Spencer delta-cohomology, restrictions, characteristics and involutive symbolic PDEs
Boris Kruglikov, Valentin Lychagin
TL;DR
The paper extends the classical Guillemin–Quillen theory from first-order PDE systems to general symbolic systems of arbitrary order by refining notions of non-characteristicity (strong vs weak) and characteristicity. It develops a Spencer δ-cohomology framework and a Leray–Serre spectral sequence to relate the cohomology of a system g to that of its restriction bar{g} on a non-characteristic subspace, establishing precise decompositions and exact sequences under involutivity. A key contribution is showing that, for strongly non-characteristic restrictions, involutivity of g implies involutivity of bar{g} and provides a reverse implication in the pure-order case, along with a generalization of Guillemin’s theorems to higher-order systems. The work also introduces equivalent reductions er_k to first-order systems, analyzes descended systems ∂g, and supplies counterexamples to illuminate the necessity of hypotheses, culminating in a broad, cohesive extension of the classical theory to the multi-order, algebraic-symbolic setting.
Abstract
We generalize the notion of involutivity to systems of differential equations of different orders and show that the classical results due to Guillemin and Quillen relating involutivity, restrictions, characteristics and characteristicity, known for first order systems, extend to the general context, though in a modified form. This involves, in particular, a new definition of strong characteristicity. The proof exploits a spectral sequence relating Spencer delta-cohomology of a symbolic system and its restriction to a non-characteristic subspace.