[Paper Review] Model structure on differential graded commutative algebras over the ring of differential operators

Gennaro Di Brino, Damjan Pištalo+1 more|arXiv (Cornell University)|May 28, 2015

Homotopy and Cohomology in Algebraic Topology17 references3 citations

TL;DR

This paper establishes a cofibrantly generated model structure on the category of non-negatively graded differential graded commutative algebras over the ring of differential operators on a smooth affine algebraic variety. Using Quillen's transfer theorem, it lifts the projective model structure on differential graded modules over the ring of differential operators, enabling a derived ${\cal D}$-geometric framework for the Batalin-Vilkovisky formalism and nonlinear PDE solutions.

ABSTRACT

We construct a cofibrantly generated model structure on the category of differential non-negatively graded quasi-coherent commutative $D_X$-algebras, where $D_X$ is the sheaf of differential operators of a smooth afine algebraic variety X. The paper contains an extensive appendix on D-modules, sheaves versus global sections, some more technical model categorical issues, as well as on relative Sullivan algebras. This article is the first of a series of works -located at the interface of homotopical algebra, algebraic geometry, and mathematical physics - on a derived D-geometric approach to the BV-formalism.

Motivation & Objective

• To develop a homotopical framework for studying solutions of nonlinear PDEs using derived ${\cal D}$-geometry.
• To construct a cofibrantly generated model structure on differential graded commutative ${\cal D}_X$-algebras for a smooth affine variety $X$.
• To provide a foundational model structure that supports the derived interpretation of on-shell function algebras via Koszul-Tate resolutions.
• To enable the treatment of derived ${\cal D}$-stacks as functors $\tt{DGAlg}({\cal D}) \to \tt{SSet}$ via fibrant object conditions.

Proposed method

• Lifts the cofibrantly generated projective model structure on $\tt{DGMod}({\cal D}(X))$ to $\tt{DGAlg}({\cal D}(X))$ using Quillen's transfer theorem.
• Applies the transfer theorem by verifying the necessary conditions on generating sets and smallness, relying on the existence of enough projectives in the module category.
• Uses relative Sullivan algebras over the ring of differential operators as a key tool to construct cofibrations in the category of ${\cal D}$-algebras.
• Establishes a monoidal categorical equivalence between $\tt{DGMod}({\cal D}_X)$ and $\tt{DGMod}({\cal D}_X(X))$ to reduce sheaf-theoretic problems to global sections.
• Employs the small object argument and transfinite composition of pushouts to ensure cofibrant replacements exist.
• Relies on results from [GS06] and [Hov07] for the underlying model structure on $\tt{DGMod}({\cal D}(X))$, adapted to the noncommutative ring ${\cal D}(X)$.

Experimental results

Research questions

• RQ1 How can a model structure be constructed on differential graded commutative algebras over the ring of differential operators on a smooth affine variety?
• RQ2 What conditions must be satisfied to transfer a model structure from $\tt{DGMod}({\cal D}(X))$ to $\tt{DGAlg}({\cal D}(X))$ via Quillen's transfer theorem?
• RQ3 How do relative Sullivan algebras over ${\cal D}(X)$ facilitate the construction of cofibrant replacements in the category of ${\cal D}$-algebras?
• RQ4 To what extent does the affine restriction preserve the essential homotopical structure needed for derived ${\cal D}$-geometry?
• RQ5 How does this model structure support the derived interpretation of the Batalin-Vilkovisky formalism and on-shell function algebras?

Key findings

• A cofibrantly generated model structure is successfully constructed on the category of non-negatively graded differential graded commutative ${\cal D}_X$-algebras for a smooth affine variety $X$.
• The model structure is transferred from the projective model structure on $\tt{DGMod}({\cal D}(X))$ via Quillen's transfer theorem, relying on the existence of enough projectives.
• Relative Sullivan algebras over ${\cal D}(X)$ are defined and used as a key technical tool to generate cofibrations in $\tt{DGAlg}({\cal D}(X))$.
• The construction is valid despite the noncommutativity of ${\cal D}(X)$, requiring careful analysis of local subtleties not present in commutative settings.
• The model structure provides a homotopical foundation for defining derived ${\cal D}$-stacks as functors $\tt{DGAlg}({\cal D}) \to \tt{SSet}$, with fibrancy corresponding to the sheaf condition.
• The Koszul-Tate resolution of an on-shell function algebra corresponds to a cofibrant replacement in this model category, linking physical constructions to homotopical algebra.