The Hilbert series of a graded vector space $V=\bigoplus_{n\mathbb Z}V_n$ is the (ordinaty) generating function of the dimensions of its homogeneous components, $h_V(t)=\sum_{n\in\mathbb Z}t^n\dim V_n$.

If now we have a finite length complex of graded vector spaces and homogeneous maps $$0\to V^0\to V^1\to\cdots\to V^\ell\to0$$ we can compute the Euler characteristic of the Hilbert series, $$\chi=\sum_{i=0}^\ell(-1)^i h_{V^i}(t).$$

Is there a standard name for «the Euler characteristic of/for the Hilbert series»?

This is a silly question, but I've written that phrase too many times :-/

The Hilbert series of a graded vector space $V=\bigoplus_{n\mathbb Z}V_n$ is the (ordinary) generating function of the dimensions of its homogeneous components, $h_V(t)=\sum_{n\in\mathbb Z}t^n\dim V_n$.

If now we have a finite length complex of graded vector spaces and homogeneous maps $$0\to V^0\to V^1\to\cdots\to V^\ell\to0$$ we can compute the Euler characteristic of the Hilbert series, $$\chi=\sum_{i=0}^\ell(-1)^i h_{V^i}(t).$$

Is there a standard name for «the Euler characteristic of/for the Hilbert series»?

This is a silly question, but I've written that phrase too many times :-/