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
Frequently used to study the global sections of semimodule sheaves on tropical varieties. 3. Semicontinuity and Stability
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. Homological Algebra of Semimodules and Semicont...
The "Semicontinuity" aspect typically refers to the behavior of dimensions (like the rank of a semimodule) under deformations.
Unlike traditional modules over a ring, are defined over semirings (like the algebra)
Constructing resolutions using free semimodules or injective envelopes (like the "max-plus" analogues of vector spaces).
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. homological algebra here often uses:
A key feature is the adaptation of and Tor functors. Since you cannot always "subtract" to find boundaries, homological algebra here often uses: