Algebra & Number Theory
- Algebra Number Theory
- Volume 6, Number 7 (2012), 1315-1348.
Néron's pairing and relative algebraic equivalence
Let be a complete discrete valuation ring with algebraically closed residue field and fraction field . Let be a proper smooth and geometrically connected scheme over . Néron defined a canonical pairing on between -cycles of degree zero and divisors which are algebraically equivalent to zero. When is an abelian variety, and if one restricts to those -cycles supported on -rational points, Néron gave an expression of his pairing involving intersection multiplicities on the Néron model of over . When is a curve, Gross and Hriljac gave independently an analogous description of Néron’s pairing, but for arbitrary -cycles of degree zero, by means of intersection theory on a proper flat regular -model of .
We show that these intersection computations are valid for an arbitrary scheme as above and arbitrary -cycles of degree zero, by using a proper flat normal and semifactorial model of over . When is an abelian variety, and is a semifactorial compactification of its Néron model , these computations can be used to study the relative algebraic equivalence on . We then obtain an interpretation of Grothendieck’s duality for the Néron model , in terms of the Picard functor of over . Finally, we give an explicit description of Grothendieck’s duality pairing when is the Jacobian of a curve of index one.
Algebra Number Theory, Volume 6, Number 7 (2012), 1315-1348.
Received: 19 February 2011
Revised: 21 December 2011
Accepted: 18 January 2012
First available in Project Euclid: 20 December 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 14K30: Picard schemes, higher Jacobians [See also 14H40, 32G20]
Secondary: 14G40: Arithmetic varieties and schemes; Arakelov theory; heights [See also 11G50, 37P30] 14K15: Arithmetic ground fields [See also 11Dxx, 11Fxx, 11G10, 14Gxx] 11G10: Abelian varieties of dimension > 1 [See also 14Kxx]
Pépin, Cédric. Néron's pairing and relative algebraic equivalence. Algebra Number Theory 6 (2012), no. 7, 1315--1348. doi:10.2140/ant.2012.6.1315. https://projecteuclid.org/euclid.ant/1513729887