Deformation Theory of Schemes and Obstruction Spaces

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. A surjective homomorphism $u\text{\hspace{0.17em}⁣}:A^{\mathrm{\prime }}\to A$ of commutative unital rings (Commutative ring, Ring homomorphism: additive, multiplicative, and required to send $1$ to $1$) with kernel $I=\ker u$ satisfying $I^2=0$ is a square-zero extension. Then $I$ carries an $A$-module structure: $I$ is an ideal of $A^{\mathrm{\prime }}$, hence an $A^{\mathrm{\prime }}$-module, and because $I\cdot I=0$ the action of $A^{\mathrm{\prime }}$ on $I$ factors through $A^{\mathrm{\prime }}\mathrm{/}I\cong A$. The trivial square-zero extension of $A$ by an $A$-module $I$ is the ring $A[I]=A\oplus I$ with multiplication $(a,x)(b,y)=(ab,ay+bx);$ the axioms hold because $I$ is an $A$-module and $I^2=0$, and the projection $A[I]\to A$ is a surjective ring map with kernel $0\oplus I$, so that $A[I]$ is a square-zero extension of $A$ by $I$. If $u\text{\hspace{0.17em}⁣}:A^{\mathrm{\prime }}\to A$ is any square-zero extension and $u$ admits a unital ring section $s\text{\hspace{0.17em}⁣}:A\to A^{\mathrm{\prime }}$, the map $A\oplus I\to A^{\mathrm{\prime }}$, $(a,x)\mapsto s(a)+x$, is an isomorphism of $A$-algebras onto $A^{\mathrm{\prime }}$ carrying the multiplication of $A[I]$; no such choice is part of the definition.

Small extensions. The standard deformation-theoretic convention fixes a base category ${\mathcal{C}}_k$ of local Artin $k$-algebras with residue field $k$ (Left and right Artinian rings, A local ring is a nonzero commutative ring with a unique maximal ideal), and a square-zero extension $u\text{\hspace{0.17em}⁣}:A^{\mathrm{\prime }}\to A$ with $A,A^{\mathrm{\prime }}\in {\mathcal{C}}_k$ is small when $I=\ker u$ is annihilated by the maximal ideal ${\mathfrak{m}}_{A^{\mathrm{\prime }}}$ of $A^{\mathrm{\prime }}$. The finiteness built into ${\mathcal{C}}_k$ is part of the convention: assuming as in the deformation-theoretic setup that $A$ and $A^{\mathrm{\prime }}$ are finite-dimensional over $k$, the $A^{\mathrm{\prime }}$-module structure on $I$ factors through $k=A^{\mathrm{\prime }}\mathrm{/}{\mathfrak{m}}_{A^{\mathrm{\prime }}}$ because ${\mathfrak{m}}_{A^{\mathrm{\prime }}}I=0$, so $I$ is a $k$-subspace of the finite-dimensional $k$-vector space $A^{\mathrm{\prime }}$; thus $I$ is a finite-dimensional $k$-vector space. The kernel is allowed to be zero, in which case $u$ is an isomorphism. This notion is weaker than the convention that additionally requires $\ker u$ to be nonzero and principal; the factorization statement below holds in the weaker form used here.

