Let \(Y \subseteq X\) be a closed subscheme, where \(X\) is a scheme of finite type over a field \(k\). Let \(D = k[t]/t^2\) be the ring of dual numbers, and define an infinitesimal deformation of \(Y\) as a closed subscheme of \(X\) to be a closed subscheme \(Y' \subseteq \fiberprod{X}{k}{D}\) which is flat over \(D\) and whose closed fibre is \(Y\).
Show that these \(Y'\) are classified by \(H^0(Y, \mcn_{Y/X})\), where \[ \mcn_{Y/X} = \sheafhom_{\mco_Y}(\mci_Y / \mci_Y^2, \mco_Y) .\]
Solution
Write \(\epsilon\) for the class of \(t\) in \[ D=k[\epsilon]/(\epsilon^2). \] For an affine open \(U=\Spec A\subseteq X\), put \[ I=\Gamma(U,\mci_Y),\qquad A_D=A[\epsilon]/(\epsilon^2). \] Then a closed subscheme of \(U\times_k\Spec D\) is determined by an ideal \(J\subseteq A_D\).
1. Suppose \(J\subseteq A_D\) has closed fibre \(I\), i.e.\[ (J+(\epsilon))/(\epsilon)=I\subseteq A. \]Then \(B=A_D/J\) is flat over \(D\) if and only if\[ \boxed{J\cap \epsilon A=\epsilon I}. \]
First note that \(\epsilon I\subseteq J\). Indeed, for \(i\in I\) choose a lift \(i+\epsilon a\in J\); multiplying by \(\epsilon\) gives \(\epsilon i\in J\).
In the quotient \(B\), multiplication by \(\epsilon\) has kernel \[ \ker(\epsilon:B\to B) =\{\bar a+\epsilon\bar b:\epsilon a\in J\}. \] The equality \(J\cap\epsilon A=\epsilon I\) is therefore equivalent to \[ \ker(\epsilon:B\to B)=\epsilon B. \]
This last condition is equivalent to flatness over \(D\). One direction is immediate by tensoring the exact periodic resolution of the residue field \(k=D/(\epsilon)\). Conversely, assume \(\ker\epsilon=\epsilon B\). Choose a \(k\)-basis \((\bar b_\lambda)\) of \(B/\epsilon B\) and lifts \(b_\lambda\in B\). The induced map \[ \bigoplus_\lambda D\longrightarrow B \] is surjective: reduce any element modulo \(\epsilon B\), lift the resulting finite linear combination, and then lift once more inside \(\epsilon B\). If \[ \sum_\lambda (c_\lambda+\epsilon d_\lambda)b_\lambda=0, \] reduction modulo \(\epsilon B\) gives all \(c_\lambda=0\). Then \[ \epsilon\sum_\lambda d_\lambda b_\lambda=0, \] so \(\sum d_\lambda b_\lambda\in\ker\epsilon=\epsilon B\). Reducing again modulo \(\epsilon B\) gives every \(d_\lambda=0\). Thus \(B\) is a free, hence flat, \(D\)-module.
2. A flat deformation ideal \(J\) determines a unique homomorphism\[ \phi_J:I/I^2\longrightarrow A/I. \]
For \(i\in I\), choose a lift \[ i+\epsilon a\in J \] and define \[ \phi_J(i)=a\bmod I. \] This is independent of the chosen lift. If also \(i+\epsilon a'\in J\), then \[ \epsilon(a-a')\in J\cap\epsilon A=\epsilon I \] by step 1, so \(a-a'\in I\).
