## Picard categories and Poincaré duality for curves with continuous coefficients

### Joost van Hamel – 20 June 2001

Let f: X -> S be a smooth projective curve over a base S, let Gm be the fppf sheaf represented by the multiplicative group. I will discuss the result of Grothendieck and Deligne, dating from the end of the 1960s, that the truncation up to degree 1 of Rf*Gm is Cartier dual to itself. Their proof essentially boils down to the correspondence between sections of f and relative divisors on X, but in order to make this run smoothly, they use the fact that the truncation up to degree 1 of Rf*Gm can be identified in a natural way with the 'Picard stack' (symmetric group-like stack) on S of invertible sheaves on X.

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