Equivariant Euler Characteristic

The equivariant Euler characteristic of an equivariant coherent sheaf is the alternating sum of its cohomology representations, assuming these are finite-dimensional and vanish in sufficiently high degree. When these representations are completely reducible, the invariant Euler characteristic takes the multiplicity of the trivial representation in this sum, equivalently the alternating sum of the dimensions of the invariant subspaces.