NEW CLASSES OF MODULE AXE-FILTRATIONS, HILBERT’S THEOREM AND ANALYTIC SPREAD OF A MODULE AXE-FILTRATION

In this work, we develop an asymptotic theory for axe-filtrations on modules, a structure that generalizes classical filtrations, quasi-graduations, and axe quasi-graduations. Our objective is to extend the theory of analytic spread, originally introduced by Northcott and Rees for ideals, to this broader framework. Our approach follows the successive generalizations investigated by Diagana, Brou, and Kablam [1-5, 8]. Our first contribution is the classification of these structures into $I$‑adic, $I$-good, and $f$-good types, building upon the fundamental results on quasi-graduations [9, 10]. The main result of this paper establishes a Hilbert-Samuel type theorem for these new structures: we prove that the Hilbert function associated with a good axe-filtration on a finitely generated module is of polynomial type for sufficiently large integers. This fundamental property allows us to rigorously define the analytic spread of an axe-filtration, denoted by $\lambda (\varphi)$, and to establish its primary properties, including its relationship with the module height and the classical analytic spread. This contribution offers new perspectives for asymptotic analysis in algebraic structures governed by constraints where standard filtration theory is no longer applicable.