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We extend Lawrence's representations of the braid groups to relative homologymodules, and we show that they are free modules over a Laurent polynomialsring. We define homological operators and we show that they actually provide arepresentation for an integral version for $U_q \mathfrak{sl}(2)$. We suggestan isomorphism between a given basis of homological modules and the standardbasis of tensor products of Verma modules, and we show it to preserve theintegral ring of coefficients, the action of $U_q \mathfrak{sl}(2)$, the braidgroup representations and their grading. This recovers an integral version forKohno's theorem relating absolute Lawrence representations with quantum braidrepresentation on highest weight vectors. It is an extension of the lattertheorem as we get rid of generic conditions on parameters, and as we recoverthe entire product of Verma-modules as a braid group and a $U_q\mathfrak{sl}(2)$-module.

A homological model for $U_q \mathfrak{sl}(2)$ Verma-modules and their braid representations