# Koszul–Tate resolutions as cofibrant replacements of algebras over differential operators

- 5 Downloads

## 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).

## References

- 1.Barnich, G.: Global and gauge symmetries in classical field theories, Series of lectures, Seminar ‘Algebraic Topology, Geometry and Physics’, University of Luxembourg. homepages.ulb.ac.be/\(\sim \)gbarnichGoogle Scholar
- 2.Beilinson, A., Drinfeld, V.: Chiral algebras, American Mathematical Society Colloquium Publications, 51. Amer. Math. Soc, Providence, RI (2004)Google Scholar
- 3.Borceux, F.: Handbook of Categorical Algebra 1. Volume 50 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (1994)Google Scholar
- 4.Borceux, F.: Handbook of Categorical Algebra 2. Volume 51 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (1994)Google Scholar
- 5.di Brino, G., Pištalo, D., Poncin, N.: Model Structure on Differential Graded Commutative Algebras Over the Ring of Differential Operators. arXiv:1505.07720
- 6.di Brino, G., Pištalo, D., Poncin, N.: Model Categorical Koszul–Tate Resolution for Algebras Over Differential Operators. arXiv:1505.07964
- 7.di Brino, G., Pištalo, D., Poncin, N.: Homotopical Algebraic Context Over Differential Operators. arXiv:1706.05922
- 8.Costello, K.: Renormalization and Effective Field Theory. Mathematical Surveys and Monographs Volume, 170. American Mathematical Society, Providence (2011)Google Scholar
- 9.Dwyer, W.G., Spalinski, J.: Homotopy Theories and Model Categories. Springer, Berlin (1996)MATHGoogle Scholar
- 10.Félix, Y., Halperin, S., Thomas, J.-C.: Rational Homotopy Theory. Graduate Texts in Mathematics, vol. 205. Springer, Berlin (2001)CrossRefGoogle Scholar
- 11.Gillespie, J.: The flat model structure on complexes of sheaves. Trans. Am. Math. Soc.
**358**(7), 2855–2874 (2006)MathSciNetCrossRefMATHGoogle Scholar - 12.Goerss, P., Schemmerhorn, K.: Model categories and simplicial methods. In: Interactions Between Homotopy Theory and Algebra. American Mathematical Society, Providence, Contemp. Math., vol. 436, pp. 3–49 (2007)Google Scholar
- 13.Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics 52. Springer, Berlin (1997)Google Scholar
- 14.Henneaux, M., Teitelboim, C.: Quantization of Gauge Systems. Princeton University Press, Princeton (1992)MATHGoogle Scholar
- 15.Hess, K.: A history of rational homotopy theory. In: History of Topology, North-Holland, Amsterdam, pp. 757–796 (1999)Google Scholar
- 16.Hirschhorn, P.: Model Categories and Their Localizations Mathematical Surveys and Monographs 99. American Mathematical Society, Providence (2000)Google Scholar
- 17.Hirschhorn, P.: Overcategories and Undercategories of Model Categories. http://www-math.mit.edu/~psh/undercat.pdf (2005)
- 18.Hotta, R., Takeuchi, K., Tanisaki, T.: \(\cal{D}\)-Modules, Perverse Sheaves, and Representation Theory. Progress in Mathematics, 236. Birkhäuser, Basel (2008)CrossRefMATHGoogle Scholar
- 19.Hovey, M.: Model Categories. American Mathematical Society, Providence (2007)CrossRefMATHGoogle Scholar
- 20.Kashiwara, M., Schapira, P.: Sheaves on Manifolds. Springer, Berlin (1990)CrossRefMATHGoogle Scholar
- 21.Kauffman, L.H., Radford, D.E., Oliveira Souza, F.J.: Hopf algebras and generalizations. Contemp. Math.,
**441**(2007)Google Scholar - 22.Mac Lane, S.: Categories for the Working Mathematician, Volume 5 of Graduate Texts in Mathematics, 2nd edn. Springer, New York (1998)MATHGoogle Scholar
- 23.Pištalo, D., Poncin, N.: On Koszul–Tate Resolutions and Sullivan Models. arXiv:1708.05936 (to appear in Dissertationes Math)
- 24.Pištalo, D., Poncin, N.: Homotopical Algebraic Geometry over Differential Operators and Applications (2018) (to appear in ArXiv)Google Scholar
- 25.Quillen, D.: Homotopical Algebra. Lecture Notes in Mathematics, 43. Springer, Berlin (1967)CrossRefGoogle Scholar
- 26.Schapira, P.: D-Modules, Lecture Notes. http://www.math.jussieu.fr/~schapira/lectnotes/Dmod.pdf (2013)
- 27.Schneiders, J.-P.: An introduction to \({\cal{D}}\)-modules Algebraic analysis meeting (Liège, 1993). Bull. Soc. R. Sci. Liège
**63**(3–4), 223–295 (1994)MathSciNetMATHGoogle Scholar - 28.Tate, J.: Homology of Noetherian rings and local rings. Ill. J. Math.
**1**(1), 14–27 (1957)MathSciNetMATHGoogle Scholar - 29.Toën, B., Vezzosi, G.: Homotopical Algebraic Geometry II: Geometric Stacks and Applications. Mem. Am. Math. Soc., 193(902), (2008)Google Scholar
- 30.Verbovetsky, A.: Remarks on two approaches to the horizontal cohomology: compatibility complex and the Koszul–Tate resolution. Acta Appl. Math.
**72**(1), 123–131 (2002)MathSciNetCrossRefMATHGoogle Scholar - 31.Weibel, C.A.: An Introduction to Homological Algebra Cambridge Studies in Advanced Mathematics, vol. 38. Cambridge University Press, Cambridge (1993)Google Scholar