The basic example is the dual numbers $k[ϵ]=k[ϵ]\mathrm{/}({ϵ}^2)=k\oplus kϵ$, ${ϵ}^2=0$, with the augmentation $k[ϵ]\to k$ sending $ϵ$ to $0$; its kernel $kϵ$ is annihilated by $\mathfrak{m}=(ϵ)$. More generally, for a $k$-vector space $I$ the projection $k[I]=k\oplus I\to k$ exhibits $k[I]$ as a square-zero extension of $k$ with kernel $I$, and it is small exactly when $I$ is finite-dimensional over $k$, since $k[I]$ is then a finite-dimensional local Artin $k$-algebra with residue field $k$ (and conversely a finite-dimensional $k[I]$ forces ${\dim }_{k}I<\mathrm{\infty }$). Every surjection $A^{\mathrm{\prime }}\to A$ in ${\mathcal{C}}_k$ factors as a composition of small extensions: the maximal ideal $\mathfrak{m}={\mathfrak{m}}_{A^{\mathrm{\prime }}}$ is nilpotent because $A^{\mathrm{\prime }}$ is Artinian, say ${\mathfrak{m}}^n=0$, so with $I=\ker u$ the chain $A^{\mathrm{\prime }}=A^{\mathrm{\prime }}\mathrm{/}I{\mathfrak{m}}^{n-1}\twoheadrightarrow A^{\mathrm{\prime }}\mathrm{/}I{\mathfrak{m}}^{n-2}\twoheadrightarrow \cdots \twoheadrightarrow A^{\mathrm{\prime }}\mathrm{/}I\cong A$ factors $u$ into surjections whose successive kernels $I{\mathfrak{m}}^k\mathrm{/}I{\mathfrak{m}}^{k+1}$ are annihilated by $\mathfrak{m}$; each intermediate ring is a quotient of $A^{\mathrm{\prime }}$, hence again in ${\mathcal{C}}_k$, and each step is small in the sense above. Thus deformations over Artin rings are built from small extensions.

First-order thickenings. On schemes, a closed immersion $i\text{\hspace{0.17em}⁣}:S\to S^{\mathrm{\prime }}$ (Closed immersions of schemes) whose ideal sheaf $J=\ker ({\mathcal{O}}_{S^{\mathrm{\prime }}}\to i_{\ast }{\mathcal{O}}_S)$ satisfies $J^2=0$ is a first-order thickening. Here $J^2=0$ means that the product ideal generated by local sections of $J$ is zero; local sections of $J$ are therefore nilpotent, and a nilpotent element of a ring lies in every prime ideal, so every prime of $S^{\mathrm{\prime }}$ contains the stalk of $J$. Hence the underlying continuous map of $i$ is a homeomorphism onto $S^{\mathrm{\prime }}$, as required of a thickening, and $J$ is in particular locally nilpotent. The quotient ${\mathcal{O}}_{S^{\mathrm{\prime }}}\to i_{\ast }{\mathcal{O}}_S$ makes $J$ a quasi-coherent ${\mathcal{O}}_S$-module (Quasi-coherent ideal sheaves, Quasi-coherent module on a scheme): the closed immersion corresponds to a quasi-coherent sheaf of ideals on $S^{\mathrm{\prime }}$ (Quasi-coherent ideals and closed subschemes), and a ${\mathcal{O}}_{S^{\mathrm{\prime }}}$-module annihilated by $J$ is the same thing as an ${\mathcal{O}}_S$-module. Because $J^2=0$, the canonical surjection $J\twoheadrightarrow J\mathrm{/}{J}^2$ is an isomorphism, so $J$ is identified with the conormal sheaf $C_{S\mathrm{/}{S}^{\mathrm{\prime }}}=J\mathrm{/}{J}^2$ of the immersion. For a field $k$ and a $k$-vector space $I$, the morphism $\mathrm{Spec}k\hookrightarrow \mathrm{Spec}k[I]$ is the trivial first-order thickening, with ideal sheaf $I{\otimes }_k{\mathcal{O}}_{\mathrm{Spec}k}$. A morphism of square-zero extensions (respectively of first-order thickenings) is a commuting square of ring maps (respectively of scheme morphisms) respecting the structure maps, in the evident sense.

