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derived group scheme

derived group scheme

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

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

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

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

This is (adapted from) definition 3.1 of

warning careful, 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.

special cases

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)