Let \(A\) be a finitely generated \(k\dash\)algebra. Write \(A\) as a quotient of a polynomial ring \(P\) over \(k\), and let \(J\) be the kernel: \[ 0 \to J \to P \to A \to 0 .\] Consider the exact sequence of (II, 8.4A), \[ J/J^2 \to \Omega_{P/k} \tensor_P A \to \Omega_{A/k} \to 0 .\] Apply the functor \(\Hom_A(\cdot, A)\), and let \(T^1(A)\) be the cokernel: \[ \Hom_A(\Omega_{P/k} \tensor A, A) \to \Hom_A(J/J^2, A) \to T^1(A) \to 0 .\]
Now use the construction of (II, Ex. 8.6) to show that \(T^1(A)\) classifies infinitesimal deformations of \(A\), that is, algebras \(A'\) flat over \(D = k[t]/t^2\) with \(A' \tensor_D k \cong A\). It follows that \(T^1(A)\) is independent of the given representation of \(A\) as a quotient of a polynomial ring \(P\).
Solution
Write \[ D=k[\epsilon]/(\epsilon^2), \qquad P_D=P\tensor_kD. \] An infinitesimal deformation is understood together with an identification \(A'/\epsilon A'\cong A\), and two deformations are identified when there is a \(D\)-algebra isomorphism inducing the identity on \(A\).
1. Every infinitesimal deformation \(A'\) of \(A\) admits a surjective \(D\)-algebra map\[ q':P_D\surjects A' \]whose reduction modulo \(\epsilon\) is the fixed quotient \(q:P\surjects A\).
By the infinitesimal lifting property of Exercise II.8.6, the map \[ q:P\longrightarrow A=A'/\epsilon A' \] lifts to a \(k\)-algebra map \[ \widetilde q:P\longrightarrow A'. \] Together with the structure map \(D\to A'\), this gives a \(D\)-algebra map \[ q':P_D\longrightarrow A'. \]
It is surjective. Indeed, let \(a'\in A'\). Its reduction in \(A\) is \(q(p)\) for some \(p\in P\), so \[ a'-\widetilde q(p)\in\epsilon A'. \] Write this difference as \(\epsilon b'\). If \(q(r)\) is the reduction of \(b'\), then \[ \epsilon b'=\epsilon\widetilde q(r), \] because \(b'-\widetilde q(r)\in\epsilon A'\) and \(\epsilon^2=0\). Hence \[ a'=q'(p+\epsilon r). \]
2. If \(J'=\ker q'\), then \(J'\) is an embedded first-order deformation of the ideal \(J\subseteq P\), and therefore determines an element\[ \phi_{q'}\in\Hom_A(J/J^2,A). \]
Reduction of \(q'\) modulo \(\epsilon\) is \(q\), so \[ (J'+(\epsilon))/(\epsilon)=J. \] Since \(A'=P_D/J'\) is flat over \(D\), the dual-number flatness criterion gives \[ J'\cap\epsilon P=\epsilon J. \] Thus, for each \(j\in J\), choose a lift \[ j+\epsilon p\in J'. \] The class of \(p\) in \(A=P/J\) depends only on \(j\bmod J^2\), and the assignment \[ j\bmod J^2\longmapsto p\bmod J \] is \(A\)-linear. This is exactly the embedded-deformation construction of Exercise III.9.7, applied to \[ \Spec A\subseteq\Spec P. \] It yields \(\phi_{q'}\in\Hom_A(J/J^2,A)\).
3. Conversely, every\[ \phi\in\Hom_A(J/J^2,A) \]determines an infinitesimal deformation\[ A_\phi=P_D/J_\phi, \]where\[ J_\phi =\{j+\epsilon p:j\in J,\ p\bmod J=\phi(j\bmod J^2)\}. \]
Exercise III.9.7 shows that \(J_\phi\) is an ideal of \(P_D\), that \[ (J_\phi+(\epsilon))/(\epsilon)=J, \] and that \[ J_\phi\cap\epsilon P=\epsilon J. \] Therefore \(A_\phi=P_D/J_\phi\) is flat over \(D\), and its special fibre is \[ A_\phi\tensor_Dk\cong P/J=A. \] Hence \(A_\phi\) is an infinitesimal deformation of \(A\).