Base change. Let $f\text{\hspace{0.17em}⁣}:X\to S$ be a flat morphism of schemes (Flat morphism of schemes), let $S^{\mathrm{\prime }}$ be a first-order thickening of $S$ with ideal sheaf $J$, suppose additionally that a retraction $r:S^{\mathrm{\prime }}\to S$ of the thickening is given, and let $X^{\mathrm{\prime }}=X{\times }_S{S}^{\mathrm{\prime }}$ using $r$, with its two projections (Fibre product of schemes). Then the ideal sheaf of $X$ in $X^{\mathrm{\prime }}$ is the pullback $f^{\ast }J$: the projection $X^{\mathrm{\prime }}\to S^{\mathrm{\prime }}$ is flat by stability of flatness under base change (Flatness is stable under arbitrary base change), so pulling back the short exact sequence $0\to J\to {\mathcal{O}}_{S^{\mathrm{\prime }}}\to {\mathcal{O}}_S\to 0$ of ${\mathcal{O}}_{S^{\mathrm{\prime }}}$-modules along $X^{\mathrm{\prime }}\to S^{\mathrm{\prime }}$ stays exact and yields $0\to f^{\ast }J\to {\mathcal{O}}_{X^{\mathrm{\prime }}}\to {\mathcal{O}}_X\to 0,$ which identifies the kernel of ${\mathcal{O}}_{X^{\mathrm{\prime }}}\to {\mathcal{O}}_X$ with $f^{\ast }J$. In particular $X\hookrightarrow X^{\mathrm{\prime }}$ is itself a first-order thickening.

• Conventions. The scheme-side definition fixes the Zariski case of Stacks, Deformation Theory, Section 91.3 (tags 08KY-08L1), where thickenings of ringed spaces are defined by a homeomorphism with locally nilpotent kernel and first-order thickenings require the kernel to have square zero. The small-extension convention follows Stacks, Formal Deformation Theory, Definitions 90.3.1-90.3.2 (tags 06GC-06GD) specialized to $\mathrm{\Lambda }=k$, with the factorized form of Lemma 90.3.3 (tag 06GE).
• Automorphisms of trivial extensions. For an $A^{\mathrm{\prime }}$-algebra $B^{\mathrm{\prime }}$ over $A^{\mathrm{\prime }}$ and a square-zero kernel, an $A^{\mathrm{\prime }}$-algebra endomorphism reducing to the identity on $B=B^{\mathrm{\prime }}\mathrm{/}I{B}^{\mathrm{\prime }}$ differs from the identity by an $A$-linear derivation into the kernel; this is used in the companion counterexample and is not needed for the definition itself.
• Choice. The scheme-side ideal correspondence and flat-base-change suppliers assume Choice. The displayed ring constructions are explicit; an identification with a trivial extension requires the specified section.

Assume the Axiom of Choice, inherited from the projective-space cohomology suppliers (The Axiom of Choice). Let $k$ be a field, $n\ge 2$, let $f\in S=k[x_0,\dots ,x_n]$ be homogeneous of degree $d\ge 1$ (The polynomial ring $R[x_i:i\in I]$ as finitely supported coefficient families on monomials), and let $X=Z(f)\subseteq {\mathbb{P}}_k^n$ be the associated hypersurface; assume $X$ is smooth over $k$ of pure dimension $n-1$ (Relative Jacobian criterion with its presentation hypothesis). Write $\mathcal{I}={\mathcal{I}}_X$ for its ideal sheaf and ${\mathcal{O}}_X(d)$ for the restriction of the twisting sheaf (Twisting sheaf on Proj). Then:

