Homology of systemic modules

Abstract

We develop the rudiments of a tropical homology theory, based on "triples" and "systems." Results include a version of Schanuel's lemma, projective dimension, the homology semi-module, and a weak Snake lemma. Contents 15 3.3. Chain complexes and exact sequences of systemic modules 19 4. (h, )-Projective dimension 20 4.1. Resolutions 20 5. Homological theory of systems 22 5.1. Homology of precongruences 22 5.2. Chains of systemic modules 25 6. Categorical aspects in the perspective of homological categories 29 6.1. Categories with negation 29 6.2. Towards semiexact categories and homological categories 31 References 34

Vector bundles on tropical schemes

“…One can easily see that with Definition 3.19, any free module is projective. We also note that in [JMR19] and [JMR20], more general versions of projective modules are introduced in a framework of systems which include modules over hyperfields.…”