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Classical valuation theory studies residually transcendental extensions through value-group growth, residue-field extensions, and prolongation behavior. The purpose of this paper is more modest: we introduce a homological framework for measuring layerwise nonflatness and torsion phenomena inside chosen valuation filtrations of residually transcendental valued function fields. Given a valued field $(K,v)$ and a simple transcendental extension $K(x)$ endowed with a prolongation $w$ of $v$, we construct valuation filtration modules associated to the layers of the valuation. Applying derived functor methods to these filtration layers, we define derived residue obstruction modules \[ \HRD_i(n) := \Tor_i^{\cO_v}(F_n,k_v), \] which detect failure of homological exactness in valuation descent, persistent derived obstructions arising from nonflat valuation layers, and residual transcendence torsion phenomena. We establish Tor-vanishing criteria under explicit filtration-flat hypotheses, prove persistence results under explicit torsion hypotheses in valuation-filtration layers, and attach annihilator ideals to the first derived residue obstruction layers. The examples are divided into residually transcendental model cases and boundary torsion computations over discrete valuation rings; the latter are included only to illustrate the homological obstruction mechanism and are not presented as the main geometric source of residual transcendence. The theory proposed here is therefore best viewed as a Tor-theoretic filtration theory over valuation rings, motivated by residually transcendental valued function fields, rather than as a new numerical ramification theory.
Kundnani R. T., Marimuthu V., Alam K., Kant Sh. 2026. Homological Ramification Filtrations in Residually Transcendental Valued Function Fields. PREPRINTS.RU. https://doi.org/10.24108/preprints-3115490
