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Generalized local jacobians and commutative group stacks

Bertrand Toen

TL;DR

A notion of algebraic homology associated to a scheme which is a universal object computing fppf cohomology with coefficients in group schemes is introduced, and it is proved that, when X is smooth, J_*(X) appears as the E1-page of the corresponding spectral sequence.

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.

Generalized local jacobians and commutative group stacks

TL;DR

A notion of algebraic homology associated to a scheme which is a universal object computing fppf cohomology with coefficients in group schemes is introduced, and it is proved that, when X is smooth, J_*(X) appears as the E1-page of the corresponding spectral sequence.

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.
Paper Structure (12 sections, 11 theorems, 81 equations)

This paper contains 12 sections, 11 theorems, 81 equations.

Table of Contents

  1. Commutative algebraic group stacks
  2. Reminders on abelian group schemes
  3. Reminders on purity for flat cohomology
  4. Stacks
  5. Commutative algebraic group stacks
  6. Algebraic homology
  7. Grothendieck's generalized local jacobians
  8. The filtration by (co)dimension
  9. Grothendieck's construction
  10. Partial generalization to more general schemes
  11. Affine homology
  12. Affine homology as a dg-module

Key Result

Theorem 2

If $X$ is smooth over $k$, we have a natural isomorphism of complexes of pro-algebraic groups over $k$

Theorems & Definitions (24)

  • Definition 1
  • Theorem 2
  • Theorem 3
  • Lemma 1.1
  • Definition 1.3
  • Corollary 1.5
  • Remark 1.6
  • Definition 1.7
  • Definition 2.1
  • Remark 2.2
  • ...and 14 more