2_8x Differentials

II.8: Differentials

II.8.1 #to_work

Here we will strengthen the results of the text to include information about the sheaf of differentials at a not necessarily closed point of a scheme \(X\).

• Generalize (8.7) as follows. Let \(B\) be a local ring containing a field \(k\), and assume that the residue field \(k(B)=B / \mathfrak{m}\) of \(B\) is a separably generated extension of \(k\). Then the exact sequence of (8.4A), \begin{align*} 0 \rightarrow \mathrm{m} / \mathrm{m}^2 \stackrel{\delta}{\rightarrow} \Omega_{B / k} \otimes k(B) \rightarrow \Omega_{k(B) / k} \rightarrow 0 \end{align*} is exact on the left also. ¹

• Generalize (8.8) as follows. With \(B, k\) as above, assume furthermore that \(k\) is perfect, and that \(B\) is a localization of an algebra of finite type over \(k\). Then show that \(B\) is a regular local ring if and only if \(\Omega_{B / k}\) is free of rank \(=\operatorname{dim} B+\) tr.d. \(k(B) / k\).

• Strengthen (8.15) as follows. Let \(X\) be an irreducible scheme of finite type over a perfect field \(k\), and let \(\operatorname{dim} X=n\). For any point \(x \in X\), not necessarily closed, show that the local ring \(\mathcal{O}_{x, X}\) is a regular local ring if and only if the stalk \(\left(\Omega_{X / k}\right)_x\) of the sheaf of differentials at \(x\) is free of rank \(n\).

• Strengthen (8.16) as follows. If \(X\) is a variety over an algebraically closed field \(k\), then \(U=\left\{x \in X \mathrel{\Big|}\mathcal{O}_x\right.\) is a regular local ring \(\}\) is an open dense subset of \(X\).

II.8.2. #to_work

Let \(X\) be a variety of dimension \(n\) over \(k\). Let \(\mathcal{E}\) be a locally free sheaf of rank \(>n\) on \(X\), and let \(V \subseteq \Gamma(X, \mathcal{E})\) be a vector space of global sections which generate \(\mathcal{E}\). Then show that there is an element \(s \in V\), such that for each \(x \in X\), we have \(s_x \notin \mathfrak{m}_x \mathcal{E}_x\). Conclude that there is a morphism \(\mathcal{O}_X \rightarrow \mathcal{E}\) giving rise to an exact sequence \begin{align*} 0 \rightarrow \mathcal{O}_X \rightarrow \mathcal{E} \rightarrow \mathcal{E}^{\prime} \rightarrow 0 \end{align*} where \(\mathcal{E}'\) is also locally free. ²

II.8.3. Product Schemes. #to_work

• Let \(X\) and \(Y\) be schemes over another scheme \(S\). Use (8.10) and (8.11) to show that \begin{align*} \Omega_{X \underset{\scriptscriptstyle {S} }{\times} Y/S} \cong p_1^* \Omega_{X / S} \oplus p_2^* \Omega_{Y / S} .\end{align*}

• If \(X\) and \(Y\) are nonsingular varieties over a field \(k\), show that \begin{align*} \omega_{X \times Y} \cong p_1^* \omega_X \otimes p_2^* \omega_Y .\end{align*}

• Let \(Y\) be a nonsingular plane cubic curve, and let \(X\) be the surface \(Y \times Y\). Show that \(p_g(X)=1\) but \(p_a(X)=-1\) (I, Ex. 7.2). This shows that the arithmetic genus and the geometric genus of a nonsingular projective variety may be different.

II.8.4. Complete Intersections in \(\mathbf{P}^n\). #to_work

A closed subscheme \(Y\) of \(\mathbf{P}_k^n\) is called a (strict, global) complete intersection if the homogeneous ideal \(I\) of \(Y\) in \(S=k\left[x_0, \ldots, x_n\right]\) can be generated by \(r=\operatorname{codim}\left(Y, \mathbf{P}^n\right)\) elements (I, Ex. 2.17).

• Let \(Y\) be a closed subscheme of codimension \(r\) in \(\mathbf{P}^n\). Then \(Y\) is a complete intersection if and only if there are hypersurfaces (i.e., locally principal subschemes of codimension 1) \(H_1, \ldots, H_r\), such that \(Y=H_1 \cap \ldots \cap H_r\) as schemes, i.e., \(\mathcal{I}_Y=\mathcal{I}_{H_1}+\ldots+\mathcal{I}_{H_r}\). ³

• If \(Y\) is a complete intersection of dimension \(\geqslant 1\) in \(\mathbf{P}^n\), and if \(Y\) is normal, then \(Y\) is projectively normal (Ex. 5.14). ⁴

• With the same hypotheses as (b), conclude that for all \(l \geqslant 0\), the natural map \(\Gamma\left(\mathbf{P}^n, \mathcal{O}_{\mathbf{P}^n}(l)\right) \rightarrow \Gamma\left(Y, \mathcal{O}_Y(l)\right)\) is surjective. In particular, taking \(l=0\), show that \(Y\) is connected.

