derived group scheme

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

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

warningcareful, this needs a bit more attention. The general idea is obvious, but the detaisl require care. One problem is that in the Elliptic Survey “derived scheme” really referes to Spectral Schemes, which isn’t available yet, and not to the derived schemes discussed in Structured Spaces.

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

Revised on November 3, 2009 18:24:22
by Toby Bartels
(173.51.68.54)