Koszul-Tate resolutions as cofibrant replacements of algebras over differential operators

TL;DR

This paper develops a model structure on differential graded algebras over differential operators, introduces a functorial Koszul-Tate resolution, and lays groundwork for a homotopical Batalin-Vilkovisky formalism in $ extit{D}$-geometry.

Contribution

It defines a cofibrantly generated model structure on DG algebras over differential operators and constructs an explicit functorial Koszul-Tate resolution.

Findings

Established a model structure on $ t DGAlg( extit{D})$

Constructed a functorial Koszul-Tate resolution

Laid foundation for homotopical $ extit{D}$-geometry and Batalin-Vilkovisky formalism

Abstract

Homotopical geometry over differential operators is a convenient setting for a coordinate-free investigation of nonlinear partial differential equations modulo symmetries. One of the first issues one meets in the functor of points approach to homotopical $\mathcal{D}$-geometry, is the question of a model structure on the category $\tt DGAlg(\mathcal{D})$ of differential non-negatively graded $\mathcal{O}$-quasi-coherent sheaves of commutative algebras over the sheaf $\mathcal{D}$ of differential operators of an appropriate underlying variety $(X,\mathcal{O})$. We define a cofibrantly generated model structure on $\tt DGAlg(\mathcal{D})$ via the definition of its weak equivalences and its fibrations, characterize the class of cofibrations, and build an explicit functorial `cofibration - trivial fibration' factorization. We then use the latter to get a functorial model categorical Koszul-Tate resolution for $\mathcal{D}$-algebraic `on-shell function' algebras (which contains the classical Koszul-Tate resolution). The paper is also the starting point for a homotopical $\mathcal{D}$-geometric Batalin-Vilkovisky formalism.