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

Homotopical algebraic D-geometry combines aspects of homotopical algebraic geometry [32, 33] and D-geometry [3]. It was introduced in [9] 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 [5, 25]. 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 étale topology is a kind of homotopical Grothendieck topology and that the notion of smooth morphism is, roughly speaking, local for the étale topology. The first half of this work was done in [9]. The remaining part covers the study of étale 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.