## Algebra & Number Theory

### Néron's pairing and relative algebraic equivalence

Cédric Pépin

#### Abstract

Let $R$ be a complete discrete valuation ring with algebraically closed residue field $k$ and fraction field $K$. Let $XK$ be a proper smooth and geometrically connected scheme over $K$. Néron defined a canonical pairing on $XK$ between $0$-cycles of degree zero and divisors which are algebraically equivalent to zero. When $XK$ is an abelian variety, and if one restricts to those $0$-cycles supported on $K$-rational points, Néron gave an expression of his pairing involving intersection multiplicities on the Néron model $A$ of $AK$ over $R$. When $XK$ is a curve, Gross and Hriljac gave independently an analogous description of Néron’s pairing, but for arbitrary $0$-cycles of degree zero, by means of intersection theory on a proper flat regular $R$-model $X$ of $XK$.

We show that these intersection computations are valid for an arbitrary scheme $XK$ as above and arbitrary $0$-cycles of degree zero, by using a proper flat normal and semifactorial model $X$ of $XK$ over $R$. When $XK=AK$ is an abelian variety, and $X=A¯$ is a semifactorial compactification of its Néron model $A$, these computations can be used to study the relative algebraic equivalence on $A¯∕R$. We then obtain an interpretation of Grothendieck’s duality for the Néron model $A$, in terms of the Picard functor of $A¯$ over $R$. Finally, we give an explicit description of Grothendieck’s duality pairing when $AK$ is the Jacobian of a curve of index one.

#### Article information

Source
Algebra Number Theory, Volume 6, Number 7 (2012), 1315-1348.

Dates
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
https://projecteuclid.org/euclid.ant/1513729887

Digital Object Identifier
doi:10.2140/ant.2012.6.1315

Mathematical Reviews number (MathSciNet)
MR3007151

Zentralblatt MATH identifier
1321.14038

#### Citation

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

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