Deformations of schemes and the infinitesimal deformation functor

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.