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!


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