# Euler characteristic and rational Poincaré series

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

$$\DeclareMathOperator\Tor{Tor}$$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.

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, available here.

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function (and the Ext version, which is sometimes called the Herbrand difference, first suggested in Buchweitz's famous unpublished note) have appeared many times recently from different aspects.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 and algebraic K-theory, (Mark Walker and his collaborators), topological K-theory (Buchweitz-Van Straten), Adams operations (Michael Brown) or Chern character on matrix factorizations (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. Unfortunately I never got around to publish it, but you can check the papers that cite it for more update information.

• I am not doing research on commutative algebra (and know basically nothing), but hope to develop an intersection theory for "not too singular" varieties after reading Fulton's book (which motivates my first question here). I came up with this definition after learning about Eisenbud's work in your answer, and proved some basic properties about it. I doubt someone has studied this in depth before, that's why I ask for a reference. Thank you! Feb 16, 2021 at 0:26
• It's interesting that you came up with that idea! I saw your question right away, but hesitated to answer since it would mean talking a lot about my own work. It should be noted that why the Hochster' pairing has many interesting property, it does not behave like one hope a true "intersection pairing" would do. For instance it vanishes uniformly for isolated hypersurfaces of even dimensions (at least over a field). Feb 16, 2021 at 1:54