$-module extends uniquely to $L$. Differentiate the minimal polynomial of each algebraic generator and divide by its nonzero derivative. Consequently the Transitivity exact sequence for Kähler differentials identifies ${\Omega }_{L/k}$ with $L{\otimes }_F{\Omega }_{F/k}$, and ${\Omega }_{L/F}=0$.

The Kähler differentials commute with extension of the base ring. A derivation of the base-changed algebra that kills $k^{\prime }$ is determined by its restriction to $A\otimes 1$, and the Leibniz rule provides its unique extension. No flatness hypothesis is required for this isomorphism.

Kähler differentials commute with localization of a ring. The quotient rule $d(a/s)=(sda-ads)/s^2$ uniquely extends a derivation after inverting $s$; its universal property proves the displayed module isomorphism.

The Kähler differentials of a polynomial ring are free on the differentials of its variables. Assigning arbitrary values to these generators uniquely defines a derivation. Applying the Conormal exact sequence for Kähler differentials to a quotient gives its presentation by the formal derivatives of its defining equations.

Composing an $A$-linear map with $d:A\to {\Omega }_{A/k}$ gives a $k$-derivation. Every derivation factors uniquely this way because its linearity and product rule are exactly the defining relations of the module of Kähler differentials.

For a commutative $k$-algebra $A$, let $I$ be the kernel of multiplication $\mu :A{\otimes }_{k}A\to A$. The formula $a\mapsto [1\otimes a-a\otimes 1]$ is a derivation of an algebra into the quotient module $I/I^2$, with $A$ acting through the first factor. It induces an isomorphism from the module of Kähler differentials. An inverse is induced by $q(x\otimes y)=xdy$: the identity $q(rs)=\mu (r)q(s)+\mu (s)q(r)$ shows $q(I^2)=0$. For $z={\sum }_i{x}_i\otimes y_i\in I$, the equality ${\sum }_i{x}_i{y}_i=0$ gives $[z]={\sum }_i{x}_i[1\otimes y_i-y_i\otimes 1]$, proving that the two maps are inverse. For an affine variety, $I$ is the defining ideal of its diagonal morphism.

For $A\to B\to C=B/I$, there is a right-exact sequencewhere $[i]\mapsto di\otimes 1$ and $db\otimes c\mapsto cd\bar{b}$.

For a closed immersion, this quasi-coherent sheaf measures the first-order equations of the subvariety. Its local module is $I/I^2$, and its map to the restricted Kähler differential sheaf sends the class of $f$ to $df$. This is the definition in Stacks Project, Section 29.32.

For a closed immersion $i:Z\hookrightarrow X$ with ideal sheaf $\mathcal{I}$, the normal sheaf is ${\mathcal{N}}_{Z/X}=\mathcal{H}o{m}_{{\mathcal{O}}_Z}(\mathcal{I}/{\mathcal{I}}^2,{\mathcal{O}}_Z)$. It is the dual of the conormal sheaf. For a regular embedding it is a locally free sheaf; in the smooth case it agrees with the normal bundle. Without regularity it need not be locally free, but its global sections still classify first-order embedded deformations.

If a projective complete intersection $Z\subseteq {\mathbf{P}}^n$ is cut out by a homogeneous regular sequence of degrees $d_1,\dots ,d_c$, then $\mathcal{I}/{\mathcal{I}}^2\cong {⨁}_j{\mathcal{O}}_Z(-d_j)$ and ${\mathcal{N}}_{Z/{\mathbf{P}}^n}\cong {⨁}_j{\mathcal{O}}_Z(d_j)$. The classes of the defining equations form the conormal basis: the first syzygies of a regular sequence are the Koszul complex relations, whose coefficients vanish modulo the defining ideal. Thus the generator classes have no residual linear relations. Dualizing gives the stated normal sheaf.

Let $W$ be an integral effective Cartier divisor in an integral variety $V$ over a perfect field, and suppose $W$ is not contained in the singular locus of $V$. On a dense open subset both varieties are smooth. The Zariski tangent space of $W$ there has codimension one in that of $V$, so the differential of the local defining equation is nonzero. Thus the conormal sheaf map is injective generically. Its kernel is a subsheaf of a line bundle on the integral variety $W$ and is therefore a torsion-free sheaf; generic vanishing implies zero everywhere. Right exactness is the Conormal exact sequence for Kähler differentials.

