Summary: For a family of closed sets \(\{\lambda_i\}\) stable under finite intersection, we prove that vanishing of cohomology groups with support over each \(\lambda_j\) for a complex of sheaves, implies vanishing over finite unions. By means of this we get, for an open convex set \(\Omega \subset \mathbb{R}^n\), a criterion of decomposition of the microsupport at the boundary \(SS_\Omega\) in the sense of P. Schapira [Semin., Equations Deriv. Partielles 1985-1986, Expose No. 13, 12 p. (1986; Zbl 0638.58027)] which refines former results by J.-L. Lieutnant [J. Fac. Sci., Univ. Tokyo, Sect. I A 33, 83-130 (1986; Zbl 0635.32005)] and G. Zampieri [Proc. Japan Acad., Ser. A 67, No. 6, 217-222 (1991; Zbl 0760.32006)].