Deformation cohomology of a smooth scheme: tangent, obstruction and automorphism spaces

The total space $X_A$ is smooth over $A$: it is flat and locally finitely presented by the definition of a deformation (Deformations of schemes and the infinitesimal deformation functor), and the sole geometric fibre is the smooth special fibre $X$. The fibre criterion in the smooth-morphism definition gives smoothness. Its differential sheaf is finite locally free, and its restriction to $X$ is ${\mathrm{\Omega }}_{X\mathrm{/}k}^1$ by differential base change (Kähler differentials commute with scalar base change). Since the kernel $I$ of a small extension is annihilated by the maximal ideal of $A$, the coefficient sheaf ${\mathcal{O}}_{X_A}{\otimes }_{A}I$ is canonically the pushforward of ${\mathcal{O}}_X{\otimes }_{k}I$ on the same underlying topological space. Applying the smooth-cotangent and finite locally free Ext formulas [F2] and [F3] therefore gives the degree-$0$ and degree-$2$ groups $H^i(X,T_{X\mathrm{/}k}{\otimes }_{k}I)$. The automorphism and obstruction assertions then follow from [F1].

For the dual numbers, the first-order theorem [F1] and the same smooth computation give $T_X^1\cong H^1(X,T_{X\mathrm{/}k})$. If the degree-$2$ group in step 1.1 vanishes, the obstruction is zero and [F1] supplies a lift.

If $X$ is smooth and proper over $k$, it is locally of finite presentation over the field $k$. Its affine chart rings are therefore quotients of finite-variable polynomial rings over $k$, which are Noetherian by iteration of Hilbert basis theorem: if $R$ is Noetherian then $R[x]$ is Noetherian and Every quotient and every localisation of a Noetherian ring is Noetherian. Thus $X$ is locally Noetherian, so its finite locally free tangent sheaf is coherent by the locally Noetherian clause of Coherent module sheaves. Applying [F4] to $T_{X\mathrm{/}k}$ makes $H^0,H^1,H^2$ finite-dimensional over $k$. The Axiom of Choice is inherited from the declared suppliers.