1. multiplication by $f$ gives a short exact sequence $0⟶{\mathcal{O}}_{{\mathbb{P}}^n}⟶{\mathcal{O}}_{{\mathbb{P}}^n}(d)⟶i_{\ast }{\mathcal{O}}_X(d)⟶0$ of quasi-coherent sheaves (Zero scheme of a line-bundle section, A section of an invertible sheaf has a canonical zero subscheme, Quasi-coherent module on a scheme); in particular $\mathcal{I}\cong {\mathcal{O}}_{{\mathbb{P}}^n}(-d)$ and $\mathcal{I}\mathrm{/}{\mathcal{I}}^2\cong {\mathcal{O}}_X(-d)$;
2. $H^0(X,{\mathcal{O}}_X(d))\cong (S\mathrm{/}(f){)}_d$, of dimension $\left(\genfrac{}{}{0px}{}{n+d}{d}\right)-1=\left(\genfrac{}{}{0px}{}{n+d}{n}\right)-1$ (Nonnegatively graded rings and modules, homogeneous elements, and twists);
3. $H^q(X,{\mathcal{O}}_X(d))=0$ for every $q>0$;
4. the normal sheaf of $X$ in ${\mathbb{P}}^n$ is ${\mathcal{N}}_{X\mathrm{/}{\mathbb{P}}^n}=\mathcal{H}o{m}_{{\mathcal{O}}_X}(\mathcal{I}\mathrm{/}{\mathcal{I}}^2,{\mathcal{O}}_X)\cong {\mathcal{O}}_X(d)$ (Invertible sheaves, Dual of a line bundle is its tensor inverse, Internal Hom of module sheaves).

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 flat, locally finitely presented $k$-scheme (Flat morphism of schemes, Locally finite presentation morphisms, Schemes and morphisms over a base). For an augmented commutative $k$-algebra $B\to k$, a deformation of $X$ over $B$ is a flat, locally finitely presented $B$-scheme $X_B$ together with an isomorphism $X_B{\times }_{\mathrm{Spec}B}\mathrm{Spec}k\cong X$ (Fibre product of schemes). An isomorphism of deformations is a $B$-isomorphism inducing the identity on this identified special fibre. These objects and isomorphisms form a possibly large groupoid ${\mathrm{Def}}_X(B)$; write ${\mathrm{Def}}_X(B{)}_{\mathrm{i}\mathrm{s}\mathrm{o}}$ for its collection of isomorphism classes, without asserting that this collection is a set for every augmented base (Isomorphism, groupoid, and connected category). This augmented-base convention includes local Artin $k$-algebras with residue field $k$ and arbitrary trivial square-zero algebras $k[I]=k\oplus I$, without a finite-dimensionality assumption on $I$ (Square-zero extensions, small extensions and first-order thickenings).

When the augmentation ideal is nilpotent, the groupoid is essentially small and ${\mathrm{Def}}_X(B{)}_{\mathrm{i}\mathrm{s}\mathrm{o}}$ is a set. Indeed the special fibre has the same underlying space as $X_B$, and the opens corresponding to a fixed affine cover of $X$ are affine by Stacks, Lemma 37.2.3 (tag 04EW). Their coordinate rings are finitely presented $B$-algebras by local finite presentation. Finite presentations over the fixed ring $B$ form a set, as do their special-fibre identifications and the gluing isomorphisms between open subsets of their spectra. The cover is indexed by a set, so these data give a set of representatives up to isomorphism. This applies to the local Artin bases and all $k[I]$ above.

For a small extension $A^{\mathrm{\prime }}\twoheadrightarrow A$ and a specified deformation $X_A$ over $A$, a lift of $X_A$ to $A^{\mathrm{\prime }}$ is a flat, locally finitely presented $A^{\mathrm{\prime }}$-scheme $X_{A^{\mathrm{\prime }}}$ with an isomorphism $X_{A^{\mathrm{\prime }}}{\times }_{\mathrm{Spec}{A}^{\mathrm{\prime }}}\mathrm{Spec}A\cong X_A$. Isomorphisms of lifts reduce to the identity of $X_A$. Thus the relative lifting problem fixes the whole $A$-deformation, whereas ${\mathrm{Def}}_X(A^{\mathrm{\prime }})$ fixes only the original $k$-fibre. When $A=k$ the two descriptions coincide.

