GENERALIZED LOCAL JACOBIANS AND COMMUTATIVE GROUP STACKS
Abstract
In [10, page 109] Grothendieck sketches the construction of a complex J * (X) made of 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 complex J * (X) has the property to compute Zariski cohomology of X with coefficients in algebraic groups and plays the role of a homology of X. The purpose of this note is to recast this construction in the setting of higher algebraic group stacks for the fppf topology instead. For this, we introduce a notion of algebraic homology associated to a scheme over a perfect field 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 of dimension n, it is n-connective with respect to Beilinson's t-structure on the category of filtered objects, and that J * (X) appears as its n-th truncation. In a final part we partially extend our constructions and results over arbitrary bases.
Related
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BibTeX
@misc{toen2026generalized,
title = {{GENERALIZED LOCAL JACOBIANS AND COMMUTATIVE GROUP STACKS}},
author = {Bertrand Toen},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05760320v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - GENERALIZED LOCAL JACOBIANS AND COMMUTATIVE GROUP STACKS
AU - Bertrand Toen
PY - 2026
DA - 2026/09/23
PB - HAL
UR - https://hal.science/hal-05760320v1
AB - In [10, page 109] Grothendieck sketches the construction of a complex J * (X) made of 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 complex J * (X) has the property to compute Zariski cohomology of X with coefficients in algebraic groups and plays the role of a homology of X. The purpose of this note is to recast this construction in the setting of higher algebraic group stacks for the fppf topology instead. For this, we introduce a notion of algebraic homology associated to a scheme over a perfect field 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 of dimension n, it is n-connective with respect to Beilinson's t-structure on the category of filtered objects, and that J * (X) appears as its n-th truncation. In a final part we partially extend our constructions and results over arbitrary bases.
ER -