A natural generalization of the concept of the Picard variety for a smooth algebraic variety within the framework of the theory of schemes. To define the Picard scheme for an arbitrary -scheme one considers the relative Picard functor in the category of schemes over the scheme . The value of this functor on an -scheme is the group
where is the base-change morphism and is the sheaf in the Grothendieck topology of strictly-flat quasi-compact morphisms associated with the pre-sheaf
and denotes the standard multiplicative group sheaf. If the Picard functor is representable on , then the -scheme representing it is called the relative Picard scheme for the -scheme and is denoted by . If is an algebraic scheme over a certain field having a rational -point, then
for any -scheme . In particular, can be identified with the group of -rational points of if such exists.
If is a projective morphism with geometrically-integral fibres, then the scheme exists and is a locally finitely representable separable group -scheme. If , then the connected component of the unit, , of is an algebraic -scheme, and the corresponding reduced -scheme is precisely the Picard variety . The nilpotent elements in the local rings of the scheme give much additional information on the Picard scheme and enable one to explain various "pathologies" in algebraic geometry over a field of characteristic . On the other hand, over a field of characteristic 0 the scheme is always reduced . It is also known that is a reduced scheme if is a smooth algebraic surface and .
For any proper flat morphism (finitely representable if the base is Noetherian) for which , the functor is an algebraic space over for any base-change morphism . In particular, the functor is representable if the ground scheme is the spectrum of a local Artinian ring.
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The standard multiplicative sheaf over a scheme assigns to an affine open set in the group of units of .
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Picard scheme. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Picard_scheme&oldid=21908