ag.algebraic geometry - Cohomology with compact support for coherent sheaves on a scheme

If X is a smooth scheme over complex numbers then you can consider $X_{an}$ as an complex analytic manifold and compute singular/ deRham/ simplicial cohomology with compact supports (this will be different from usual cohomology if X is not proper)
On the Algebraic side there is etale cohomology with compact supports (which is defined by embedding X into a proper scheme...).

Comparison theorems tell you that etale cohomology with torsion coefficients agree with singular cohomology (with torsion coeff).

Any reference on etale cohomology will discuss this.

Ref:SGA 4.5, Milne: Etale Cohomology.