nLab derived group scheme

[To be merged with .]

Idea

A general group scheme is a group object in generalized schemes: it is a generalization to higher geometry of a group scheme.

Definition

Let $X$ be a $\mathcal{G}$-generalized scheme for $\mathcal{G}$ the geometry (for structured (∞,1)-toposes) that defines the desired notion of derived schemes.

A commutative group $\mathcal{G}$-scheme over $X$ is an (∞,1)-functor

$$G : Sch(\mathcal{G})/X \to Ab \infty Grpd$$

from $\mathcal{G}$-schemes over $X$ to topological abelian groups, such that composition with the forgetful functor $Ab \infty Grpd \to \infty Grpd$ is representable by a derived scheme flat over $X$.

This is (adapted from) definition 3.1 of

• Jacob Lurie, A Survey of Elliptic Cohomology

warning careful, this needs a bit more attention. The general idea is obvious, but the details require care. One problem is that in the Elliptic Survey “derived scheme” really refers to Spectral Schemes and not to the derived schemes discussed in Structured Spaces.

special cases

An important special case is that of a derived elliptic curve.