Commutative formal groups arising from schemes

TL;DR

This paper establishes a criterion for when the deformation cohomology of a commutative formal Lie group, arising from schemes, is pro-representable, based on properties of morphisms and higher direct images.

Contribution

It provides a new criterion linking deformation cohomology pro-representability to conditions on morphisms and higher direct images in algebraic geometry.

Findings

Deformation cohomology is pro-representable under specified conditions.

Higher direct images of the tangent space are key to pro-representability.

The criterion applies to flat, separated morphisms with flat targets over integers.

Abstract

We prove the following criterion for the pro-representability of the deformation cohomology of a commutative formal Lie group. Let f be a flat and separated morphism between noetherian schemes. Assume that the target of f is flat over the integers. For a commutative formal Lie group E, we have the deformation cohomology of f with coefficients in E at our disposal. If the higher direct images of the tangent space of E are locally free and of finite rank then the deformation cohomology is pro-representable by a commutative formal Lie group.