Applications of the Infinitesimal Lifting Property

This question was posted to MSE but didn't get any answers, so I am posting it here. Original post

Hartshorne in his book gives the 'Infinitesimal Lifting Property' as an exercise in chapter 2, section 8 and mentions this to be very important in the deformation theory of nonsingular varieties. For completeness, I record the statement below:

Let $$k$$ be an algebraically closed field and $$A$$ a finitely generated $$k$$-algebra such that $$\operatorname{Spec} A$$ is a nonsingular variety over $$k$$. Let $$B$$ be a $$k$$-algebra and $$B'$$ a square-zero extension of $$B$$ by $$I$$, i e., there is an exact sequence $$0 \rightarrow I \rightarrow B' \xrightarrow{\pi} B \rightarrow 0$$ where $$B'$$ is a $$k$$-algebra and $$I$$ is an ideal such that $$I^2 = 0$$. Let $$f : A \rightarrow B$$ be a $$k$$-algebra homomorphism. Then there is a lift $$g : A \rightarrow B'$$, i.e. a $$k$$-algebra homomorphism $$g$$ such that $$\pi \circ g = f$$.

As someone starting with deformation theory, I would like to know how/why this result is very important as well as some applications of this result. Does this result have applications while studying moduli problems/moduli spaces?

For instance, Hartshorne gives one application: For $$X$$ a nonsingular variety over $$k$$ and $$\mathcal{F}$$ a coherent sheaf on $$X$$, the set of Infinitesimal extensions of $$X$$ by $$\mathcal{F}$$ upto isomorphism is in one to one correspondence with $$H^1(X, \mathcal{F} \otimes \mathcal{T}_X)$$ where $$\mathcal{T}_X$$ is the tangent sheaf.

This is known as the formal criterion for "formal smoothness." In this stacks project entry they prove that a morphism of schemes (in your case $$\text{Spec }A \to \text{Spec }k$$) is smooth if and only if it's formally smooth and locally finite presentation.

Aside from philosophical importance, it's often easier/more intuitive to check this formal criterion than to check the dimension of $$\Omega_{\text{Spec }A/\text{Spec }k}$$ or use a Jacobian. In the same stacks project tag, they say:

Michael Artin's position on differential criteria of smoothness (e.g., Morphisms, Lemma 01V9) is that they are basically useless (in practice).

Let's suppose $$\text{Spec }A$$ were instead a moduli space $$\overline{M}$$, e.g. of curves. Then to check $$\overline{M}$$ is smooth (if we know finite presentation), we need only consider an infinitesimal extension $$S \subseteq S'$$ coming from $$B' \to B$$ as in your question, and try to extend a curve over $$S$$ to a curve over $$S'$$.

If you want to build the cotangent complex of $$X = \text{Spec } A$$ (or equivalently the "normal sheaf") but $$A$$ is not smooth, the first step is to replace $$A$$ by a smooth $$k$$-algebra mapping to it, say $$k[A]$$. Even if some $$A \to B$$ doesn't factor through $$B'$$, $$k[A] \to A \to B$$ certainly will (by choosing a set-theoretic preimage in $$B'$$ of the image of $$A$$ in $$B$$) and so the "problem" obstructing a factorization of $$A \to B$$ can be traced to the kernel of $$k[A] \to A$$. I highly recommend this stacks project article that carries this out as concretely as possible.

You can get specific cohomological obstructions to such a factorization by continuing with a simplicial resolution: $$\cdots k[k[A]] \rightrightarrows k[A] \to A$$. One can even think of this as "covering $$A$$ by smooth algebras" in a topological sense using this Jonathan Wise article.

EDIT: I can be a bit more precise about the connection to ordinary differentials. Suppose $$f : A \to B$$ is fixed and we're trying to find a map $$\widetilde{f} : A \to B'$$ such that $$A \overset{\widetilde{f}}{\to} B' \to B$$ is $$f$$. First, pullback along $$f$$ to get another squarezero extension $$0 \to I \to B' \times_B A \to A \to 0.$$ Our old search for $$\widetilde{f}$$ translates to finding a section of the map $$B' \times_B A \to A$$. We've reduced to the case $$f = id_B : A = B$$.

Given one section $$s$$ of $$B' \to B$$, this splits the underlying sequence of modules and lets us write $$B' =_{modules} B \oplus I$$. But what's the ring structure? One computes $$(b, i)*(b', i') = (bb', bi' + b'i)$$, i.e. $$B' = B + I\epsilon$$ is the trivial squarezero extension!

If I have two sections $$s, t$$ of $$B' \to B$$, they'll give very different isomorphisms $$B' \simeq B + I \epsilon$$, inducing an automorphism $$\varphi$$ of $$B + I \epsilon$$ over $$B$$. Such automorphisms are precisely derivations! Indeed $$\varphi$$ must be the identity on $$B$$ and send $$I$$ to itself, but $$\varphi - id_{B'}$$ will be a map from $$B \to I$$ (using co/kernel univ props on the short exact sequence) which you can check is a derivation.

If sections exist, they all differ by a unique derivation $$s-t$$. A fancy way to say this is that "sections form a pseudo-torsor under $$\text{Der}(B, I)$$." If sections really do (locally) exist, you call it a plain torsor.

• Thank you for the answer, I will take some time to understand it. I will wait to see if there are more answers before accepting. Aug 12, 2020 at 16:50