Square-zero extensions, small extensions and first-order thickenings

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.