Rings of definition of smooth and proper dg-algebras

TL;DR

This paper proves that the homotopy theory of smooth and proper dg-algebras over a filtered colimit of rings is itself a colimit of the homotopy theories over the rings, enabling finite type definitions.

Contribution

It establishes that smooth and proper dg-algebras over a colimit ring can be approximated by those over finitely generated rings, linking their homotopy theories.

Findings

Homotopy theory of dg-algebras over colimits is a colimit of homotopy theories.

Any smooth and proper dg-algebra can be defined over a finite type Z-algebra.

The results extend to a broad class of base rings and dg-algebras.

Abstract

This is a companion paper to math.AT/0609762. For a filtered colimit of commutative rings k=colim k_i, we prove that the homotopy theory of smooth and proper dg-algebras over k is the colimit of the homotopy theories of smooth and proper dg-algebras over k_i. As a consequence, we deduce that any smooth and proper dg-algebra can be defined over a commutative Z-algebra of finite type.