[Paper Review] Rings of definition of smooth and proper dg-algebras

B. Toen|arXiv (Cornell University)|Nov 17, 2006

Homotopy and Cohomology in Algebraic Topology2 references3 citations

TL;DR

This paper establishes that the homotopy theory of smooth and proper dg-algebras over a filtered colimit of commutative rings is equivalent to the colimit of the homotopy theories over the constituent rings. The key result is that every smooth and proper dg-algebra over a commutative ring $k$ can be defined over a $\mathbb{Z}$-algebra of finite type, confirming a conjecture by Kaledin and enabling the application of positive characteristic techniques to derived algebraic geometry.

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.

Motivation & Objective

• To establish a finiteness property for smooth and proper dg-algebras over filtered colimits of commutative rings.
• To resolve Conjecture [Ka, 5.3] on the finite definability of such dg-algebras.
• To show that the homotopy theory of smooth and proper dg-algebras commutes with filtered colimits of base rings.
• To support the geometricity of the moduli stack of smooth and proper dg-algebras by proving local finite presentation.
• To lay foundational tools for studying Hodge-to-de Rham degeneration via reduction to positive characteristic.

Proposed method

• Use of filtered colimits of commutative rings as base rings, with $k = \mathrm{colim}_i k_i$.
• Leverage known results on homotopically finitely presented dg-algebras and their relation to smoothness and properness.
• Apply the derived base change functor $- \otimes_k^\mathbb{L} k'$ to relate dg-algebras over $k_i$ to those over $k$.
• Utilize the fact that perfect complexes and perfect modules over $k$-algebras lift along filtered colimits via the derived base change.
• Employ the Karoubi envelope property of the homotopy category $Ho(k\text{-}dg\text{-}alg)$ to handle idempotent liftings.
• Use the equivalence of derived categories of perfect modules under base change to prove essential surjectivity.

Experimental results

Research questions

• RQ1 Can every smooth and proper dg-algebra over a commutative ring $k$ be defined over a finitely generated $\mathbb{Z}$-algebra?
• RQ2 Does the homotopy theory of smooth and proper dg-algebras commute with filtered colimits of base rings?
• RQ3 Is the moduli stack of smooth and proper dg-algebras locally of finite presentation?
• RQ4 Can the derived category of perfect modules over a smooth and proper dg-algebra be lifted from a finitely generated base?
• RQ5 Does the base change functor along filtered colimits preserve the property of being smooth and proper?

Key findings

• The homotopy theory of smooth and proper dg-algebras over $k = \mathrm{colim}_i k_i$ is equivalent to the colimit of the homotopy theories over the $k_i$.
• Every smooth and proper dg-algebra over a commutative ring $k$ is defined over a $\mathbb{Z}$-algebra of finite type.
• The base change functor $- \otimes_{k_i}^\mathbb{L} k$ induces an equivalence on the homotopy categories of smooth and proper dg-algebras.
• Perfect modules over $k$-dg-algebras lift along filtered colimits via derived base change.
• The derived category of perfect modules over a smooth and proper dg-algebra over $k$ is equivalent to the colimit of such categories over the $k_i$.
• The moduli stack of smooth and proper dg-algebras is locally of finite presentation, supporting its geometric nature.