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

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.