We propose a means to relate properties of an interconnected system to its separate component systems in the presence of cascade-like phenomena. Building on a theory of interconnection reminiscent of the behavioral approach to systems theory, we introduce the notion of generativity, and its byproduct, generative effects. Cascade effects, enclosing contagion phenomena and cascading failures, are seen as instances of generative effects. The latter are precisely the instances where properties of interest are not preserved or behave very badly when systems interact. The goal is to overcome that obstruction. We will show how to extract mathematical objects from the systems, that encode their generativity: their potential to generate new phenomena upon interaction. Those objects may then be used to link the properties of the interconnected system to its separate systems. Such a link will be executed through the use of exact sequences from commutative algebra.