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This is inspired by the theorem mentioned in Why is this theorem attributed to Serre?. But I'm not sure if it's research level. If not, please feel free to vote for closing.

Let $R$ be a ring and let $Mod_R$ be the category of finitely generated $R$-modules.

What are examples of additive integer-valued functions on $Mod_R$, i.e. functions $\lambda: Mod_R \to \mathbb{Z}$ satisfying $\lambda(M) = \lambda(M') + \lambda(M'')$ for short exact sequences $$0 \to M' \to M \to M'' \to 0$$ in $Mod_R$ ?

Two obvious examples that come into my mind are:

  1. $\lambda(M)=\dim_k M$ if $R=k$ is a field.
  2. $\lambda(M)=\text{length}(M)$ if $R$ is Artinian.
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If $R$ is Artinian, any such function is uniquely and freely determined by what it does to simple modules, isn't it? –  Qiaochu Yuan Jun 19 '12 at 23:30
    
@Ralph: Spelling of length is wrong in the last statement. –  Chandrasekhar Jun 20 '12 at 1:29
    
@Qiaochu: Yes it is. –  Ralph Jun 20 '12 at 6:59
    
@Chandrasekhar: Thanks for the hint. –  Ralph Jun 20 '12 at 7:01
    
@Ralph: When you think about these sort of questions and try to construct the universal example, you end up with G-theory automatically. –  Martin Brandenburg Jun 20 '12 at 7:14

5 Answers 5

up vote 8 down vote accepted

Such functions are the same as homomorphisms $G_0(R)\rightarrow\mathbb{Z}$ from the Grothendieck group of your category, the $G$-theory group of degree $0$. The answer is only trivial from this formal point of view. The computation of $G_0(R)$ is non-trivial in general. If your ring is commutative noetherian and regular then $G_0(R)=K_0(R)$ is the $K$-theory group of degree $0$, i.e. additive functions only depend on the behaviour on projectives.

Let me complete my answer with the examples you consider in your question. If $R=k$ is a field $G_0(k)=K_0(k)=\mathbb{Z}$ generated by the isomorphism class of $k$, therefore all additive functions are multiples of the dimension. If $R$ is artinian then $G_0(R)$ is the free abelian group on simple $R$-modules, hence not all additive functions are multiples of the length in general, but for each simple module $S$, $\lambda(S)=n_S\cdot\operatorname{length}(S)$, and any choice of such $n_S$ determines an additive function $\lambda$.

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1  
$G_0(R)=K_0(R)$ even holds when $R$ is a left-noetherian left-regular ring (see Rosenberg's book). –  Martin Brandenburg Jun 20 '12 at 7:13

This has been studied for group rings to some extend. It is a theorem of Wolfgang Lück that a homomorphism $\varphi \colon G_0(\mathbb Z \Gamma) \to \mathbb R$ can be constructed with the property $\varphi([\mathbb Z \Gamma]) = 1$ if $\Gamma$ is amenable. Moreover, such a homomorphism cannot exist if $\Gamma$ contains a non-abelian free group. It is conjectured that the existence is a characterization of amenability. Moreover, if $\Gamma$ is torsionfree and amenable, the conjecture is that the range of $\varphi$ is $\mathbb Z$, this is called Atiyah's conjecture.

Sometimes, maps like the one you consider exist on subcategories of the category of f.g. modules. An easy example is the category of f.g. abelian groups $A$, so that $A \otimes_{\mathbb Z} \mathbb Q=0$, i.e. torsion groups. Then, the map $A \mapsto \log |A|$ is additive.

There is also a version for f.g. modules over the group ring of an amenable group. It can be shown that assinging to a f.g. module $M$ over $\mathbb Z \Gamma$ ($\Gamma$ is amenable here) the entropy of the natural $\Gamma$-action on the Pontryagin dual of $M$ is additive. This is Yuzvinskii's Additivity Formula as proved by Hanfeng Li in

Hanfeng Li, Compact group automorphisms, addition formulas and Fuglede-Kadison determinants, Ann. of Math. (2) 176 (2012), no. 1, 303--347.