• Now suppose given integers \(d_1, \ldots, d_r \geqslant 1\), with \(r<n\). Use Bertini’s theorem (8.18) to show that there exist nonsingular hypersurfaces \(H_1, \ldots, H_r\) in \(\mathbf{P}^n\), with deg \(H_i=d_i\), such that the scheme \(Y=H_1 \cap \ldots \cap H_r\) is irreducible and nonsingular of codimension \(r\) in \(\mathbf{P}^n\).

• If \(Y\) is a nonsingular complete intersection as in (d), show that \begin{align*} \omega_Y \cong \mathcal{O}_Y\left(\sum d_i-n-1\right) .\end{align*}

• If \(Y\) is a nonsingular hypersurface of degree \(d\) in \(\mathbf{P}^n\), use (c) and (e) above to show that \(p_g(Y)={d-1\choose n}\). Thus \(p_g(Y)=p_a(Y)\) (I, Ex. 7.2). In particular, if \(Y\) is a nonsingular plane curve of degree \(d\), then \begin{align*} p_g(Y)=\frac{1}{2}(d-1)(d-2) .\end{align*}

• If \(Y\) is a nonsingular curve in \(\mathbf{P}^3\), which is a complete intersection of nonsingular surfaces of degrees \(d, e\), then \begin{align*} p_g(Y)=\frac{1}{2} d e(d+e-4)+1 .\end{align*} Again the geometric genus is the same as the arithmetic genus (I, Ex. 7.2).

II.8.5. Blowing up a Nonsingular Subvariety. #to_work

As in (8.24), let \(X\) be a nonsingular variety, let \(Y\) be a nonsingular subvariety of codimension \(r \geqslant 2\), let \(\pi: \widetilde{X} \rightarrow X\) be the blowing-up of \(X\) along \(Y\), and let \(Y^{\prime}=\pi^{-1}(Y)\).

• Show that the maps \(\pi^*:\operatorname{Pic}X \rightarrow\operatorname{Pic}\tilde{X}\), and \(\mathbf{Z} \rightarrow \operatorname{Pic}X\) defined by \(n \mapsto\) class of \(n Y^{\prime}\), give rise to an isomorphism \(\operatorname{Pic}\tilde{X} \cong \operatorname{Pic} X \oplus \mathbf{Z}\).

• Show that ⁵ \begin{align*} \omega_{\tilde{x}} \cong f^* \omega_X \otimes \mathcal{L}\left((r-1) Y^{\prime}\right) .\end{align*}

II.8.6. The Infinitesimal Lifting Property. #to_work

The following result is very important in studying deformations of nonsingular varieties. Let \(k\) be an algebraically closed field, let \(A\) be a finitely generated \(k\)-algebra such that \(\operatorname{Spec} A\) is a nonsingular variety over \(k\). Let \begin{align*} 0 \rightarrow I \rightarrow B^{\prime} \rightarrow B \rightarrow 0 \end{align*} be an exact sequence, where \(B^{\prime}\) is a \(k\)-algebra, and \(I\) is an ideal with \(I^2=0\). Finally suppose given a \(k\)-algebra homomorphism \(f: A \rightarrow B\). Then there exists a \(k\)-algebra homomorphism \(g: A \rightarrow B^{\prime}\) making a commutative diagram

Link to Diagram

We call this result the infinitesimal lifting property for \(A\). We prove this result in several steps.

• First suppose that \(g: A \rightarrow B^{\prime}\) is a given homomorphism lifting \(f\). If \(g^{\prime}: A \rightarrow B^{\prime}\) is another such homomorphism, show that \(\theta=g-g^{\prime}\) is a \(k\)-derivation of \(A\) into \(I\), which we can consider as an element of \(\operatorname{Hom}_A\left(\Omega_{A / k}, I\right)\). Note that since \(I^2=0, I\) has a natural structure of \(B\)-module and hence also of \(A\)-module. Conversely, for any \(\theta \in \operatorname{Hom}_A\left(\Omega_{A / k}, I\right), g^{\prime}=g+\theta\) is another homomorphism lifting \(f\). (For this step, you do not need the hypothesis about Spec \(A\) being nonsingular.)

• Now let \(P=k\left[x_1, \ldots, x_n\right]\) be a polynomial ring over \(k\) of which \(A\) is a quotient, and let \(J\) be the kernel. Show that there does exist a homomorphism \(h: P \rightarrow B^{\prime}\) making a commutative diagram,

  

  Link to Diagram

  and show that \(h\) induces an \(A\)-linear map \(\mkern 1.5mu\overline{\mkern-1.5muh\mkern-1.5mu}\mkern 1.5mu: J / J^2 \rightarrow I\).

