Additive integer-valued functions on the module category - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T22:03:05Z http://mathoverflow.net/feeds/question/100042 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/100042/additive-integer-valued-functions-on-the-module-category Additive integer-valued functions on the module category Ralph 2012-06-19T19:59:10Z 2012-08-09T03:08:22Z <p>This is inspired by the theorem mentioned in <a href="http://mathoverflow.net/questions/99916/why-is-this-theorem-attributed-to-serre" rel="nofollow">http://mathoverflow.net/questions/99916/why-is-this-theorem-attributed-to-serre</a>. But I'm not sure if it's research level. If not, please feel free to vote for closing. </p> <p>Let $R$ be a ring and let $Mod_R$ be the category of finitely generated $R$-modules. </p> <p>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 $Mod_R$ ? </p> <p>Two obvious examples that come into my mind are: </p> <ol> <li>$\lambda(M)=\dim_k M$ if $R=k$ is a field. </li> <li>$\lambda(M)=\text{length}(M)$ if $R$ is Artinian. </li> </ol> http://mathoverflow.net/questions/100042/additive-integer-valued-functions-on-the-module-category/100047#100047 Answer by Fernando Muro for Additive integer-valued functions on the module category Fernando Muro 2012-06-19T20:33:39Z 2012-06-20T07:50:42Z <p>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.</p> <p>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 $\lambda$.</p> http://mathoverflow.net/questions/100042/additive-integer-valued-functions-on-the-module-category/100095#100095 Answer by Andreas Thom for Additive integer-valued functions on the module category Andreas Thom 2012-06-20T08:19:29Z 2012-06-20T08:34:28Z <p>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.</p> <p>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. </p> <p>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</p> <p>Hanfeng Li, <em>Compact group automorphisms, addition formulas and Fuglede-Kadison determinants</em>, Ann. of Math. (2) 176 (2012), no. 1, 303--347.</p> <p>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.</p> http://mathoverflow.net/questions/100042/additive-integer-valued-functions-on-the-module-category/100132#100132 Answer by Mahdi Majidi-Zolbanin for Additive integer-valued functions on the module category Mahdi Majidi-Zolbanin 2012-06-20T13:27:49Z 2012-06-20T13:27:49Z <p>If R is a domain and $\lambda$ has <em>non-negative</em> values on objects of $Mod_R$, then $\lambda$ is a multiple of generic rank. See this question <a href="http://mathoverflow.net/questions/83313/nonnegative-additive-functions-on-coherent-sheaves" rel="nofollow">Nonnegative additive functions on coherent sheaves</a>.</p> http://mathoverflow.net/questions/100042/additive-integer-valued-functions-on-the-module-category/103867#103867 Answer by Hailong Dao for Additive integer-valued functions on the module category Hailong Dao 2012-08-03T14:33:57Z 2012-08-09T03:08:22Z <p>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. </p> <p>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 $G_0(R) \to \mathbb Z$:</p> <p><code>$$\chi_{F_{\bullet}}: M \mapsto \chi(F_{\bullet}\otimes M)$$</code> </p> <p>Here $\chi$ of a complex with support in $X$ is just the alternating sum of length of the homologies. </p> <p>When $R$ is artinian and $F_{\bullet}$ is just a single module $R$ one recovers the length example in your question. </p> <p>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.</p> <p>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 <a href="http://mathoverflow.net/questions/12236/serre-intersection-formula-and-derived-algebraic-geometry" rel="nofollow">Serre's intersection multiplicity</a>. Hence such an example would imply that Serre's definition does not work in a singular setting (as intersection multiplicity should not be negative!). </p> <p>This was an open problem for a while. The first example was constructed in a famous paper by <a href="http://www.springerlink.com/content/x6537160gu0v442v/" rel="nofollow">Dutta-Hochster-McLaughlin</a>. 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 <a href="http://www.springerlink.com/content/4h3pqre2t80q378g/" rel="nofollow"><em>numerically nontrivial</em></a> elements of Grothendieck groups of local rings. </p> http://mathoverflow.net/questions/100042/additive-integer-valued-functions-on-the-module-category/103878#103878 Answer by Simone Virili for Additive integer-valued functions on the module category Simone Virili 2012-08-03T15:46:35Z 2012-08-03T15:46:35Z <p>Let me suggest you some references:</p> <p>Northcott Reufel, "A generalization of the concept of length" (http://qjmath.oxfordjournals.org/content/16/4/297.full.pdf)</p> <p>Vamos, "ADDITIVE FUNCTIONS AND DUALITY OVER NOETHERIAN RINGS" (http://qjmath.oxfordjournals.org/content/19/1/43.full.pdf)</p> <p>Zanardo, "Multiplicative invariants and length functions over valuation domains" (http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.jca/1323364358)</p> <p>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. </p> <p>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.</p>