If $\Gamma$ is finite, then this entropy is essentially the logarithm of the cardinality of $M$. For infinite $\Gamma$, this invariant of $M$ is equal to the so-called $\ell^2$-Torsion of $M$, if it can be defined. For $\Gamma = \mathbb Z^d$, this invariant is related to the Mahler measure and of number theoretic significance.

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Thanks for this very interesting answer. A comment and a question: 1) The log-example can also be used if $R$ is finite (i.e. $\lambda(M)=\log|M|$), giving some variant of the length. 2) In your explanation you relate the Atiyah conjecture and Lück's map $\varphi$. From a historical point of view, was the Atiyah conjecture first ? –  Ralph Jun 20 '12 at 9:37
    
Atiyah asked in the last 70s if $\ell^2$-Betti numbers are always rational. This question has turned into a conjecture over the years, with various modifications for torsionfree groups and groups with bounded torsion. Finally, some have been disproved by Grigorchuk-Zuk, and later Austin, Grabowski and Schick-Zuk (in various papers). The torsionfree case is still open. If true, then Lück's map above is integer-valued. Lück defined the map in his study of dimension-functions in the 90's, also Elek has a definition of rank of modules over the group ring of an amenable group. –  Andreas Thom Jun 21 '12 at 9:28

If R is a domain and $\lambda$ has non-negative values on objects of $Mod_R$, then $\lambda$ is a multiple of generic rank. See this question Nonnegative additive functions on coherent sheaves.

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Let me suggest you some references:

Northcott Reufel, "A generalization of the concept of length" (http://qjmath.oxfordjournals.org/content/16/4/297.full.pdf)

Vamos, "ADDITIVE FUNCTIONS AND DUALITY OVER NOETHERIAN RINGS" (http://qjmath.oxfordjournals.org/content/19/1/43.full.pdf)

Zanardo, "Multiplicative invariants and length functions over valuation domains" (http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.jca/1323364358)

If your interest goes in this direction I know that people in Padova is working on generalizations of the work of Zanardo to classify length functions of Prufer domains.

The classification given by Vamos on Noetherian rings was generalized by him in his (non-pubblished) PhD thesis to a classification for rings with Gabriel-Krull dimension. I recently gave an alternative poof of Vamos' result for Grothendieck categories with Gabriel-Krull dimension based on the formalism of torsion theories, contact me if you are interested.

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In general there are many such functions and the non-trivial ones are pretty interesting. I will focus on the commutative case, as that's what I know best.

So let $R$ be a commutative noetherian ring and $X \subseteq \text{Spec}(R)$ be an artinian subscheme. Let $F_{\bullet}$ be a perfect complex such that the homologies are supported on $X$. Then it defines a function from $G_0(R) \to \mathbb Z$:

$$\chi_{F_{\bullet}}: M \mapsto \chi(F_{\bullet}\otimes M) $$

Here $\chi$ of a complex with support in $X$ is just the alternating sum of length of the homologies.

When $R$ is artinian and $F_{\bullet}$ is just a single module $R$ one recovers the length example in your question.

The interesting problem is: when such a function is a "new" one? As Mahdi pointed out, if the function is non-negative on the modules, then it would just be a multiple of rank (suitably defined). Thus to make it interesting one would need it to be negative on some modules.

The issue now has deep consequences in intersection theory. Namely, if such complex exists one can often (say if $R$ is local and Cohen-Macaulay) replace it with the resolution of an artinian module of finite projective dimension. But then the definition would agree with Serre's intersection multiplicity. Hence such an example would imply that Serre's definition does not work in a singular setting (as intersection multiplicity should not be negative!).

This was an open problem for a while. The first example was constructed in a famous paper by Dutta-Hochster-McLaughlin. The construction is very complicated (involving the construction of a $60\times60$ matrix essentially by hand). This result has been extended by Levine, Roberts-Srinivas, Miller-Singh, Kurano and others. In fact, such an example is now understood in a more general framework of numerically nontrivial elements of Grothendieck groups of local rings.

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That's quite interesting. Thanks for the example. –  Ralph Aug 9 '12 at 6:44

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