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Classical Krasner theory controls the stability of algebraic extensions through valuation inequalities: if an algebraic element is approximated sufficiently closely, then the generated extension stabilizes. This article develops a Tor-theoretic obstruction formalism associated with Krasner-type approximation over Henselian valued fields. Given algebraic elements \(\alpha,\beta \in \overline{K}\) algebraic over \(K\), we attach to the pair an approximation quotient module \[ A_{\alpha,\beta} := \frac{\OK[\alpha,\beta]} {\OK[\alpha]+\OK[\beta]}. \] We define the Tor-theoretic Krasner defect modules \[ \DKD_i(\alpha,\beta) := \Tor_i^{\OK}(A_{\alpha,\beta},k_K). \] In the discretely valued setting, this construction has homological amplitude at most one: \(\DKD_0\) records the residual approximation quotient, while \(\DKD_1\) records the \(\pi\)-torsion of \(A_{\alpha,\beta}\). The paper therefore does not claim a higher obstruction theory over discrete valuation rings. Its main contribution is to isolate, after classical Krasner field-theoretic stabilization, the remaining integral saturation obstruction \[ \OK[\alpha]+\OK[\beta]\subseteq \OK[\alpha,\beta]. \] Tame defectless hypotheses alone are not used to prove flatness of \(A_{\alpha,\beta}\); rather, flatness or saturation is stated as the precise additional integral condition under which the positive Tor-defect vanishes. Wild and inseparable examples are presented as sources of possible non-vanishing unless an explicit torsion class is exhibited.
Kundnani R. T., Marimuthu V., Alam K., Kant Sh. 2026. Tor-theoretic Krasner Constants and Torsion-Theoretic Approximation Defects over Henselian Valued Fields. PREPRINTS.RU. https://doi.org/10.24108/preprints-3115322
