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Derivations and Alberti representations

We relate generalized Lebesgue decompositions of measures in terms of curve fragments (Alberti representations) and Weaver derivations. This correspondence leads to a geometric characterization of the local norm on the Weaver cotangent bundle of a metric measure space $(X,μ)$: the local norm of a form $df$ sees how fast $f$ grows on curve fragments seen by $μ$. This implies a new characterization of differentiability spaces in terms of the $μ$-a.e.~equality of the local norm of $df$ and the local Lipschitz constant of $f$. As a consequence, the Lip-lip inequality of Keith must be an equality. We also provide dimensional bounds for the module of derivations in terms of the Assouad dimension of $X$.