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We calculate the derivations and the first Hochschild cohomology group of thequantum grassmannian over a field of characteristic zero in the generic casewhen the deformation parameter is not a root of unity. Using graded techniquesand two special homogeneous normal elements of the quantum grassmannian, wereduce the problem to computing derivations of the quantum grassmannian thatact trivially on these two normal elements. We then use the dehomogenisationequality which shows that a localisation of the quantum grassmannian is equalto a skew Laurent extension of quantum matrices. This equality is used toconnect derivations of the quantum grassmannian with those of quantum matrices.More precisely, again using graded techniques, we show that derivations of thequantum grassmannian that act trivially on our two normal elements restrict tohomogeneous derivations of quantum matrices. The derivations of quantummatrices are known in the square case and technical details needed to deal withthe general case are given in an appendix. This allows us to explicitlydescribe the first Hochschild cohomology group of the quantum grassmannian.

Derivations and the first Hochschild cohomology group of the quantum grassmannian