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Deformations of algebras: obstruction in degree two and torsor structure in degree one

Statement

Assume the Axiom of Choice as inherited from the resolution comparisons (The Axiom of Choice). Let $A^{'} \to A$ be a surjective ring map with square-zero kernel $I$ (Square-zero extensions, small extensions and first-order thickenings), let $A \to B$ be a ring map with $B$ flat over $A$ (Flat and faithfully flat modules and ring homomorphisms), let $N$ be a $B$ -module and $c ⁣ : I \to N$ an $A$ -module map. Consider the problem of finding a surjection of $A^{'}$ -algebras $B^{'} \to B$ whose kernel is a square-zero ideal identified with $N$ and which induces $c$ ; let $S o l$ be the set of isomorphism classes of solutions. Then:

there is a canonical element $ξ \in {Ext}_{B}^{2} (L_{B / A}, N)$ (Ext groups of the cotangent complex) whose vanishing is necessary and sufficient for $S o l \neq \emptyset$ ;
if $S o l \neq \emptyset$ , then $S o l$ is a torsor (principal homogeneous space) under ${Ext}_{B}^{1} (L_{B / A}, N)$ ;
for a solution $B^{'}$ , the group of automorphisms of $B^{'}$ over $B$ compatible with the data is canonically ${Ext}_{B}^{0} (L_{B / A}, N) = {Hom}_{B} (Ω_{B / A}, N) = {Der}_{A} (B, N)$ (Derivation of an algebra, Universal Kähler differential module).

Specializing to deformations of a flat $A$ -algebra $B$ over any square-zero extension $A^{'} \to A$ (take $N = B \otimes_{A} I$ and for $c$ the canonical $A$ -module map $I \to B \otimes_{A} I$ , $i \mapsto i \cdot 1$ ; e.g. $A$ and $A^{'}$ local Artin $k$ -algebras and $A^{'} \to A$ small): a flat deformation of $B$ over $A^{'}$ exists if and only if the obstruction class of $B$ vanishes, and then the set of isomorphism classes of flat deformations of $B$ over $A^{'}$ is a torsor under ${Ext}_{B}^{1} (L_{B / A}, B \otimes_{A} I)$ , with automorphism group ${Ext}_{B}^{0} (L_{B / A}, B \otimes_{A} I)$ . If $B$ is finitely presented over $A$ , every such flat lift is finitely presented over $A^{'}$ , without a finite-generation assumption on $I$ .

Facts & Assumptions

Given: a surjective ring map $A^{'} \to A$ with square-zero kernel $I$ , a ring map $A \to B$ with $B$ flat over $A$ , a $B$ -module $N$ , an $A$ -module map $c : I \to N$ , and the Axiom of Choice.

[F1]

${Ext}_{B}^{i} (L_{B / A}, N) = H^{i} (R {Hom}_{B} (L_{B / A}, N))$ for the ring-map cotangent complex $L_{B / A}$ , a complex concentrated in degrees $\leq 0$ ; for $i = 0, 1, 2$ these groups are computed by the Lichtenbaum-Schlessinger complex, with $T^{1}$ and $T^{2}$ as displayed in the Lichtenbaum-Schlessinger item. (Ext groups of the cotangent complex, The Lichtenbaum-Schlessinger complex computes Ext of the cotangent complex in degrees at most two)

[F2]

${Ext}_{B}^{0} (L_{B / A}, N) ≅ {Hom}_{B} (Ω_{B / A}, N) = {Der}_{A} (B, N)$ , because $τ_{\geq - 1} L_{B / A}$ is the naive cotangent complex whose degree- $0$ cohomology is $Ω_{B / A}$ . (Truncation, differentials and the cotangent complex of a smooth morphism, Universal Kähler differential module, Derivation of an algebra)

[F3]

The obstruction and torsor theorem for deformations of ring maps: for a square-zero extension $A^{'} \to A$ , an $A$ -algebra $B$ , a $B$ -module $N$ and the prescribed map $c : I \to N$ , the solutions form either the empty set or a torsor under ${Ext}_{B}^{1} (L_{B / A}, N)$ with automorphism group ${Ext}_{B}^{0} (L_{B / A}, N)$ , and the obstruction is a canonical element of ${Ext}_{B}^{2} (L_{B / A}, N)$ . This is Stacks, The Cotangent Complex, Lemma 92.16.1 (tag 08SP), with the affine case of Stacks, Deformation Theory, Lemmas 91.2.1-91.2.3 as its affine inputs; Hartshorne Theorem 10.1(a),(b) independently treats the flat Artin specialization. This is an exact application of the cited source theorem with the given $c$ , not merely its flat specialization. (Square-zero extensions, small extensions and first-order thickenings)

[F4]

