Embedded deformations of a closed subscheme

Assume the Axiom of Choice as inherited from the cited scheme and flat-base-change suppliers (The Axiom of Choice). Let $k$ be a field and let $X$ be a smooth projective $k$-scheme (Smooth morphism of schemes) with a closed subscheme $Y\subseteq X$ (Closed immersions of schemes) that is flat over $k$ (Flat morphism of schemes). For a local Artin $k$-algebra $R$ with residue field $k$, fix the trivial ambient deformation $X_R=X{\times }_k\mathrm{Spec}R$. An embedded deformation of $Y$ in $X$ over $R$ is a closed subscheme $Y_R\subseteq X_R$ that is flat and locally finitely presented over $R$, with its special fibre identified with $Y\subseteq X$: $Y_R{\times }_{\mathrm{Spec}R}\mathrm{Spec}k\cong Y\mathrm{.}$ This is the embedded version of Deformations of schemes and the infinitesimal deformation functor. An isomorphism of embedded deformations is an $X_R$-isomorphism inducing the identity on the identified special fibre. These objects and isomorphisms form the groupoid ${\mathrm{ED}}_{Y\subseteq X}(R)$; its set of isomorphism classes is ${\mathrm{ED}}_{Y\subseteq X}(R{)}_{\mathrm{i}\mathrm{s}\mathrm{o}}$. An augmented $k$-algebra map $R\to R^{\mathrm{\prime }}$ induces base change of these closed subschemes inside $X_{R^{\mathrm{\prime }}}$. Closed immersions, flatness and local finite presentation persist under base change (Base change of immersions, Flatness is stable under arbitrary base change, Local finiteness conditions under base change), and the identified special fibre stays $Y$, so these groupoids and their isomorphism classes define the embedded deformation functor on local Artin $k$-algebras with residue field $k$. The trivial embedded deformation is $Y{\times }_k\mathrm{Spec}R\subseteq X_R$.

For a small extension $A^{\mathrm{\prime }}\twoheadrightarrow A$ (Square-zero extensions, small extensions and first-order thickenings) and a fixed embedded deformation $Y_A\subseteq X_A$, a relative embedded lift is a flat, locally finitely presented closed subscheme $Y_{A^{\mathrm{\prime }}}\subseteq X_{A^{\mathrm{\prime }}}$ with an identification $Y_{A^{\mathrm{\prime }}}{\times }_{\mathrm{Spec}{A}^{\mathrm{\prime }}}\mathrm{Spec}A\cong Y_A$ as closed subschemes of $X_A=X{\times }_k\mathrm{Spec}A$ (Fibre product of schemes). Isomorphisms of relative lifts induce the identity on $Y_A$. In particular, when $Y_A$ is the trivial embedded deformation, the prescribed reduction is $Y{\times }_k\mathrm{Spec}A$; when $A=k$, it is $Y$ itself.

When $Y=V(f)\subseteq {\mathbb{P}}_k^n$ is a hypersurface, this is the functor of deformations of $Y$ inside the fixed projective space, whose tangent space is computed by the normal sheaf; the normal sheaf of a closed immersion with ideal sheaf $\mathcal{I}$ is ${\mathcal{N}}_{Y\mathrm{/}X}=\mathcal{H}o{m}_{{\mathcal{O}}_Y}(\mathcal{I}\mathrm{/}{\mathcal{I}}^2,{\mathcal{O}}_Y)$ (Quasi-coherent ideal sheaves, The internal Hom sheaf of two module sheaves, Internal Hom of module sheaves), where $\mathcal{I}\mathrm{/}{\mathcal{I}}^2$ is the conormal sheaf of the immersion; for a closed immersion $i:Y\hookrightarrow X$ with both schemes smooth over $k$, the conormal sheaf is locally free and the conormal sequence $0\to \mathcal{I}\mathrm{/}{\mathcal{I}}^2\to i^{\ast }{\mathrm{\Omega }}_{X\mathrm{/}k}^1\to {\mathrm{\Omega }}_{Y\mathrm{/}k}^1\to 0$ is exact: Stacks tag 06AA applies because $Y$ is smooth, and the sequence locally splits since ${\mathrm{\Omega }}_{Y\mathrm{/}k}^1$ is finite locally free. Its kernel is therefore a direct summand of the finite locally free $i^{\ast }{\mathrm{\Omega }}_{X\mathrm{/}k}^1$, hence finite locally free (Differentials of a smooth morphism). For the projective-space case this also follows from Smooth closed immersion is regular with exact conormal sequence; the general right-exact sequence is Conormal sequence for a closed immersion. For a hypersurface $V(f)\subseteq {\mathbb{P}}^n$ with $f$ a nonzerodivisor the ideal sheaf is invertible with $\mathcal{I}\cong \mathcal{O}(-d)$, so the conormal sheaf is a line bundle on $Y$ and the normal sheaf is its dual.

• Reference conventions. The definition is the fixed-ambient (Hilbert scheme) form of the embedded deformation problem, following Hartshorne, Lectures on Deformation Theory, Chapter 1 Theorem 1.1 and Chapter 1 Section 2, Situation A, where closed subschemes of a fixed nonsingular projective $X$ are deformed inside $X$; the tangent space of that problem is $H^0(Y,{\mathcal{N}}_{Y\mathrm{/}X})$. The same problem is described in Sernesi, An overview of classical deformation theory, Section 2.
• Comparison with abstract deformations. An embedded deformation of $Y$ in $X$ is in particular a deformation of $Y$ as a $k$-scheme in the sense of Deformations of schemes and the infinitesimal deformation functor, but the two functors are different in general: embedded deformations come with a closed immersion into the fixed thickening $X_{A^{\mathrm{\prime }}}$, which is additional structure not present in an abstract deformation. No injectivity or surjectivity of the comparison is asserted here.
• Choice. Choice is inherited from the flat-base-change and smooth-differential suppliers; the normal sheaf itself is given by its displayed internal Hom.