# ag.algebraic geometry – Euler characteristic and rational Poincaré series

$$DeclareMathOperatorlen{len}DeclareMathOperatorTor{Tor}$$Let $$(A,mathfrak{m})$$ be a regular local ring, and $$x in mathfrak{m}^2$$ be a non-zero prime element. So $$R:=A/(x)$$ is a non-regular Cohen-Macaulay local domain.

By Eisenbud’s famous work “Homological algebra on a complete intersection, with an application to group representations”, the (infinite) minimal free resolution of any finitely generated $$R$$-module $$M$$ becomes periodic of period at most $$2$$ after at most $$dim R$$ steps.

For two finitely generated $$R$$-module $$M, N$$ such that $$Tor_i^R(M,N)$$ has finite length for all $$i$$, we define their Poincaré series as

$$P_{M,N}(t)=sum_{i=0}^{infty}len_{R}(Tor_i^R(M,N))t^{i}$$.

By Eisenbud’s result, $$P_{M,N}(t)$$ is a rational function with only possible poles at $$t=1$$. We then define the Euler characteristic as $$chi(M,N):=P_{M,N}(-1)$$.

Questions: has such generalized Euler characteristic been studied before? Is it well-behaved e.g satisfying usual properties of Euler characteristic? Do we know analogs of Serre’s homological conjecture for it?