$\DeclareMathOperator\len{len}\DeclareMathOperator\Tor{Tor}$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?



Euler characteristic and rational Poincaré series
$\DeclareMathOperator\Tor{Tor}$I studied this function in my thesis (gee, typing this answer makes me feel old!). In fact, this function can be defined even when the Tor modules have finite length *eventually*. One simply takes the difference $\theta^R(M,N):=\ell(\Tor\_{2e}(M,N)) - \ell(\Tor\_{2e-1}(M,N))$ for some $e\gg0$. When $M$ has finite projective dimension on the punctured spectrum of $R$, then $\theta^R(M,N)$ is defines for all $N$, and gives a map from the Grothendieck group of $\mathrm{mod}(R)$ to $\mathbb Z$.


The idea was suggested in an old paper by Hochster, and this is now widely called Hochster's theta invariant. You can find some basic information in [my paper here](https://www.ams.org/journals/tran/2013-365-06/S0002-9947-2012-05574-7/S0002-9947-2012-05574-7.pdf).


The most interesting case is when $R$ has isolated singularity, then one has a pairing on the Grothendieck group. I made a number of conjectures along the line of the Serre's homological conjectures. They are too numerous to list, you can read about them in Section 3 of [this paper with Kurano](https://www.cambridge.org/core/journals/journal-of-k-theory/article/abs/hochsters-theta-pairing-and-numerical-equivalence/21AC7DE6E78191A6AA69F646811009B0), available [here](http://www.isc.meiji.ac.jp/%7Ekurano/ronbun/kurano32.pdf).


This function (and the Ext version, which is sometimes called the Herbrand difference, first suggested in Buchweitz's famous unpublished note) have appeared many times recently from different aspects. Here is a sample of some of my favorite: [projective geometry](https://arxiv.org/abs/0910.1289) and [algebraic K-theory](https://arxiv.org/abs/1402.0805), (Mark Walker and his collaborators), [topological K-theory](https://arxiv.org/abs/1103.5574) (Buchweitz-Van Straten), [Adams operations](https://arxiv.org/abs/1608.07116) (Michael Brown) or [Chern character on matrix factorizations](https://projecteuclid.org/euclid.dmj/1340801626) (Polishchuck-Vaintrob).


If you are actually doing research on this topic, feel free to send me an email and I can load you up with more specific information.


PS: a version for complete intersection exists, and was written down in [the second part of my thesis](https://arxiv.org/abs/0710.5818). Unfortunately I never got around to publish it, but you can check [the papers that cite it](https://scholar.google.com/scholar?client=firefox-b-1-d&um=1&ie=UTF-8&lr&cites=6295381473396041052) for more update information.


