Semimodules And Semicont... | Homological Algebra Of

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

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 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 Applications: Tropical Geometry algebra)

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. 3. Semicontinuity and Stability

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:

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