The action of Adeles on the residue complex

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2 Scopus citations


Let X be a scheme of finite type over a perfect field double-struck k sign. In this paper we study the relation between two objects associated to X: the Grothendieck residue complex. script K signX. and the Beilinson adeles complex double-struck A sign-red .((script O signX). The latter is a differential graded algebra (DGA). Our first main result (Theorem 0.1) is that script K sign X. is a right differential graded (DG) module over double-struck A sign-red.(script O signX). We give an application to de Rham theory. Define graded sheaves ℱ X.:= ℋomscript O sign XX/double-struck k sign., script K signX .) and script A signX.:= double-struck A sign-red.(script O signX) ⊗script O sign X ΩX/double-struck k sign.. It is known that script A signX. is a DGA. Our second main result (Theorem 0.2) is that ℱX. is a right DG script A signX.-module. When X is smooth then ℱ X. calculates de Rham homology, script A sign X. calculates cohomology, and the action induces the cap product. We extend these constructions to singular schemes in characteristic 0 using smooth formal embeddings.

Original languageEnglish
Pages (from-to)4131-4151
Number of pages21
JournalCommunications in Algebra
Issue number8
StatePublished - 1 Aug 2003


  • Beilinson adeles
  • Grothendieck duality residues

ASJC Scopus subject areas

  • Algebra and Number Theory


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