Let \(X\) be a nonsingular variety over an algebraically closed field \(k\), and let \(\mcf\) be a coherent sheaf on \(X\). Show that there is a one-to-one correspondence between the set of infinitesimal extensions of \(X\) by \(\mcf\) (II, Ex. 8.7) up to isomorphism of extensions, and the group \(H^1(X,\mcf\tensor\mct)\), where \(\mct\) is the tangent sheaf of \(X\), see (II, §8). An extension here includes its identification of the quotient with \(X\) and its square-zero ideal with \(\mcf\); an isomorphism of extensions induces the identity on both \(X\) and the identified sheaf \(\mcf\).
(Forgetting the kernel identification changes the classification)
For an automorphism \(v\) of \(\mcf\), the map \((a,m)\mapsto(a,v(m))\) conjugates \(u_D\) to \(u_{v\circ D}\). Thus forgetting the identification of the kernel with \(\mcf\), while fixing \(X\), identifies classes under the induced action of \(\Aut_{\OO_X}(\mcf)\). For example, on \(X=\PP_k^1\) with \(\mcf=\OO(-4)\), one has \(\mathcal T\cong\OO(2)\) and \(H^1(X,\mcf\otimes\mathcal T)\cong H^1(\PP_k^1,\OO(-2))\cong k\) [@Har10a, Example II.8.20.1 and Theorem III.5.1]. Scalar automorphisms of \(\mcf\) identify distinct nonzero scalar multiples of a class when the kernel identification is forgotten. The correspondence proved here retains that identification, as required for isomorphisms of extensions.
Hint
Use (II, Ex. 8.6) and Theorem III.4.5.
Solution
Write \(\mathcal T=\sheafhom_{\OO_X}(\Omega_{X/k},\OO_X)\) and \[ \mathcal D=\sheafhom_{\OO_X}(\Omega_{X/k},\mcf). \] Regard an extension as a sheaf of \(k\)-algebras \(\mathcal A\) on the underlying space of \(X\), with an exact sequence \[ 0\longrightarrow\mcf\xrightarrow{\iota}\mathcal A\xrightarrow{\rho}\OO_X\longrightarrow0, \] where \(\iota(\mcf)\) is an ideal of square zero and its induced quotient-module structure is the given one on \(\mcf\). This is legitimate because a nilpotent closed immersion is a homeomorphism on the underlying spaces. The ringed space \((X,\mathcal A)\) is required to be a scheme, as in P-AGH287INFEXT.
1. The abelian sheaf \(\mathcal D\) is the sheaf of \(k\)-derivations \(\OO_X\to\mcf\), and there is a natural isomorphism \(\mathcal D\cong\mcf\otimes\mathcal T\).
The universal property of Kähler differentials identifies a sheaf morphism \(\Omega_{X/k}\to\mcf\) with its composite with the universal derivation. This works on every open subset and commutes with restriction, giving the first assertion. Since \(X\) is nonsingular over the algebraically closed field, \(\Omega_{X/k}\) is locally free of finite rank [@Har10a, Theorem II.8.15]. For a finite locally free sheaf \(E\), the evaluation map \(\mcf\otimes E^\vee\to\sheafhom(E,\mcf)\) is an isomorphism, as is checked in a local basis. Apply this with \(E=\Omega_{X/k}\) to obtain the stated isomorphism. In particular \(\mathcal D\) is coherent.
2. The automorphisms of the split extension \(\OO_X\oplus\mcf\) that fix the quotient and kernel are exactly\[ u_D(a,v)=(a,v+D(a)),\qquad D\in\mathcal D(X), \]and the same description holds over every open subset.
The split multiplication is \((a,v)(b,w)=(ab,aw+bv)\). An additive map inducing the identity on both the quotient and the kernel must have the displayed form for a unique additive map \(D:\OO_X\to\mcf\). Being a \(k\)-algebra homomorphism requires \(D\) to annihilate \(k\) and satisfy \[ D(ab)=aD(b)+bD(a). \] Conversely, these conditions make the displayed map multiplicative and unital, by expanding the two products and using the square-zero ideal, as in P-AGH286INFLIFT, part (a). It has inverse \(u_{-D}\). Composition satisfies \(u_Du_E=u_{D+E}\), proving the identification with the additive derivation sheaf. All formulas commute with restriction.
3. Each extension determines a well-defined class in \(H^1(X,\mathcal D)\).
Choose a finite affine open cover \(\mathfrak U=(U_i)\) of \(X\). The affine splitting theorem P-AGH287INFEXT, proved from infinitesimal lifting in P-AGH286INFLIFT, gives isomorphisms of identified extensions \[ \theta_i:(\OO_X\oplus\mcf)|_{U_i}\xrightarrow{\cong}\mathcal A|_{U_i}. \] On each overlap, step 2 writes the transition uniquely as \[ \theta_i^{-1}\theta_j=u_{D_{ij}},\qquad D_{ij}\in\mathcal D(U_i\cap U_j). \] The identity \(\theta_i^{-1}\theta_j\theta_j^{-1}\theta_h=\theta_i^{-1}\theta_h\) gives \(D_{ij}+D_{jh}=D_{ih}\), so these sections form a Čech one-cocycle.
Replacing \(\theta_i\) by \(\theta_i u_{b_i}\) changes this cocycle to \(D_{ij}+b_j-b_i\). It therefore changes it by a Čech coboundary. An isomorphism of extensions, composed with all the \(\theta_i\), gives the same cocycle for the other extension. Thus its Čech class depends only on the extension isomorphism class. The scheme \(X\) is noetherian and separated, and \(\mathcal D\) is quasi-coherent, so the affine-cover comparison identifies this class with an element of \(H^1(X,\mathcal D)\) [@Har10a, Theorem III.4.5]. Refining the cover restricts the cocycle, and the comparison commutes with refinement by P-AGH344REFINEMENTH1, part (b). Hence the cohomology class is also independent of the cover.
4. Every class in \(H^1(X,\mathcal D)\) is obtained by gluing split extensions, and the glued object is a scheme.
Use the same finite affine cover. By the affine-cover comparison, represent the class by a Čech cocycle \((D_{ij})\). Glue the sheaves of \(k\)-algebras \((\OO_X\oplus\mcf)|_{U_i}\) using the maps \(u_{D_{ij}}\) from the \(j\)th chart to the \(i\)th chart. Their composition rule in step 2 and the cocycle identity give the required gluing identities. Explicitly, the resulting sheaf consists of local tuples whose components satisfy \(s_i=u_{D_{ij}}(s_j)\) on overlaps. It restricts on each \(U_i\) to that split extension.
1.Thus the glued ringed space is locally a scheme and is a scheme.Every transition fixes the quotient and the kernel, so the quotient maps glue to \(\rho\) and the kernel inclusions glue to the specified identification with \(\mcf\).Its kernel is square-zero and has the required module structure, as can be checked on each chart.It is an extension of the desired kind, and its class from step 3 is the original class.
5. Two extensions determine the same cohomology class exactly when they are isomorphic as identified extensions.
3.This proves injectivity of the classification, while step 4 proves surjectivity.
6. Q.E.D.
Steps <1>3–<1>5 give a bijection between extension isomorphism classes and \(H^1(X,\mathcal D)\). Step 1 identifies the latter with \(H^1(X,\mcf\otimes\mathcal T)\), proving the stated correspondence. The zero cocycle glues the split extension by identity maps and therefore corresponds to the trivial extension.