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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    $\begingroup$ 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. $\endgroup$
    – anon
    Mar 18, 2013 at 17:37
  • $\begingroup$ See also mathoverflow.net/questions/8918 for a related question. $\endgroup$ Mar 20, 2013 at 8:30

2 Answers 2


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.

  • $\begingroup$ I fully agree with this remark. $\endgroup$ Mar 18, 2013 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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