Jacobi-Bernoulli cohomology and deformations of schemes and maps

TL;DR

This paper introduces Jacobi-Bernoulli cohomology for semi-simplicial Lie algebras and demonstrates its application in understanding infinitesimal deformations of algebraic schemes over the complex numbers.

Contribution

It defines a new cohomology theory for semi-simplicial Lie algebras and connects it to deformation theory of schemes.

Findings

Jacobi-Bernoulli cohomology is constructed for SELA.

The tangent SELA encodes infinitesimal deformations of schemes.

A relationship between cohomology and scheme deformations is established.

Abstract

We introduce a notion of Jacobi-Bernoulli cohomology associated to a semi-simplicial Lie algebra (SELA). For an algebraic scheme $X$ over $\C$, we construct a tangent SELA $\t_X$ and show that the Jacobi-Bernoulli cohomology of $\t_X$ is related to infinitesimal deformations of $X$.