Homological Algebra Of Semimodules And Semicont... Here

Unlike traditional modules over a ring, are defined over semirings (like the

Frequently used to study the global sections of semimodule sheaves on tropical varieties. 3. Semicontinuity and Stability Homological Algebra of Semimodules and Semicont...

algebra). Because semimodules lack additive inverses, they do not form an abelian category. This necessitates a shift from exact sequences to and kernel-like structures based on congruences. 2. Derived Functors in Non-Additive Settings Unlike traditional modules over a ring, are defined

The rank or homological dimension of a semimodule often drops at specific points of a parameter space, mirroring the behavior of coherent sheaves in algebraic geometry. Because semimodules lack additive inverses, they do not

This framework provides the "linear algebra" for tropical varieties. Just as homological algebra helps classify manifolds, semimodule homology helps classify and understand the intersections of tropical hypersurfaces.

Constructing resolutions using free semimodules or injective envelopes (like the "max-plus" analogues of vector spaces).