ag.algebraic geometry – Euler characteristic and rational PoincarĂ© 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


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?