# Representability of sheaves of groups

There are lots of natural functors (that define sheaves in the fppf topology) that are not representable by schemes. For example, hilbert schemes of proper non-projective schemes in general need algebraic spaces. However, I know of no examples of such subtleties with group schemes. Every sheaf of groups that I know of is already representable.

Is this a consequence of general theorems? Is it considered easier to show that a sheaf of groups (in the fppf topology) is representable by a scheme than a sheaf of sets? For example, are necessary and sufficient criteria known for a sheaf of groups in the fppf topology to be representable. How about for sheaves of abelian groups?

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It is easier to show that a sheaf of groups is representable than a sheaf of sets --- see for example Matsumura, Hideyuki; Oort, Frans. Representability of group functors, and automorphisms of algebraic schemes. Invent. Math. 4 1967 1--25. –  anon Mar 18 '13 at 17:37
See also mathoverflow.net/questions/8918 for a related question. –  Dan Petersen Mar 20 '13 at 8:30

I disagree with the premise of your question. There are many natural non-representable sheaves of groups. For example, the formal additive group $\widehat{\mathbb{G}_a}$ is a sheaf, as the colimit of (the sheaves represented by) spectra of $\mathbb{Z}[x]/(x^n)$, but it is not representable as a scheme.

I suspect your experience is a result of mathematicians being generally more likely to encounter group sheaves that are given by quasicoherent sheaves of finite type Hopf algebras, than by weird moduli functors.

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I fully agree with this remark. –  Matthieu Romagny Mar 18 '13 at 20:16

For a functor with values in abelian groups there are nice representability criteria. Specifically, in the paper

Murre, J. P.: On contravariant functors from the category of pre-schemes over a field into the category of abelian groups (with an application to the Picard functor). Inst. Hautes Études Sci. Publ. Math., No. 23, 1964, pp. 5–43.

the author gives a list of (7) conditions that ensure representability. Notice that conditions P4 & P5 just say that your functor is a fpqc sheaf.

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