论文标题
分配格和澳大利亚的常规代数
Distributive lattices and Auslander regular algebras
论文作者
论文摘要
令$ l $表示至少两个点的有限晶格，让$ a $表示$ l $的发病率代数。我们证明，$ l $是分配的，并且仅当$ a $是Auslander常规环，这给出了分布式晶格的同源表征。在这种情况下，$ a $具有明确的最低注射核心，其$ i $ then的期限由$ l $ of $ i $ y $元素的元素给出。我们给出了$ a $的低音编号组合公式。我们应用结果表明，分布晶格$ l $的订单维度与$ l $的发病率代数的全球维度一致。另外，我们还使用更高的Auslander-Reiten翻译简单模块对分布晶格进行了分配晶格的分类。
Let $L$ denote a finite lattice with at least two points and let $A$ denote the incidence algebra of $L$. We prove that $L$ is distributive if and only if $A$ is an Auslander regular ring, which gives a homological characterisation of distributive lattices. In this case, $A$ has an explicit minimal injective coresolution, whose $i$-th term is given by the elements of $L$ covered by precisely $i$ elements. We give a combinatorial formula of the Bass numbers of $A$. We apply our results to show that the order dimension of a distributive lattice $L$ coincides with the global dimension of the incidence algebra of $L$. Also we categorify the rowmotion bijection for distributive lattices using higher Auslander-Reiten translates of the simple modules.
