Generalized local jacobians and commutative group stacks

TL;DR

This paper reinterprets Grothendieck's construction of local generalized jacobians within the framework of higher algebraic group stacks, introducing algebraic homology to connect with fppf cohomology and spectral sequences.

Contribution

It introduces a new notion of algebraic homology for schemes, linking local jacobians to higher algebraic group stacks and spectral sequences in algebraic geometry.

Findings

J_*(X) appears as the E1-page of a spectral sequence for smooth X.

Algebraic homology computes fppf cohomology with group scheme coefficients.

Partial extension of constructions over arbitrary bases.

Abstract

In [CS01, Page 109] Grothendieck sketches the construction of a complex J_*(X) or commutative pro-algebraic groups, associated to a smooth variety X, and for which each J_i(X) is a product of local factors called the local generalized jacobians. The purpose of this note is to recast this construction in the setting of higher algebraic group stacks for the fppf topology. For this, we introduce a notion of algebraic homology associated to a scheme which is a universal object computing fppf cohomology with coefficients in group schemes. We endow this algebraic homology with a filtration by dimension of supports, and prove that, when X is smooth, J_*(X) appears as the E1-page of the corresponding spectral sequence. In a final part we partially extends our constructions and results over arbitrary bases.