Cohomological length functions (Q2828022)

Let \(\mathcal{C}\) be a triangulated category with suspension \(\Sigma\). In this paper, the author investigates certain length functions on \(\chi: \mathrm{Ob}\,\mathcal{C}\to\mathbb{N}\), which are called \textit{cohomological} if they are additive with respect to direct sums, for each \(C\in \mathrm{Ob}\,\mathcal{C}\) there is \(n\in\mathbb{Z}\) such that \(\chi(\Sigma^nC)=0\), and satisfy a condition of compatibility with exact triangles. A cohomological functor \(H:\mathcal{C}^{\mathrm{op}}\to\mathcal{A}b\) which is \textit{endofinite} (that is, for each \(C\), \(H(C)\) has finite length over the ring \(\mathrm{End}(H)\), and \(H(\Sigma^n(C)=0\) for some \(n\in\mathbb{Z}\)) yields the cohomological function \(\chi_H:\mathrm{Ob}\,\mathcal{C}\to\mathbb{N}\), \(C\mapsto\mathrm{length}_{\mathrm{End}(H)}H(C)\). Locally finite sums of cohomological functions and irreducible cohomological functions are defined in a natural way, and their basic properties are studied. The main result of the paper states that if \(\mathcal{C}\) is essentially small, then every endofinite cohomological functor decomposes uniquely into a direct sum of functors with local endomorphism rings, every cohomological function decomposes uniquely into a locally finite sum of irreducible cohomological functions, and the assignment \(H\mapsto \chi_H\) induces a bijection between the isomorphism classes of endofinite cohomological functors and the irreducible cohomological functions. Several interesting examples and explicit calculations for the category of perfect complexes over some specific rings are also given.