发表机构
Parakeet Inc.（鹦鹉公司）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在代数闭域上证明了关于模范畴 magnitude 的猜想：其值至少为简单模数，等号恰在特殊双列代数时成立，并通过吊床理论将一般情形化归为表示有向情形。
AI 中文摘要
Børve、Horiatakis 和 Kalck 将表示有限代数的模范畴的 magnitude 定义为不可分解模之间 Hom 维数矩阵的逆矩阵各项之和。他们证明了它是欧拉示性数，即 Auslander–Reiten 箭图的顶点、箭头和网格的交替计数，并猜想它至少等于简单模的个数，且等号恰在特殊双列代数时成立。我们在代数闭域上证明了这一猜想。对于表示有向代数，我们建立了模范畴、不含给定简单合成因子的子范畴及其理想商之间的 magnitude 关系。根据 Ringel 和 Vossieck 的吊床理论，该商等价于一个有限偏序集表示的范畴，我们计算了其 magnitude。一般情形通过标准形式的代数分级化归结为表示有向情形。
英文摘要
Børve, Horiatakis, and Kalck defined the magnitude of the module category of a representation-finite algebra as the sum of the entries of the inverse of the matrix of Hom dimensions between indecomposable modules. They showed it is an Euler characteristic, the alternating count of vertices, arrows, and meshes of the Auslander--Reiten quiver, and conjectured that it is at least the number of simple modules, with equality precisely for special biserial algebras. We prove this conjecture over an algebraically closed field. For a representation-directed algebra, we relate the magnitudes of the module category, the subcategory of modules without a given simple composition factor, and the ideal quotient by it. By the theory of hammocks of Ringel and Vossieck, this quotient is equivalent to a category of representations of a finite poset, whose magnitude we compute. The general case reduces to the representation-directed case through algebras obtained by grading a standard form.
Comments33 pages, comments welcome! Lean formalization of the main result is available at https://haruhisa-enomoto.github.io/magnitude-conjecture/
