Optimal algebraic tangent cone of torsion-free sheaves via valuations

We develop a valuation-theoretic framework for studying tangent cones of torsion-free sheaves on algebraic varieties. To analyze these objects, we introduce a slope stability theory, including the Harder-Narasimhan filtrations, for finitely generated $\mathbb{R}$-graded modules over finitely generated ${\mathbb{R}}_{\ge 0}$-graded algebras. Using it, we show that there is a canonically determined tangent cone of torsion-free sheaves, up to the expected equivalence ambiguity, for quasi-regular valuations, which generalize Chen-Sun [3].