The hierarchies of identities and closed products for multiple complexes

TL;DR

This paper develops a hierarchical framework for identities and closed products in complex spaces with parameters, using algebraic structures and differential conditions to generate multi-graded differential algebras.

Contribution

It introduces a novel hierarchy of differential identities and closed products for parameter-dependent complexes, expanding the algebraic understanding of such structures.

Findings

Hierarchies of differential identities derived for complex spaces.

Generation of multi-graded differential algebras from maximal differential orders.

Establishment of conditions for the existence of vanishing ideals and differential powers.

Abstract

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.