4. If \(q'_1,q'_2:P_D\surjects A'\) are two lifts of the same quotient map \(q:P\surjects A\) to one deformation \(A'\), then\[ \phi_{q'_2}-\phi_{q'_1} \]lies in the image of\[ \Hom_A(\Omega_{P/k}\tensor_PA,A) \longrightarrow \Hom_A(J/J^2,A). \]
For \(p\in P\), the two lifts have the same reduction modulo \(\epsilon\), so there is a unique element \(\delta(p)\in A\) such that \[ q'_2(p)-q'_1(p)=\epsilon\delta(p). \] Flatness of \(A'\) identifies \(\epsilon A'\) with \(A\) via \[ a\longmapsto\epsilon\widetilde a, \] so \(\delta\) is well defined. Comparing products shows \[ \delta(pp')=q(p)\delta(p')+q(p')\delta(p), \] and therefore \(\delta:P\to A\) is a \(k\)-derivation. By the universal property of Kähler differentials, it corresponds to an \(A\)-linear map \[ \theta_\delta:\Omega_{P/k}\tensor_PA\longrightarrow A. \]
For \(j\in J\), choose \[ j+\epsilon p\in\ker q'_1. \] Then \[ q'_2(j+\epsilon p) =\epsilon\bigl(\delta(j)+q(p)\bigr). \] Thus replacing the lift \(q'_1\) by \(q'_2\) changes the homomorphism \(J/J^2\to A\) by minus the restriction of the derivation \(\delta\), namely by the negative of the image of \(\theta_\delta\) under \[ \Hom_A(\Omega_{P/k}\tensor_PA,A) \longrightarrow \Hom_A(J/J^2,A). \] In particular, the two homomorphisms have the same class in the cokernel.
5. Conversely, if two elements\[ \phi_1,\phi_2\in\Hom_A(J/J^2,A) \]differ by the restriction of a derivation \(\delta:P\to A\), then the deformations\(A_{\phi_1}\) and \(A_{\phi_2}\) are isomorphic over \(D\) by an isomorphism inducing the identity on \(A\).
Choose polynomial generators \(x_1,\ldots,x_m\) of \(P\) and choose lifts \(h_i\in P\) of the elements \(\delta(x_i)\in A\). Define a \(D\)-algebra automorphism \[ \alpha:P_D\longrightarrow P_D, \qquad x_i\longmapsto x_i+\epsilon h_i, \qquad \epsilon\longmapsto\epsilon. \] Its inverse is obtained by replacing \(\epsilon h_i\) with \(-\epsilon h_i\), so \(\alpha\) reduces to the identity on \(P\) modulo \(\epsilon\).
For \(j\in J\), \[ \alpha(j)\equiv j+\epsilon\delta(j)\pmod{\epsilon J}. \] Hence, if \(\phi_2-\phi_1\) is the restriction of \(\delta\), the defining formula for the ideals in step 3 gives \[ \alpha(J_{\phi_1})=J_{\phi_2} \] exactly. Therefore \(\alpha\) descends to a \(D\)-algebra isomorphism \[ A_{\phi_1}\iso A_{\phi_2} \] which induces the identity on the special fibre \(A\).
6. Two elements of \(\Hom_A(J/J^2,A)\) determine isomorphic infinitesimal deformations if and only if they have the same image in \(T^1(A)\).
Steps 4 and 5 show that changing a polynomial lift, or equivalently changing the presentation of the same deformation inside \(\Spec P_D\), changes the embedded-deformation class precisely by an element in the image of \[ \Hom_A(\Omega_{P/k}\tensor_PA,A). \]
Conversely, suppose \[ \Psi:A_{\phi_1}\iso A_{\phi_2} \] is a \(D\)-algebra isomorphism inducing the identity on \(A\). Compose the quotient map \(P_D\to A_{\phi_1}\) with \(\Psi\). This gives a second lift of \(P\to A\) to \(A_{\phi_2}\). Its kernel is \(J_{\phi_1}\), while the standard quotient map has kernel \(J_{\phi_2}\). Step 4 therefore shows that \(\phi_1\) and \(\phi_2\) differ by the image of a derivation of \(P\).
Thus the set of isomorphism classes is exactly the cokernel \[ \operatorname{coker}\!\left( \Hom_A(\Omega_{P/k}\tensor_PA,A) \longrightarrow \Hom_A(J/J^2,A) \right). \]
7. The infinitesimal deformations of \(A\) are classified by\[ \boxed{T^1(A)}. \]
By definition, the cokernel displayed in step 6 is \(T^1(A)\). Steps <1>1–<1>3 associate an element of that cokernel to every deformation and construct a deformation from every representative, while steps <1>4–<1>6 show that precisely the representatives in one cokernel class give isomorphic deformations.
8. The module \(T^1(A)\) is independent of the chosen polynomial presentation of \(A\).
Step 7 identifies \(T^1(A)\) intrinsically with the set of first-order deformations of the \(k\)-algebra \(A\) up to isomorphism fixing the special fibre. This classification depends only on \(A\), not on a presentation \(P\surjects A\). Hence the classifying module denoted \(T^1(A)\) is independent, up to the canonical identification supplied by deformation classes, of the chosen polynomial presentation.
9. Q.E.D.
Steps <1>1–<1>7 prove the classification, and step 8 gives the stated independence of the polynomial presentation.