  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.
