Homology of systemic modules (Q2114195)

The main purpose of the paper under review is to lay the foundations of a consistent, effective homology theory for systemic modules. What are these? Although this notion should be considered very well established, due to the many papers already appeared on the subject, it is less known among the non experts. Therefore the purpose of the present review will be to speak a bit about the author's jargon, putting it in a proper historical and mathematical context, trying to feel the juice of the main results. The spectacular achievements obtained within the last three decades by the so called tropical algebraic geometry has induced a massive development of what, accordingly, one refers to as tropical algebra. The latter is concerned, so to speak, with traditional semi-structures: semi-groups instead of groups, semi-rings instead of rings. Semi-rings are sets endowed with two binary operations satisfying all the axioms of a ring with the possible exception of the existence of an additive inverse. Due to its importance in tropical geometry, the most popular example of semi-ring which is not a ring, is \(\mathbb{R}_{\max}=(\mathbb{R}\cup\{-\infty\}, \max, +)\), the set of positive real numbers union \(-\infty\) (the neutral element for the additive structure), where ``\(\max\)'' plays the role of the sum, and ``+'' plays the role of the multiplication, However, once one focuses on the purely algebraic theory, it is soon realized that the lack of negatives in semi-rings does not allow to go too far, especially with respect to building a homology theory working for (super)tropical algebras. This latter task, that is anyhow interesting in its own, enjoys of additional motivation coming from some important relatively recent work due to \textit{A. Connes} and \textit{C. Consani} [High. Struct. 3, No. 1, 155--247 (2019; Zbl 1411.18006)], who studied similar problems especially related to the theory of fields with one element, to which the authors of the paper under review compare their own. The idea to circumvent the difficulties arising by lack of negatives is to consider \textit{systems}, arising as follows. In the most general situation one is given with a \textit{magma} \(\mathcal{T}\), which, for simplicity, the reader can think of as a multiplicative monoid endowed of an absorbing element, to play the role of the set of \textit{tangible elements}. The other ingredients are a semi-module \(\mathcal{A}\) over \(\mathcal{T}\) containing \(\mathcal{T}\), and a \textit{negation}, namely a \(\mathcal{T}\) map \((-):\mathcal{A}\to \mathcal{A}\) such that its square is the identity. If \((\mathcal{T}\cup\{0\})\cap \mathcal{A}\) is empty, then the data \((\mathcal{T}, \mathcal{A}, (-))\) is said to be a \textit{triple}. A \textit{system} is a quadruple \((\mathcal{T}, \mathcal{A}, (-), \preceq)\), where \((\mathcal{T}, \mathcal{A}, (-))\) is a triple and \(\preceq\) is a \textit{surpassing} relation, which often substitutes the equality. One example of surpassing relation, though not all are of this kind, is the one induced by a negation, by declaring \(a\preceq b\) if \(a+c=b\), for some \textit{quasi-zero} \(c\) which, by definition, is is a quasi-zero, i.e. a \(\mathcal{T}\)-linear combination of elements of the form \(a(-)a\). Basing on the notion of systems, the authors define \(\preceq\)-morphisms betwewn systemic modules. In this context a substitutive notion of projective dimension of systemic modules is achieved. This is probably the point where the reader may wonder how much of the elementary traditional apparatus is possible to recover from the classical theory. More concretely: what about, for example, the snake lemma? or what about the Schanuel lemma, which in the classical situation says that if \(0\to M' \to M \to M'' \to 0\) and \(0 \to N' \to N \to N'' \to 0\) are short exact sequences of \(R\)-modules and \(N\) and \(N'\) are projective, then \(M'\oplus N\) is isomorphic to \(N' \oplus M\)? As the authors point out, in spite it does not look easy to get the full systemic analog even of standard homological results, the approach based on systemic modules is effective in providing additional insight that is not available in the categorical approach. In the paper the main actors are \(\preceq\)-morphism \(f:M\to M'\). With respect to them, the authors succeed to define the basic notions on which constructing the apparatus, namely the \textit{null-module kernel}, the \(\succeq\)-\textit{module image} and the \textit{systemic cokernel}. Basing on these crucial bricks, the paper releases the seeds of a promising and fruitful theory based on five fundamental theorems which can be considered the foundation of this new intriguing subject. Among them, all rather technical, a semi-Schanuel lemma (Theorem B) and a semi-snake lemma (Theorem C). Theorem A, instead, constructs a faithful functor from the category of semirings into the category of semirings with a negation map (and preserving additive idempotence). The paper consists in six sections, whose first is a very enlightening introduction clarifying motivations, aims and scopes. All the basic notion which this review attempted to give a flavor of, are collected in Section 2. That helps to keep the paper fairly self contained. A special attention is devoted to the examples collected in Section 2.8. Section 3 draws a parallel of semi-rings theory with classical module theory, The beautiful paper under review culminates with the very three last sections. The fourth one is devoted, as the title puts it, to the notion of projective dimension for the \(\preceq\)-morphism: Section 5, probably the true hearth of all the article, is about the \textit{Homology of Systemic Modules}. Last Section 6, \textit{Categorical aspects in the perspective of homological categories}, is devoted to some general considerations which, more than being mere speculations are concrete suggestions on how to push the research forward. The paper under review is really and exceptionally well written, with an enthusiastic mood capturing the reader's attention. However the subject is rather new and barely the non-specialist audience can capture the subtleties without an adequate bibliographic support. That is why, like being aware of this need, the paper ends itself by a rich list of comprehensive references, through which no reader can ever feel lost.