A theorem of Poincar\'e-Hopf type

We compute (algebraically) the Euler characteristic of a complex of sheaves with constructible cohomology. A stratified Poincar'e-Hopf formula is then a consequence of the smooth Poincar'e-Hopf theorem and of additivity of the Euler-Poincar'e characteristic with compact supports, once we have a suitable definition of index.