Self-intersection 0-cycles and coherent sheaves on arithmetic schemes
type: Item
instance of: scholarly article
title: Self-intersection 0-cycles and coherent sheaves on arithmetic schemes
zbMATH Open document ID: 0687.14004
DOI: 10.1215/S0012-7094-88-05725-0
author: Takeshi Saito
published in: Duke Mathematical Journal
publication date: 1988-01-01T00:00:00Z
review text: Let S be the spectrum of a discrete valuation ring with perfect residue field and X be a regular flat proper S-scheme purely of relative dimension 1 with smooth general fiber. Bloch has defined a number \((\Delta_ X,\Delta_ X)_ S\) which is an equivalent of the usual intersection number for the equal characteristic case. The coherent sheaves \(\Omega^ p_{X/S,tors}\) are concentrated in the closed fiber of X. This makes a definition possible for the Euler characteristic of the sheaf. The author proves that \[ \deg (\Delta_ X,\Delta_ X)_ S=\sum_{p}(-1)^ p\chi (X,\Omega^ p_{X/S,tors}). \] This gives an equivalence of two expressions for the conductor of X over S conjectured by Bloch and Kato.
reviewed by: A. N. Parshin
Mathematics Subject Classification ID: 14C17
Mathematics Subject Classification ID: 14G99
Mathematics Subject Classification ID: 14F45
zbMATH DE Number: 4125535
zbMATH Keywords: intersection number
zbMATH Keywords: Euler characteristic
zbMATH Keywords: conductor
MaRDI profile type: Publication
cites work: Q2546548
cites work: Q3800176
cites work: De Rham cohomology and conductors of curves
cites work: Q5611784
cites work: Q3328651
cites work: Complexe cotangent et déformations. I. (The cotangent complex and deformations. I.)
full work available at URL: s0012-7094-88-05725-0
OpenAlex ID: W2067685126
description: scientific article; zbMATH DE number 4125535