# Intuition for the Euler form in a finitary category

Suppose that $$\mathcal{C}$$ is a finitary category, so for any two objects $$A$$ and $$B$$ we have that $$|\mathrm{Ext}^i(A,B)| < \infty$$ for $$i\geq 0$$, suppose $$\mathcal{C}$$ has finite global dimension, so $$\mathrm{Ext}^i(A,B) = \{0\}$$ for all sufficiently large $$i$$*. In such a category $$\mathcal{C}$$ we can define the Euler form of two objects $$\langle A,B \rangle_m = \left( \prod_{i=0}^{\infty} \Big|\mathrm{Ext}^i(A,B)\Big|^{(-1)^i}\right)^{\frac{1}{2}}\,.$$

This is a definition I've encountered a few times while studying quiver representations and Hall algebras. It's a constant we need to introduce when giving a Hall algebra an associative algebra structure (see these lecture notes by Olivier Schiffmann for details). I've yet to get a good grasp on what exactly this is counting in terms of the objects and morphisms in the category though. Is this related to the good ol' Euler characteristic of a topological space in some non-superficial way? How should I be thinking about this form?

* Typically I've only seen this in the context of hereditary categories, where the $$\mathrm{Ext}^i$$ groups vanish for $$i>1$$, and so we only need to consider $$\mathrm{Hom}(A,B)$$ and $$\mathrm{Ext}^1(A,B)$$ for any objects $$A$$ and $$B$$.

This answer perhaps says things that are all obvious to the OP. $$\textrm{Ext}^i(A,B)$$ is a vector space over the ground field, so its cardinality is $$q^d$$ where $$d$$ is the dimension of the vector space and $$q$$ is the cardinality of the field. The quantity in question is therefore $$(q^{1/2})^{\langle A,B\rangle}$$, where $$\langle A,B\rangle = \sum_i (-1)^i \textrm{dim } \textrm{Ext}^i(A,B)$$. This quantity depends only on the dimension vectors of $$A$$ and $$B$$, and not on the actual module structures of $$A$$ and $$B$$. I have seen the induced form on the Grothendieck group (or equivalently the space of dimension vectors) called the Euler form, or the Euler-Ringel form. It is the alternating sum of the dimensions of the homology of a chain complex (obtained by taking a resolution of one of $$A$$ or $$B$$); I think this is what is Euler-like about it.