## Abstract

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 sign_{X}^{.} and the Beilinson adeles complex double-struck A sign_{-red} ^{.}((script O sign_{X}). 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 sign_{X}). We give an application to de Rham theory. Define graded sheaves ℱ _{X}^{.}:= ℋom_{script O sign X} (Ω _{X/double-struck k sign}^{.}, script K sign_{X} ^{.}) and script A sign_{X}^{.}:= double-struck A sign_{-red}^{.}(script O sign_{X}) ⊗_{script O sign X} Ω_{X/double-struck k sign}^{.}. It is known that script A sign_{X}^{.} is a DGA. Our second main result (Theorem 0.2) is that ℱ_{X}^{.} is a right DG script A sign_{X}^{.}-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 language | English |
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Pages (from-to) | 4131-4151 |

Number of pages | 21 |

Journal | Communications in Algebra |

Volume | 31 |

Issue number | 8 |

DOIs | |

State | Published - 1 Aug 2003 |

## Keywords

- Beilinson adeles
- Grothendieck duality residues

## ASJC Scopus subject areas

- Algebra and Number Theory