For ring maps $A\to B\to C$, the first or transitivity exact sequence isThe maps send $db\otimes c$ to $cdb$ and then kill differentials of elements coming from $B$.

For a morphism of schemes $X\to Y$, the sheaf of relative Kähler differentials is obtained by sheafifying the modules ${\Omega }_{B/A}$ on affine charts $\mathrm{Spec}B\to \mathrm{Spec}A$.

The relative tangent sheaf is the dual of a sheaf of Kähler differentials. Its sections are derivations of the structure sheaf that kill base scalars. For a smooth morphism it is locally free and describes first-order deformations of sections into the scheme.

For $n\ge 1$, the cotangent form of the Euler sequence is $0\to {\Omega }_{{\mathbb{P}}^n/k}^1\to \mathcal{O}(-1{)}^{\oplus (n+1)}\to \mathcal{O}\to 0$. The cohomology of twisting sheaves on projective space and the long exact sequence in sheaf cohomology give $H^1({\mathbb{P}}_k^n,{\Omega }^1)\cong k$, with every other cohomology group zero. Therefore the Euler characteristic of a coherent sheaf is $-1$. The nonzero class is the connecting image of the constant section $1$.

For a variety over $k$, the module of Kähler differentials ${\Omega }_{A/k}$ on each affine chart gives a quasi-coherent sheaf. Compatibility with localization glues these into ${\Omega }_{X/k}^1$. A finite presentation of an algebra gives a finite presentation of a module for its differentials, so this is a coherent sheaf on a variety.

For a smooth algebraic variety of dimension $r$ over a field, the sheaf of Kähler differentials over a field is locally free of rank $r$ and defines its algebraic cotangent bundle. Its dual is the tangent sheaf of a scheme. For a smooth hypersurface of degree $d$ in ${\mathbb{P}}^n$, the Conormal exact sequence for Kähler differentials is the short exact sequence $0\to {\mathcal{O}}_X(-d)\to {\Omega }_{{\mathbb{P}}^n}^1{|}_X\to {\Omega }_X^1\to 0$.

An invertible maximal-rank minor in the Jacobian matrix eliminates generators and gives a local surjection from ${\mathcal{O}}_X^n$ to the Kähler differential sheaf. For an integral smooth variety the map is an isomorphism at the generic point. Its kernel is a submodule of a free module over an integral domain, so it is a torsion-free module; vanishing generically forces it to be zero. This gives local free frames for differentials.

For a smooth $n$-dimensional variety $X$ over a field, its canonical line bundle is ${\omega }_X={\bigwedge }^n{\Omega }_{X/k}^1$. It is the bundle of top-degree algebraic differential forms. A nonzero rational section defines a canonical divisor of a smooth variety. This algebraic definition works in arbitrary characteristic and is distinct from restricting the analytic canonical-bundle definition to complex manifolds.

A canonical divisor $K_X$ on a smooth integral variety is the Cartier divisor of a nonzero rational top-degree differential form; ${\mathcal{O}}_X(K_X)\cong {\omega }_X$. Its linear-equivalence class is independent of the chosen form. The plurigenus is the dimension of the space of sections of a positive tensor power of this bundle.

For a smooth projective variety, its $m$th plurigenus is $P_m(X)=h^0(X,{\omega }_X^{\otimes m})$, for $m\ge 1$. These dimensions describe pluricanonical linear systems. Blowup invariance of plurigenera proves their invariance under a point blowup on a smooth surface.

For a point blowup of a smooth algebraic surface $\psi :\tilde{X}\to X$, where $X$ is a smooth projective surface over an algebraically closed field, multiplication by the exceptional section to power $m$ identifies $H^0(\tilde{X},m{\psi }^{\ast }{K}_X)$ with $H^0(\tilde{X},m{K}_{\tilde{X}})$. Every effective pluricanonical representative must contain $mE$, since after subtracting $j<m$ copies its intersection with $E$ is $-m+j<0$. The projection formula for sheaves and ${\psi }_{\ast }\mathcal{O}=\mathcal{O}$ identify the remaining sections with those on $X$. No effectivity of $K_X$ itself is assumed.