# 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.

## Keywords

Differential operator \({{{\mathcal {D}}}}\)-module Model category Relative Sullivan \({{{\mathcal {D}}}}\)-algebra Homotopical geometry \({{{\mathcal {D}}}}\)-geometry Functor of points Koszul–Tate resolution Batalin–Vilkovisky formalism## Mathematics Subject Classification

18G55 16E45 35A27 32C38 16S32 18G10## Notes

### Acknowledgements

The current paper combines [5] and [6]. The authors are grateful to the referee for valuable comments and constructive suggestions. Further, they are indebted to Jim Stasheff for his careful reading of the first version of this text and his worthwhile remarks. The recommendations of both of these people allowed for significant improvements of the original text(s).

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