论文标题
第二类和Zeta函数的nilpotent环的合并力的理想增长
Ideal growth in amalgamated powers of nilpotent rings of class two and zeta functions of quiver representations
论文作者
论文摘要
让$ l $成为紧凑型离散估值环$ a $特征零或足够大的积极特征的二级代数。令$ Q $是$ a $的残留基数。 $ l $的理想Zeta功能是列举$ l $的有限索引理想的Dirichlet系列。我们证明，$ q $，$ q^m $，$ q^{ - s} $和$ q^{ - ms} $具有$ m $ copies $ l $的理想Zeta函数，$ m $ cop of $ l $ cop the Perived Supring，对于每个$ M M \ geq 1 $，最多可示出explacit centlicit consexplacit clacticit因素。更普遍地，我们证明了这一点是为Lee和Voll定义的第二类颤抖表示的Zeta函数，尤其是针对分级$ a $ -MODULE的分级子模型计算的Dirichlet系列。如果代数$ l $或QUIVIL表示，则在$ \ Mathbb {Z} $上定义，则我们获得统一的合理性结果。
Let $L$ be a nilpotent algebra of class two over a compact discrete valuation ring $A$ of characteristic zero or of sufficiently large positive characteristic. Let $q$ be the residue cardinality of $A$. The ideal zeta function of $L$ is a Dirichlet series enumerating finite-index ideals of $L$. We prove that there is a rational function in $q$, $q^m$, $q^{-s}$, and $q^{-ms}$ giving the ideal zeta function of the amalgamation of $m$ copies of $L$ over the derived subring, for every $m \geq 1$, up to an explicit factor. More generally, we prove this for the zeta functions of nilpotent quiver representations of class two defined by Lee and Voll, and in particular for Dirichlet series counting graded submodules of a graded $A$-module. If the algebra $L$, or the quiver representation, is defined over $\mathbb{Z}$, then we obtain a uniform rationality result.
