nLab spectral group scheme

Idea

A spectral group scheme is a group scheme in E-∞ geometry. In the affine case then the formal dual notion is a Hopf E-∞ algebra.

(Integrate comment here about smooth and flat versions.)

Example

• $GL$: Consider $R$ an E-∞ ring, $CAlg_R$, the ∞-category of $R$-algebras, $Mod_R$ an ∞-category of $R$-modules. Let $Sym_R$ be left adjoint to the forgetful functor $CAlg_R \to Mod_R$. Set $M_{n,R} = Spec Sym_R End_R(R^{\oplus n})$, a spectral monoid scheme which represents $R \mapsto End_R(R^{\oplus n})$. Let $GL_{n,R}$ be the spectral group scheme obtained by inverting the determinant element in $\pi_o M_{n,R}$ (Ohara).

References

• Mariko Ohara, On the K-theory of finitely generated projective modules over a spectral ring, slide 27 of slides.