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

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

G:Sch(𝒒)/Xβ†’Ab∞Grpd G : Sch(\mathcal{G})/X \to Ab \infty Grpd

from 𝒒\mathcal{G}-schemes over XX to topological abelian groups, such that composition with the forgetful functor Ab∞Grpdβ†’βˆžGrpdAb \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 β€œderived 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)