Dg models for derived differential operators
Abstract
We show that for any algebra R that is essentially of finite type over a field k, the derived functors of Smith-Van den Bergh admit a dg enhancement. In particular, for any R-module M one can associate a dg algebra of differential operators EM whose cohomology is a nilpotent thickening of the (classical) ring of k-linear differential operators on M. We show that these dg algebras fit into a framework of Koszul duality due to Dwyer-Greenlees and specializes to a derived ring of differential operators studied by Yang when M=R and k=ℂ. In particular, upon fixing a presentation R=S/I with S essentially smooth, we show a derived Kashiwara's equivalence, namely that whenever suppS(M)=V(I) then there is an equivalence of triangulated categories between the derived category of dg EM-modules and the derived category of D-modules on S that are set-theoretically supported on I. Finally, in the case of positive characteristic, we show that the dg algebra EM admits a description in terms of the Frobenius endomorphism akin to the classical description of differential operators, and the proof employs simplicial techniques to obtain a result of independent interest in commutative algebra.
Related
- Rigid dualizing complex for quantum enveloping algebras and algebras of generalized differential operators
- The Ring of Differential Operators on a Nodal Curve is not a Bialgebroid
- Study of the algebra of smooth integro-differential operators with applications
- Eigenvectors and Eigenvalues of p-Curvature Operators for KZ--type Differential Connections
- Stable Symmetric Series, Differential Operators, and Jack Deformations
BibTeX
@misc{jeffries2026models,
title = {{Dg models for derived differential operators}},
author = {Jack Jeffries and Devlin Mallory and Claudia Miller and Josh Pollitz and Eamon Quinlan-Gallego},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.23778},
eprint = {2610.23778},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.23778},
}
RIS
TY - UNPB
TI - Dg models for derived differential operators
AU - Jack Jeffries
AU - Devlin Mallory
AU - Claudia Miller
AU - Josh Pollitz
AU - Eamon Quinlan-Gallego
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.23778
UR - https://arxiv.org/abs/2610.23778
AB - We show that for any algebra R that is essentially of finite type over a field k, the derived functors of Smith-Van den Bergh admit a dg enhancement. In particular, for any R-module M one can associate a dg algebra of differential operators E_M whose cohomology is a nilpotent thickening of the (classical) ring of k-linear differential operators on M. We show that these dg algebras fit into a framework of Koszul duality due to Dwyer-Greenlees and specializes to a derived ring of differential operators studied by Yang when M=R and k=ℂ. In particular, upon fixing a presentation R=S/I with S essentially smooth, we show a derived Kashiwara's equivalence, namely that whenever supp_S(M)=V(I) then there is an equivalence of triangulated categories between the derived category of dg E_M-modules and the derived category of D-modules on S that are set-theoretically supported on I. Finally, in the case of positive characteristic, we show that the dg algebra E_M admits a description in terms of the Frobenius endomorphism akin to the classical description of differential operators, and the proof employs simplicial techniques to obtain a result of independent interest in commutative algebra.
ER -