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We consider families of reductive complexes related by level-raising operators and originating from an associative algebra. In the main theorem it is shown that the multiple cohomology of that complexes is given by the factor space of products of reduction operators. In particular, we compute explicit torsor structure of the genus $g$ multiple cohomology of the families of horizontal complexes with spaces of of canonical converging reductive differential forms for a $C_2$-cofinite quasiconformal strong-conformal field theory-type vertex operator algebra associated to a complex curve. That provides an equivalence of multiple cohomology to factor spaces of products of sums of reduction functions with actions of the group of local coordinates automorphisms.

The idea of transversality is explored in the construction of cohomology theory associated to regularized sequences of multiple products of rational functions associated to vertex algebra cohomology of codimension one foliations on complex curves. Explicit formulas for cohomology invariants results from consideration transversality conditions applied to sequences of multiple products for elements of chain-cochain transversal complexes defined for codimension one foliations.

We consider infinite $\Z_\Z$-index complexes $\mathcal C$ of spaces with elements depending on a number of parameters, complete with respect to a linear associative regular inseparable multilinear product. The existence of nets of vanishing ideals of orders of and powers of differentials is assumed for subspaces of $\mathcal C$-spaces. In the polynomial case of orders and powers of the differentials, we derive the hierarchies of differential identities and closed multiple products. We prove that a set of maximal orders and powers for differentials, differential conditions, together with coherence conditions on indices of a complex $\mathcal C$ elements generate families of multi-graded differential algebras.

We consider families of chain-cochain infinite complexes $\mathcal C$ of spaces with elements depending on a number of parameters, and endowed with a converging associative multiple product. The existence of left/right local/non-local square-vanishing ideals is assumed for subspaces of $\mathcal C$-spaces. We show that a set of differential and orthogonality relations together with coherence conditions on indices of a chain-cochain complex $\mathcal C$ elements generates families of graded differential algebras. With the appropriate orthogonality conditions on completions of $\mathcal C$ elements in the multiple product, we define the equivalence classes of cohomology invariants.