The Tor Spectral Sequence and Flat Morphisms in Homotopical $D$-Geometry

TL;DR

This paper develops foundational tools for homotopical $D$-geometry, focusing on the Tor spectral sequence to analyze flat and étale morphisms within differential graded $D$-algebras, bridging homological and homotopical methods.

Contribution

It introduces the use of the Tor spectral sequence in homotopical $D$-geometry to study flat and étale morphisms, establishing key algebraic properties in this framework.

Findings

Established that the Tor spectral sequence connects graded Tor functors with derived tensor products.

Proved that étale topology behaves as a homotopical Grothendieck topology.

Demonstrated that smooth morphisms are local for the étale topology.

Abstract

Homotopical algebraic $D$-geometry combines aspects of homotopical algebraic geometry of Toen and Vezzosi and $D$-geometry of Beilinson and Drinfeld. It was introduced by the paper's last two authors and di Brino as a suitable framework for a coordinate-free study of the Batalin-Vilkovisky complex and more generally for the study of non-linear partial differential equations and their symmetries. In order to consolidate the foundation of the theory, we have to prove that the standard methods of linear and commutative algebra are available in the context of homotopical algebraic $D$-geometry, and we must show that in this context the \'etale topology is a kind of homotopical Grothendieck topology and that the notion of smooth morphism is, roughly speaking, local for the \'etale topology. The first half of this work was done. The remaining part covers the study of \'etale and flat morphisms in the category of differential graded $D$-algebras and is based on the Tor spectral sequence which connects the graded Tor functors in homology with the homology of the derived tensor product of two differential graded $D$-modules over a differential graded $D$-algebra.