Homological Algebra Of Semimodules And Semicont... – No Sign-up

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

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

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...

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.

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

Constructing resolutions using free semimodules or injective envelopes (like the "max-plus" analogues of vector spaces). 3. Semicontinuity and Stability

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