Suppose $M$M is an arbitrary smooth manifold and $D$D is its bundle of $1$1-densities. On the category of finite-dimensional vector bundles over $M$M and linear differential operators between them. There there is a contravariant endofunctor that sends a vector bundle $E$ to $E*⊗D$ and a differential operator $f: E→F$E to the adjoint differential operator $f*: F*⊗D→E*⊗D$E⊗D and a differential operator f: E→F to the adjoint differential operator f: F⊗D→E⊗D.
Applying this endofunctor to the standard de Rham (cochain) complex $$0→Ω^0(M)→Ω^1(M)→⋯→Ω^n(M)→0$$0→Ω^0(M)→Ω^1(M)→⋯→Ω^n(M)→0 with morphisms being de Rham differentials we obtain another (chain) complex $$0←Λ^0(M)⊗D←Λ^1(M)⊗D←⋯←Λ^n(M)⊗D←0$$0←Λ^0(M)⊗D←Λ^1(M)⊗D←⋯←Λ^n(M)⊗D←0 with morphisms being codifferentials. Here $Λ^k(M)$Λ^k(M) denotes the bundle of $k$k-polyvectors ($k$thkth exterior power of the tangent bundle).
What is the exact relationship between the homology of this complex and the usual singular (co)homology of $M$M?
Using Hodge duality we can rewrite this complex as $$0←Ω^n(M)⊗W←Ω^{n-1}(M)⊗W←⋯←Ω^0(M)⊗W←0$$0←Ω^n(M)⊗W←Ω^{n-1}(M)⊗W←⋯←Ω^0(M)⊗W←0, where $W$W is the orientation bundle.
It looks like the answer should be some standard fact from the 1950s, therefore any references will be appreciated.