The trivial deformation is $X{\times }_k\mathrm{Spec}B$, with its canonical special-fibre identification. Its flatness and local finite presentation follow from base-change stability (Flatness is stable under arbitrary base change, Local finiteness conditions under base change). Write $k[ϵ]=k[ϵ]\mathrm{/}({ϵ}^2)$. The tangent space is the pointed set $T_X^1={\mathrm{Def}}_X(k[ϵ]{)}_{\mathrm{i}\mathrm{s}\mathrm{o}}$, pointed by the trivial class. The infinitesimal automorphism group is the group of $k[ϵ]$-automorphisms of $X{\times }_k\mathrm{Spec}k[ϵ]$ reducing to the identity on $X$; this reduction condition is part of the definition of ${\mathrm{Inf}}_X$.

Base change. An augmented $k$-algebra map $B_1\to B_2$ induces $\mathrm{Spec}{B}_2\to \mathrm{Spec}{B}_1$ (Morphisms of schemes) and the functor ${\mathrm{Def}}_X(B_1)⟶{\mathrm{Def}}_X(B_2),X_{B_1}⟼X_{B_1}{\times }_{\mathrm{Spec}{B}_1}\mathrm{Spec}{B}_2\mathrm{.}$ The special fibre is canonically $X$, and flatness and local finite presentation persist by the cited base-change results. Base change carries isomorphisms to isomorphisms and composes through the canonical fibre-product identifications. Equivalently, these deformation groupoids form a category fibred in groupoids over the category of augmented affine bases (Categories fibred in groupoids over a site).

A deformation of a morphism $f:Y\to X$ consists of specified deformations $Y_B$, $X_B$ and a $B$-morphism $f_B:Y_B\to X_B$ reducing to $f$. A bare morphism $Y\to X$ supplies neither these deformations nor a lift and therefore does not define a general map between their deformation groupoids.

• The local Artin case and relative lifting convention are the deformation categories of Stacks, Deformation Problems, Section 93.9, Example 9.1 (tag 0DY7). The augmented-base convention above explicitly also defines the groupoid on arbitrary square-zero bases required by the first-order classification.
• The groupoids need not be discrete: an object can have nonidentity automorphisms reducing to the identity on its special fibre. Isomorphism classes and automorphism groups are distinct invariants.
• Choice is inherited from the flat-base-change supplier and the square-zero convention; no simultaneous choice of base-change objects is required.

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.

Assume the Axiom of Choice (The Axiom of Choice). Let $A^{\mathrm{\prime }}\to A$ be a square-zero extension of rings (Square-zero extensions, small extensions and first-order thickenings), let $X$ be a flat $A$-scheme (Flat morphism of schemes) and let $\{U_i\hookrightarrow X\}$ be a Zariski open cover. Here a flat deformation is a flat $A^{\mathrm{\prime }}$-scheme $X^{\mathrm{\prime }}$ with a specified identification $X^{\mathrm{\prime }}{\times }_{\mathrm{Spec}{A}^{\mathrm{\prime }}}\mathrm{Spec}A\cong X$, and its isomorphisms are isomorphisms over $A^{\mathrm{\prime }}$ inducing the identity on $X$ (the deformation convention of Deformations of schemes and the infinitesimal deformation functor, extended here to arbitrary square-zero base extensions). The closed fibre inclusion identifies the underlying spaces of $X$ and $X^{\mathrm{\prime }}$; for an open $V\subset X$ write $X^{\mathrm{\prime }}{\mathrm{\mid }}_V$ for the open subscheme of $X^{\mathrm{\prime }}$ corresponding to $V$ under this identification. Then the groupoid of flat deformations of $X$ over $A^{\mathrm{\prime }}$ is equivalent to the groupoid of descent data consisting of flat deformations $U_i^{\mathrm{\prime }}$ of $U_i$ over $A^{\mathrm{\prime }}$ and isomorphisms $U_i^{\mathrm{\prime }}{\mathrm{\mid }}_{U_i\cap U_j}\cong U_j^{\mathrm{\prime }}{\mathrm{\mid }}_{U_i\cap U_j}$ over $A^{\mathrm{\prime }}$ inducing the identity on $U_i\cap U_j$, satisfying the identity and cocycle conditions on triple overlaps, with compatible isomorphisms as morphisms of descent data. Thus flat deformations and their isomorphisms satisfy effective Zariski descent (a sheaf of groupoids in this sense), and affine local deformation data compute the global deformation groupoid by Cech descent. The same statement holds for a first-order thickening $S\hookrightarrow S^{\mathrm{\prime }}$ and a flat $S$-scheme $X$, where a deformation means a flat $S^{\mathrm{\prime }}$-scheme $X^{\mathrm{\prime }}$ together with an identification $X^{\mathrm{\prime }}{\times }_{S^{\mathrm{\prime }}}S\cong X$; no retraction $S^{\mathrm{\prime }}\to S$ is required. If the deformation convention also requires local finite presentation (Locally finite presentation morphisms), the equivalence restricts to those objects. No separatedness or quasi-finiteness hypothesis is required.

