论文标题
双层代数和通用砖
Biserial algebras and generic bricks
论文作者
论文摘要
我们考虑通用砖，并在代数封闭的磁场上对任意双层代数的研究中使用它们。对于双重代数$λ$，我们表明$λ$在且仅当它承认通用砖时，也就是说，存在通用的$λ$ -Module $ g $，而$end_λ（g）= k（x）$。此外，我们给出了双重代数的砖浸入砖的明确数值条件：如果$λ$是等级$ n $的，则$λ$是砖额iNfinite，并且仅当存在一个无限的$ d $的无限砖家族，对于某些$ 2 \ leq d \ leq d \ leq d \ leq d \ leq leq leq leq 2n $。这也会导致$τ$的代数几何实现这个家庭的有限：$λ$是$τ$ - 当且仅当$λ$是砖 - $ brick-discrete时，这意味着每个代表性$ mod $ mod（λ，\ useverline {d}）$，只有有限的bricks of bricks of bricks of bricks of Bricks of Bricks of Bricks of Bricks of Bricks of Bricks of Bricks of Bricks of Bricks of Bricks of Bricks of bricks上只有很多。 我们的结果依赖于我们在最小砖饮用的双层代数方面的全部分类。这是林德尔给出的最小表示（特殊）双方代数的最小代表（特殊）双层代数的现代类似物。特别是，我们表明，每个最小砖内的双式代数都很温和，并且完全承认一个通用的砖。此外，我们描述了这种代数的范围，该代数与驯服的世袭代数非常相似。换句话说，$砖（λ）$是一个独特的通用砖的脱节结合，具有可数的无限长度的无限砖，并且是由地面场参数参数的相同有限长度的砖一家。
We consider generic bricks and use them in the study of arbitrary biserial algebras over algebraically closed fields. For a biserial algebra $Λ$, we show that $Λ$ is brick-infinite if and only if it admits a generic brick, that is, there exists a generic $Λ$-module $G$ with $End_Λ(G)=k(x)$. Furthermore, we give an explicit numerical condition for brick-infiniteness of biserial algebras: If $Λ$ is of rank $n$, then $Λ$ is brick-infinite if and only if there exists an infinite family of bricks of length $d$, for some $2\leq d\leq 2n$. This also results in an algebro-geometric realization of $τ$-tilting finiteness of this family: $Λ$ is $τ$-tilting finite if and only if $Λ$ is brick-discrete, meaning that in every representation variety $mod(Λ, \underline{d})$, there are only finitely many orbits of bricks. Our results rely on our full classification of minimal brick-infinite biserial algebras in terms of quivers and relations. This is the modern analogue of the recent classification of minimal representation-infinite (special) biserial algebras, given by Ringel. In particular, we show that every minimal brick-infinite biserial algebra is gentle and admits exactly one generic brick. Furthermore, we describe the spectrum of such algebras, which is very similar to that of a tame hereditary algebra. In other words, $Brick(Λ)$ is the disjoint union of a unique generic brick with a countable infinite set of bricks of finite length, and a family of bricks of the same finite length parametrized by the ground field.
