A proof of Solomon's second conjecture on local zeta functions of orders
Let \(R\) be a complete discrete valuation domain with quotient field \(K\) and residue field \(k\) such that \(|k|=q<\infty\). Let \(\Lambda\) be an \(R\)-order in a semisimple \(K\)-algebra \(A\), and let \(V\) be a finite dimensional \(A\)-module. By the Jordan-Zassenhaus theorem, there is a finite representing system \(I\) of full \(\Lambda\)-lattices in \(V\). For \(L,M\in I\), consider the zeta function \[ Z_{LM}(s)=\sum(L:N)^{-s}, \] where \(N\) runs through the \(\Lambda\)-submodules of finite index in \(L\) which are isomorphic to \(M\). Solomon has shown that the \(|I|\times|I|\) matrix \(\mathbb{Z}_\Lambda(V;s)=(Z_{LM}(s))\) has an inverse with coefficients in \(\mathbb{Z}[q^{-s}]\). His second conjecture (1978) states that the determinant of \(\mathbb{Z}_\Lambda(V;s)\) is of the form \[ \prod(1-q^{a_i-b_is})^{-1} \] with \(a_i\in\mathbb{N}\) and \(b_i\in\mathbb{N}\setminus\{ 0\}\). The author proves this conjecture by an inductive method which already led to a solution of two other important problems in representation theory. Firstly, he obtained a characterization of the Auslander-Reiten quivers of representation-finite orders [\(\tau\)-categories, I, II, III, 72pp., Algebr. Represent. Theory (to appear)]. Secondly, he proved a conjecture of Ringel-Yamagata, which implies that the representation dimension of any Artinian algebra is finite [Proc. Am. Math. Soc. 131, No. 4, 1011-1014 (2003; Zbl 1018.16010)]. For the proof of Solomon's second conjecture, the author considers the full subcategory \(\mathcal C_V=\text{add }\mathcal X\) of \(\Lambda\text{-}\mathbf{lat}\), where \(\mathcal X\) consists of the finitely generated \(\Lambda\)-submodules of \(V\). By the Jordan-Zassenhaus theorem, there are only finitely many isomorphism-classes of indecomposable objects in \(\mathcal C_V\). He proves that there exists a chain \(\mathcal C_V=\mathcal C_m\supset\mathcal C_{m-1}\supset\cdots\supset\mathcal C_0=\Gamma\text{-}\mathbf{lat}\) of full subcategories with a hereditary overorder \(\Gamma\) of \(\Lambda\), such that \(\mathcal C_{i-1}\) has exactly one indecomposable \(\Lambda\)-lattice less than \(\mathcal C_i\), up to isomorphism. Note that for an additive generator \(M\) of \(\mathcal C_V\), the endomorphism ring \(\text{End}_\Lambda(M)\) is quasi-hereditary, and the chain of \(\mathcal C_i\) is closely related to a heredity chain. The author makes use of this chain to prove a more general formula for zeta function determinants by induction. A specialization to Solomon's context yields an explicit formula for \(\det\mathbb{Z}_\Lambda(V;s)\).
scientific article; zbMATH DE number 1891314
scientific article