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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 :-/

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  • $\begingroup$ Knot theory people use «graded Euler characteristic» if I recall correctly, but it seems restricted to that crowd. $\endgroup$
    – Mariano Suárez-Álvarez
    Dec 11, 2014 at 22:00
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    $\begingroup$ "Euler characteristic of/for Hilbert series" sounds a bit off to me; I would expect some space or complex to have Euler characteristic, not the Hilbert series. I also only heard the term "graded Euler characteristic" in the knot context. In various disjoint crowds where I saw this thing used, people frequently say "generating function of Euler characteristics" (which effectively means they prefer to change the order of summation in your formula for $\chi$ ;-) ). Another usage I encountered was "Poincare series", implying that "Hilbert" counts dimensions, and "Poincare" - Euler characteristic. $\endgroup$
    – Vladimir Dotsenko
    Dec 11, 2014 at 23:05

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