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At the end of V.3.4 in Algebra: Chapter 0, Aluffi describes the construction of a Grothendieck group over the category of finite dimensional $\operatorname{k}$-vector spaces, $K(\operatorname{k-vect^f})$. You first construct $F^{\operatorname{Ab}}([\operatorname{k-vect^f}])$, the free abelian group on the isomorphism classes of objects in $\operatorname{k-vect^f}$. Then you quotient out the subgroup generated by $[V]-[U]-[W]$ whenever there is an exact \begin{equation} 0\longrightarrow U\longrightarrow V\longrightarrow W\longrightarrow 0. \end{equation}

Although the construction works in more general settings (over exact categories), but let's just focus on this $K(\operatorname{k-vect^f})$. In particular, this Grothendieck group gives a generalized Euler characteristic \begin{equation} \chi_K(V_{\bullet})=\sum(-1)^j[V_j], \end{equation} where $V_{\bullet}$ is the complex\begin{equation} 0\longrightarrow V_{N}\longrightarrow V_{N-1}\longrightarrow V_{N-2}\longrightarrow\cdots\longrightarrow V_0\longrightarrow 0. \end{equation}

Allufi says this this universal in the sense that if $\delta:\operatorname{k-vect^f}\to G$ is a function from $\operatorname{k-vect^f}$ to an abelian group satisfying two natural conditions: $\delta(V)=\delta(V')$ if $V\cong V'$, and $\delta(V/U)=\delta(V)-\delta(U)$, then there is a unique group homomorphism $\operatorname{k-vect^f}\to G$ that maps \begin{equation} \chi_{K}(V_\bullet)\mapsto \chi_{G}(V_\bullet):=\sum(-1)^j\delta(V_j). \end{equation}

Note that the original Euler characteristic is obtained by taking $\delta(V)=\operatorname{dim}(V)$.

All this seems impressive, but it would not be powerful if we did not have a supply of other $\delta$ functions other than $\operatorname{dim}$. Unfortunately I could not think of interesting examples of such functions.

Can someone give some nice examples? Maybe $\operatorname{k-vect^f}$ is not so good because its Grothendieck group is too simple, so examples from other categories are also welcome.

Thanks so much!

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This is essentially what K-theory does: Classify generalized Euler characteristics / dimensions. And since the reduced K-theory of a field vanishes, there is no other generalized Euler characteristic except the one induced by the dimension (or multiples of it). More precisely, let $A$ be an arbitrary abelian group and $a \in A$. Then there is a unique homomorphism $\mathbb{Z} \to A$ mapping $1 \mapsto a$, which corresponds to a unique generalized Euler characteristic mapping the $1$-dimensional vector space to $a$, or more generally $[V]$ to $\mathrm{dim}(V) \cdot a$. Only the dimension matters here, and no new interesting invariant can arise.

Similarily, the isomorphism $K(\mathsf{FinAb}) \cong (\mathbb{Q}^+,*)$ shows that the order is essentially the only generalized Euler characteristic defined on finite abelian groups.

Other exact categories provide more interesting invariants. But I won't even try to enumerate them here, because it is not possible and there are, of course, a lots of interesting books on K-theory which you may consult (Rosenberg, Bass, Weibel, etc.). Perhaps you can focus your question a little bit, or ask a new question.

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Thanks very much! This is exactly what I thought: there is no other Euler characteristics on finite dimension vector spaces and finite abelian groups. So K theory does not show its power on these simple categories. –  Hui Yu Feb 7 '13 at 1:59

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