The construction is additive. For \(c\in A\), multiplying a lift by \(c\) gives \[ ci+\epsilon ca\in J, \] hence \[ \phi_J(ci)=c\phi_J(i) \] in \(A/I\). Thus \(\phi_J:I\to A/I\) is \(A\)-linear. Since \(I\) acts trivially on \(A/I\), it follows that \(\phi_J(I^2)=0\), so it factors uniquely through the \(A/I\)-linear map \[ I/I^2\longrightarrow A/I. \]
3. Conversely, every\[ \phi\in\operatorname{Hom}_{A/I}(I/I^2,A/I) \]determines a flat deformation ideal\[ J_\phi =\{i+\epsilon a\in A_D:i\in I,\ a\bmod I=\phi(i\bmod I^2)\}. \]
The set \(J_\phi\) is additive. For \(c+\epsilon d\in A_D\) and \(i+\epsilon a\in J_\phi\), \[ (c+\epsilon d)(i+\epsilon a) =ci+\epsilon(ca+di). \] Modulo \(I\), \[ ca+di\equiv c\phi(i) =\phi(ci), \] because \(di\in I\) and \(\phi\) is \(A/I\)-linear. Hence \(J_\phi\) is an ideal.
Its reduction modulo \(\epsilon\) is exactly \(I\), so its closed fibre is \(Y\cap U\). If \(\epsilon a\in J_\phi\), then its first component is zero and the defining condition says \[ a\bmod I=\phi(0)=0. \] Thus \[ J_\phi\cap\epsilon A=\epsilon I. \] Step 1 shows that \(A_D/J_\phi\) is flat over \(D\). Therefore \(J_\phi\) defines an infinitesimal embedded deformation of \(Y\cap U\) in \(U\).
4. The constructions in steps 2 and 3 are mutually inverse.
Starting from \(\phi\), the lift \(i+\epsilon a\in J_\phi\) satisfies \[ a\bmod I=\phi(i\bmod I^2), \] so step 2 recovers exactly \(\phi\).
Conversely, let \(J\) be flat and put \(\phi=\phi_J\). Every element \(i+\epsilon a\in J\) satisfies the defining congruence for \(J_\phi\), hence \[ J\subseteq J_\phi. \] For the reverse inclusion, let \(i+\epsilon a\in J_\phi\). Choose \(i+\epsilon a'\in J\). The equality of their images under \(\phi\) gives \(a-a'\in I\), so \[ \epsilon(a-a')\in\epsilon I\subseteq J. \] Therefore \(i+\epsilon a\in J\) and \(J_\phi=J\).
5. The affine correspondence is compatible with localization and therefore glues canonically over \(X\).
Every operation used in steps <1>1–<1>4 commutes with localization: reduction modulo \(\epsilon\), the ideal quotient \(I/I^2\), the module \(A/I\), and the formula defining \(J_\phi\). Thus on overlapping affine opens, the homomorphisms obtained from a global deformation ideal sheaf agree after restriction. They glue to a global section \[ \phi_{Y'}\in H^0\!\left(Y, \sheafhom_{\mco_Y}(\mci_Y/\mci_Y^2,\mco_Y) \right). \]
3.The corresponding ideals \(J_\phi\) agree on overlaps by the localization compatibility, so they glue to a quasicoherent ideal sheaf on \(X\times_k\Spec D\).The resulting closed subscheme has closed fibre \(Y\) and is flat over \(D\) by the local criterion in step
1.
6. The set of infinitesimal embedded deformations of \(Y\) in \(X\) is therefore\[ \boxed{H^0(Y,\mcn_{Y/X})}. \]
By the definition of the normal sheaf, \[ \mcn_{Y/X} =\sheafhom_{\mco_Y}(\mci_Y/\mci_Y^2,\mco_Y). \] Step 5 gives mutually inverse global constructions between its sections and flat closed subschemes \[ Y'\subseteq X\times_k\Spec D \] whose closed fibre is \(Y\). Hence \(H^0(Y,\mcn_{Y/X})\) classifies exactly the infinitesimal deformations requested in the problem.
7. Q.E.D.
Steps <1>1–<1>4 establish the affine classification, and steps <1>5–<1>6 globalize it and identify the parameter space with global sections of the normal sheaf.