Connected affine groups are irreducible

Let H be a finite-type commutative Hopf algebra over a perfect field k. If the prime spectrum of H is connected, it is irreducible.

Over a perfect field the reduced quotient H ⧸ nilradical H is again a Hopf algebra (TauCeti.HopfIdeal.reduction): its tensor square is reduced. It is geometrically reduced, hence smooth over k (TauCeti.smoothCommHopfAlgProperty_of_geometricallyReduced), so it is a regular ring and the local rings of its spectrum are regular local rings, in particular domains. A connected Noetherian scheme whose local rings are domains is irreducible (TauCeti.AlgebraicGeometry.irreducibleSpace_of_connected_of_isDomain_stalk). Finally, the quotient map by the nilradical induces a homeomorphism of prime spectra, which carries both connectedness and irreducibility across.

Perfectness of k is used twice: to make the nilradical a Hopf ideal and to make the reduced quotient geometrically reduced. It holds in particular over an algebraically closed field.

Over an arbitrary field, geometric connectedness suffices: a geometrically connected finite-type commutative Hopf algebra has irreducible prime spectrum, with no perfectness hypothesis.

Main declarations

• CommHopfAlgCat.irreducibleSpace_primeSpectrum_of_connectedSpace: a finite-type commutative Hopf algebra over a perfect field with connected prime spectrum has irreducible prime spectrum.
• TauCeti.geometricallyConnectedCommHopfAlgProperty.irreducibleSpace: a geometrically connected finite-type commutative Hopf algebra over any field has irreducible prime spectrum.

References

• M. Brion, Some structure theorems for algebraic groups, Proc. Sympos. Pure Math. 94 (2017), Theorem 2.4.1(3) and Proposition 2.1.12.

theorem CommHopfAlgCat.irreducibleSpace_primeSpectrum_of_connectedSpace (k : Type u) [Field k] [PerfectField k] (H : CommHopfAlgCat k) [Algebra.FiniteType k ↑H] [ConnectedSpace (PrimeSpectrum ↑H)] : IrreducibleSpace (PrimeSpectrum ↑H)

A finite-type commutative Hopf algebra over a perfect field whose prime spectrum is connected has irreducible prime spectrum: a connected affine group of finite type over a perfect field is irreducible.

theorem TauCeti.geometricallyConnectedCommHopfAlgProperty.irreducibleSpace (k : Type u) [Field k] (H : CommHopfAlgCat k) [Algebra.FiniteType k ↑H] (hH : geometricallyConnectedCommHopfAlgProperty k H) : IrreducibleSpace (PrimeSpectrum ↑H)

A geometrically connected finite-type commutative Hopf algebra over any field has irreducible prime spectrum.