• Now use the hypothesis Spec \(A\) nonsingular and (8.17) to obtain an exact sequence \begin{align*} 0 \rightarrow J / J^2 \rightarrow \Omega_{P / k} \otimes A \rightarrow \Omega_{A / k} \rightarrow 0 . \end{align*} Show furthermore that applying the functor \(\operatorname{Hom}_A(\cdot, I)\) gives an exact sequence \begin{align*} 0 \rightarrow \operatorname{Hom}_A\left(\Omega_{A / k}, I\right) \rightarrow \operatorname{Hom}_P\left(\Omega_{P / k}, I\right) \rightarrow \operatorname{Hom}_A\left(J / J^2, I\right) \rightarrow 0 . \end{align*} Let \(\theta \in \operatorname{Hom}_P\left(\Omega_{P / k}, I\right)\) be an element whose image gives \(\mkern 1.5mu\overline{\mkern-1.5muh\mkern-1.5mu}\mkern 1.5mu\in \operatorname{Hom}_A\left(J / J^2, I\right)\). Consider \(\theta\) as a derivation of \(P\) to \(B^{\prime}\). Then let \(h^{\prime}=h-\theta\), and show that \(h^{\prime}\) is a homomorphism of \(P \rightarrow B^{\prime}\) such that \(h^{\prime}(J)=0\). Thus \(h^{\prime}\) induces the desired homomorphism \(g: A \rightarrow B^{\prime}\).

II.8.7. #to_work

As an application of the infinitesimal lifting property, we consider the following general problem. Let \(X\) be a scheme of finite type over \(k\), and let \(\mathcal{F}\) be a coherent sheaf on \(X\). We seek to classify schemes \(X^{\prime}\) over \(k\), which have a sheaf of ideals \(\mathcal{I}\) such that \(\mathcal{I}^2=0\) and \(\left(X^{\prime}, \mathcal{O}_X / \mathcal{I}\right) \cong\left(X, \mathcal{O}_X\right)\), and such that \(\mathcal{I}\) with its resulting structure of \(\mathcal{O}_X\)-module is isomorphic to the given sheaf \(\mathcal{F}\). Such a pair \(X^{\prime}, \mathcal{I}\) we call an infinitesimal extension of the scheme \(X\) by the sheaf \(\mathcal{F}\).

One such extension, the trivial one, is obtained as follows. Take \(\mathcal{O}_{X^{\prime}}=\mathcal{O}_X \oplus \mathcal{F}\) as sheaves of abelian groups, and define multiplication by \begin{align*} (a \oplus f) \cdot\left(a^{\prime} \oplus f^{\prime}\right)=a a^{\prime} \oplus \left(a f^{\prime}+a^{\prime} f\right) .\end{align*} Then the topological space \(X\) with the sheaf of rings \(\mathcal{O}_{X^{\prime}}\) is an infinitesimal extension of \(X\) by \(\mathcal{F}\).

The general problem of classifying extensions of \(X\) by \(\mathcal{F}\) can be quite complicated. So for now, just prove the following special case: if \(X\) is affine and nonsingular, then any extension of \(X\) by a coherent sheaf \(\mathcal{F}\) is isomorphic to the trivial one. See (III, Ex. 4.10) for another case.

II.8.8. Plurigenus and (some) Hodge numbers are birational invariants. #to_work

Let \(X\) be a projective nonsingular variety over \(k\). For any \(n>0\) we define the \(n\)th plurigenus of \(X\) to be \begin{align*} P_n=\operatorname{dim}_k \Gamma\left(X, \omega_X^{\otimes n}\right) .\end{align*} Thus in particular \(P_1=p_g\). Also, for any \(q, 0 \leqslant q \leqslant \operatorname{dim} X\) we define an integer \begin{align*} h^{q, 0}=\operatorname{dim}_k \Gamma\left(X, \Omega_{X / k}^q\right) \quad\text{where}\quad \Omega_{X / k}^q=\bigwedge^q \Omega_{X / k} .\end{align*} is the sheaf of regular \(q\)-forms on \(X\). In particular, for \(q=\operatorname{dim} X\), we recover the geometric genus again. The integers \(h^{q, 0}\) are called Hodge numbers.

Using the method of \((8.19)\), show that \(P_n\) and \(h^{q, 0}\) are birational invariants of \(X\), i.e., if \(X\) and \(X^{\prime}\) are birationally equivalent nonsingular projective varieties, then \(P_n(X)=P_n\left(X^{\prime}\right)\) and \(h^{q, 0}(X)=h^{q, 0}\left(X^{\prime}\right)\).