Homological Algebra Of Semimodules And Semicont... Now

The "Semicontinuity" aspect typically refers to the behavior of dimensions (like the rank of a semimodule) under deformations.

A key feature is the adaptation of and Tor functors. Since you cannot always "subtract" to find boundaries, homological algebra here often uses:

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.

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 over semirings (like the

It connects to the Lusternik-Schnirelmann category in idempotent analysis, where semicontinuity helps track the stability of eigenvalues in max-plus linear systems. 4. Applications: Tropical Geometry

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

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