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Homology of systemic modules

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Abstract

In this paper, we develop the rudiments of a tropical homology theory. We use the language of “triples” and “systems” to simultaneously treat structures from various approaches to tropical mathematics, including semirings, hyperfields, and super tropical algebra. We enrich the algebraic structures with a negation map where it does not exist naturally. We obtain an analogue to Schanuel’s lemma which allows us to talk about projective dimension of modules in this setting. We define two different versions of homology and exactness, and study their properties. We also prove a weak Snake lemma type result.

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Notes

  1. There is a related weaker relation \(\nabla \), where \(a_1 \nabla a_2\) if and only if \(\mathbb {0}\preceq a_1 (-) a_2\), which also restricts to equality on \({\mathcal {T}}\). The relation \(\nabla \) is reflexive and symmetric, but not transitive. According to experience, \(\nabla \) has been more effective in linear algebra [2, 3], but \(\preceq \) serves better in developing representation theory, and features in [31] as well as this paper.

  2. In order not to get caught up in later complications, but not following the convention in [8, Definition 1.1], we adjoin a formal absorbing element \(\mathbb {0}\) to \({\mathcal {T}}\), i.e., \(a\mathbb {0}= \mathbb {0}a = \mathbb {0}\), for all \(a \in {\mathcal {T}}\).

  3. One could also require \(b^\circ = b\). This has categorical advantages seen in Sect. 6 but has algebraic drawbacks.

  4. This slightly strengthens the version of “uniquely negated,” for triples, used in [45], which says that there is a unique element b of \({\mathcal {T}}\) for which \( a+b \in \mathcal {A}^\circ .\)

  5. Note that this is just an equalizer of f.

  6. It might be more natural to use \(\preceq \) instead of \(\succeq \); for example, when \(f=0\) we would get \(b'(-)b \) a quasi-zero. But then we woulg lose the important Proposition 3.25.

  7. Even if we use the term \(\succeq \)-chain and \(\succeq \)-exact in this definition, all morphisms are assumed to be \(\preceq \)-morphisms unless otherwise stated.

  8. [23, § 1.3.1] calls this an “ideal” but we prefer to reserve this terminology for semirings.

  9. By this notation, we mean \(f+(-)f\). This makes sense since we assume that \({\text {Hom}}(A,B)\) is an additive semigroup. Alternatively, more in line with [11], one could take N to be \(\{ f: f = (-)f \}.\) This is more inclusive since \(f^\circ = (-) f^\circ .\) On the other hand when \((-)\) is the identity, \(f = (-)f\) always.

  10. See [11, Equation (22)]. This is a categorical version of the twist product defined for modules in Definition 2.36. For application to tropical geometry, see [30].

  11. In [23], this is called the kernel with respect to N.

  12. Once it exists, g as in the above diagram is unique. See [23, §1.5.5].

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Acknowledgements

J.J. was supported by an AMS-Simons travel grant.

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Authors and Affiliations

  1. Department of Mathematics, State University of New York at New Paltz, New Paltz, NY, 12561, USA

    Jaiung Jun

  2. Department of Mathematics, Tulane University, New Orleans, LA, 70118, USA

    Kalina Mincheva

  3. Department of Mathematics, Bar-Ilan University, 52900, Ramat-Gan, Israel

    Louis Rowen

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  1. Jaiung Jun
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  2. Kalina Mincheva
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Correspondence to Kalina Mincheva.

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Jun, J., Mincheva, K. & Rowen, L. Homology of systemic modules. manuscripta math. 167, 469–520 (2022). https://doi.org/10.1007/s00229-021-01272-z

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