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 \({\mathtt{DGAlg({{{\mathcal {D}}}})}}\) 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 \({\mathtt{DGAlg({{{\mathcal {D}}}})}}\) 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.