For a polynomial presentation $R \to B$ with kernel $K$ , choose a free $R$ -module $F ↠ K$ with kernel $Q$ and Koszul relation submodule $F_{0} \subseteq Q$ . The Lichtenbaum-Schlessinger complex has terms $Q / F_{0}$ , $F \otimes_{R} B$ , and $Ω_{R / A} \otimes_{R} B$ in degrees $- 2, - 1, 0$ . The conormal module $K / K^{2}$ is the cokernel of $Q / F_{0} \to F \otimes_{R} B$ , rather than its degree- $- 1$ term; it is the degree- $- 1$ term of the naive cotangent complex. (Existence and generators of Kähler differentials, Conormal exact sequence for an algebra quotient, The Lichtenbaum-Schlessinger complex computes Ext of the cotangent complex in degrees at most two, Truncation, differentials and the cotangent complex of a smooth morphism)

Proof

technique · identify the deformation problem with the deformation theory of the ring map $A\to B$, transport the Stacks 92.16 result through the Lichtenbaum-Schlessinger computation of Ext, and specialize to flat deformations

1.1F3given

A solution $B^{'} \to B$ of the stated problem is exactly a deformation of the $A$ -algebra $B$ over the square-zero extension $A^{'}$ , in the sense of the ring-map deformation problem of [F3]: the kernel of $B^{'} \to B$ is a square-zero ideal identified with $N$ and the induced map $I \to N$ is $c$ . Flatness of $B$ over $A$ is the hypothesis under which the problem is the correct deformation problem (a flat deformation of $B$ over $A^{'}$ has $B = B^{'} \otimes_{A^{'}} A$ and $N = B \otimes_{A} I$ ).

1.2F1F2F3F4given

Applying the obstruction theorem [F3] to this problem gives at once the canonical obstruction class $ξ \in {Ext}_{B}^{2} (L_{B / A}, N)$ of part (1), the torsor structure under ${Ext}_{B}^{1} (L_{B / A}, N)$ of part (2), and the automorphism group ${Ext}_{B}^{0} (L_{B / A}, N)$ of part (3). The identification of the automorphisms with ${Der}_{A} (B, N)$ is [F2], and the identification of the $Ext$ groups with the Lichtenbaum-Schlessinger $T^{i}$ is [F1] with the presentations of [F4].

1.3F3given

For the flat specialization, take $N = B \otimes_{A} I$ and $c (i) = 1 \otimes i$ . In a solution, multiplication induces the identity $I \otimes_{A} B \to N$ ; hence its image is the whole kernel and $B^{'} / I B^{'} = B$ . The square-zero flatness criterion, Stacks tag 063Y (affine case), makes $B^{'}$ flat over $A^{'}$ since $B$ is flat over $A$ . Conversely flatness identifies $I B^{'}$ with $I \otimes_{A} B$ , so every flat lift is a solution with this canonical kernel identification. Thus steps 1.1 and 1.2 apply to exactly the flat lifts.

2.1step 1.3algebra∎

Suppose additionally $B$ is finitely presented over $A$ . Lift a finite set of algebra generators to $B^{'}$ and map $P^{'} = A^{'} [t_{1}, \dots, t_{r}] \to B^{'}$ . Its image $C$ satisfies $B^{'} = C + I B^{'}$ ; since $I^{2} = 0$ and $C$ contains $A^{'}$ , this implies $I B^{'} = I C \subset C$ and the map is surjective. Let $J$ be its kernel. Flatness of $B^{'}$ gives $J \cap I P^{'} = I J$ : a tensor in $I \otimes_{A^{'}} P^{'}$ whose product lies in $J$ maps to zero in $I \otimes_{A^{'}} B^{'}$ by injectivity of multiplication for the flat module $B^{'}$ , and right exactness lifts it from $I \otimes_{A^{'}} J$ . Hence $J / I J$ is the kernel of $A [t_{1}, \dots, t_{r}] \to B$ , so is finitely generated. Lift its finitely many generators to $J$ , with generated ideal $J_{0}$ . Then $J / J_{0} = I (J / J_{0}) = I^{2} (J / J_{0}) = 0$ , so $J = J_{0}$ and $B^{'}$ is finitely presented. This proves the added finiteness assertion without assuming finite generation of $I$ . Choice is inherited from the cotangent and derived-Hom suppliers.

Source application. The exact source theorem Stacks tag 08SP, read with its full proof, applies to arbitrary $N$ and $c$ . Its proof constructs the obstruction as the image of the extension datum under the long exact Ext sequence of the transitivity triangle; its torsor and automorphism assertions use the naive-cotangent comparison. The flat specialization additionally uses the square-zero flatness criterion: with $B$ flat over $A$ and kernel $B \otimes_{A} I$ , the induced multiplication $I \otimes_{A} B \to \ker (B^{'} \to B)$ is the identity, which is the flatness criterion for $B^{'}$ over $A^{'}$ .

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Sources

The Stacks Project, flatness across a square-zero extension (standard reference, not scraped)
The Stacks Project, The Cotangent Complex, complete chapter (Chapter 92) (standard reference, not scraped)
The Stacks Project, Deformation Theory, complete chapter (Chapter 91) (standard reference, not scraped)
Robin Hartshorne, Lectures on Deformation Theory (Berkeley Math 274 draft, 2004/2005) (standard reference, not scraped)