Assume the Axiom of Choice, inherited from the projective-space cohomology suppliers (The Axiom of Choice). Let $k$ be a field, $n\ge 2$, $d\ge 1$, let $f\in S=k[x_0,\dots ,x_n]$ be homogeneous of degree $d$, let $X=Z(f)\subseteq {\mathbb{P}}_k^n$, and assume $X$ smooth over $k$ of pure dimension $n-1$. Let $A$ be a local Artin $k$-algebra with residue field $k$ and let $A^{\mathrm{\prime }}\to A$ be a small extension with kernel $I$ (Square-zero extensions, small extensions and first-order thickenings). Then:

1. for every lift $F\in (S{\otimes }_k{A}^{\mathrm{\prime }}{)}_d$ of $f$ (that is, $F\equiv f$ modulo the maximal ideal), the closed subscheme $Z(F)\subseteq {\mathbb{P}}_{A^{\mathrm{\prime }}}^n$ is flat over $A^{\mathrm{\prime }}$ and $Z(F){\times }_{\mathrm{Spec}{A}^{\mathrm{\prime }}}\mathrm{Spec}A$ is the equation family $Z(F_A)$, where $F_A$ is the reduction of $F$ modulo $I$. Its special fibre is $X$. If $F\equiv f$ modulo $I$, the intermediate reduction is the trivial embedded family $X{\times }_k\mathrm{Spec}A$;
2. conversely, fix an equation $F_A\in (S{\otimes }_{k}A{)}_d$ reducing to $f$ modulo the maximal ideal. Every flat closed subscheme $Y\subseteq {\mathbb{P}}_{A^{\mathrm{\prime }}}^n$ whose reduction to $A$ is the embedded family $Z(F_A)$ is a relative effective Cartier divisor of degree $d$ and equals $Z(F)$ for a lift $F$ of $F_A$, uniquely up to a unit of $A^{\mathrm{\prime }}$. This includes the trivial-reduction case $F_A=f$;
3. consequently the functor of embedded deformations of $X$ in ${\mathbb{P}}^n$ (Embedded deformations of a closed subscheme) is canonically isomorphic to the functor $R\mapsto \{F\in (S{\otimes }_{k}R{)}_d:F\equiv f<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>{\mathfrak{m}}_R\}\mathrm{/}(R^{\times })$ on local Artin $k$-algebras, where two residue-normalized lifts are equivalent when one is a unit multiple of the other (that unit necessarily has residue $1$); it is pro-represented by the formal completion of the projective space $\mathbb{P}(S_d^{\vee })$ at the point $[f]$, and it is formally smooth and unobstructed;
4. the tangent space at the trivial deformation is $(S\mathrm{/}(f){)}_d\cong H^0(X,{\mathcal{O}}_X(d))$, of dimension $\left(\genfrac{}{}{0px}{}{n+d}{n}\right)-1$, and every first-order embedded deformation extends to every small extension.