# Properties of a generalization (regularization) of the Euler characteristic?

Intro: This question is about a version of the Euler characteristic for infinite dimensional chain complexes. I have no idea if this is a pre-existing concept, that's essentially what my query is about.

Filtered Complexes: Let me start by reviewing/introducting some terminology on filtered complexes. Let $$C$$ be a $$\mathbb{Z}/2$$-graded chain complex over a field $$k$$.

Definition 1: An $$\mathbb{R}_+$$-filtration* $$F$$ on $$C$$ is a decomposition of $$C$$ into sub-complexes $$F_{\le L}C \subset C$$ for each $$L \ge 0$$ with $$L \le K$$ implying $$F_{\le L}C \subseteq F_{\le K}C$$.

Given a filtration $$F$$ and a $$\lambda > 0$$, we let $$\lambda F$$ denote the filtration with $$\lambda F_{\le L}C := F_{\le \lambda L}C$$ for all $$L$$. We call this scaling the filtration.

A $$\mathbb{R}_+$$-filtration is proper if $$F_{\le L}C$$ is finite rank for all $$L$$ and exhaustive if $$\text{colim}_L F_{\le L}C = C$$. Both properties are preserved by scaling.

Now let $$(C,F)$$ and $$(C',F')$$ be two complexes equipped with exhaustive, proper filtrations.

Definition 2: We say that $$(C,F)$$ and $$(C',F')$$ are commensurate if there exists filtered quasi-isomorphisms $$f:(C,F) \to (C',\lambda F')$$ and $$g:(C',F') \to (C,\lambda F)$$ such that $$f \circ g$$ and $$g \circ f$$ are homotopic to the identities $$\text{Id}_C$$ and $$\text{Id}_{C'}$$ via homotopies respecting the filtrations.

Zeta Functions/Euler Characteristic: Now the subject of my questions. Let $$C = (C,F)$$ be a $$\mathbb{Z}/2$$-graded chain complex over a field $$k$$ equipped with an exhaustive, complete $$\mathbb{R}_+$$-filtration $$F$$.

Definition 1: (Poincare/Zeta Function) The filtered Poincare function or zeta function $$\zeta_C(s)$$ is (for this post) the formal function of $$s \in \mathbb{C}$$ defined as follows.

Consider the associated graded $$\text{Gr}(C) = \oplus_{L \in \mathbb{R}_+} \text{Gr}_L(C)$$. Here I am defining the graded pieces by $$\text{Gr}_L(C) := F_{\le L}(C)/F_{< L}(C)$$, and the sum in $$\text{Gr}(C)$$ is countable (with finite-dimensional summands) since $$F$$ is proper. We then define $$\zeta_C(s) := \sum_{L \in \mathbb{R}_+} \chi(\text{Gr}_L(C)) e^{-Ls}$$ Here $$\chi(-)$$ is the Euler characteristic, and all of the Euler characteristics in this sum are well-defined because (again) $$\text{Gr}_L(C)$$ is finite-dimensional for all $$L \in \mathbb{R}_+$$.

Of particular interest to me are complexes where $$\zeta_C$$ is meromorphic.

Definition 2: (Extension Property) We will say that $$C = (C,F,\partial)$$ has the extension property when it has the following properties.

• $$\zeta_C(s)$$ defines a meromorphic function for $$\text{Re}(s) \gg 0$$.
• $$\zeta_C:\{z| \text{Re}(s) \gg 0\} \to \mathbb{C}$$ extends to a meromorphic function $$\zeta_C:\mathbb{C} \to \mathbb{C}$$.

In the case where $$C$$ has the extension property, we can define an Euler characteristic.

Definition 3: (Euler Characteristic) Let $$(C,F)$$ be a filtered complex with the extension property.

The regularized Euler characteristic $$\chi(C,F) := \zeta_C(0)$$ to be the value of $$\zeta_C$$ at $$s = 0$$.

Properties: I am interested in the following kinds of questions. The first question is simply

Question 0: Has anyone studied these ideas before? I would really love some references if so!

My next two questions are whether the extension property and/or the regularized Euler characteristic are invariant under commensurate equivalence.

Question 1a: If $$(C,F) \sim (C',F')$$ are commensurate complexes, does this imply that $$(C,F)$$ has the extension property if and only if $$(C',F')$$ does?

Question 1b: If $$(C,F) \sim (C',F')$$ are commensurate complexes with the extension property, do we have $$\chi(C,F) = \chi(C',F')$$?

My last question is about criteria for recognizing the extension property.

Question 2: What properties can one impose on $$F$$ to ensure the extension property besides the obvious (e.g. finite dimensionality of $$C$$).

Thanks for reading the long post!