标题： 计算直接和分解 显示英文标题

标题： Computing Direct Sum Decompositions

摘要： 寻找具有某些性质的不可分解模的同构类，或确定模的不可分解直和项，是交换代数、群论、表示论和其他领域中普遍存在的问题。 本工作的目的是描述并证明一种实用算法的正确性，该算法用于计算在有限生成k-代数上有限生成模的不可分解直和项，其中k是正特征的域。 我们的算法适用于多分次环，这使得能够计算toric簇子簇上的凝聚层的不可分解直和项（特别是对于嵌入在射影空间中的簇）。 我们还提供了多个例子，包括一些关于Frobenius前推和Artin环上挠元的行为的先前未知现象的例子。 显示英文摘要

摘要： The problems of finding isomorphism classes of indecomposable modules with certain properties, or determining the indecomposable summands of a module, are ubiquitous in commutative algebra, group theory, representation theory, and other fields. The purpose of this work is to describe and prove correctness of a practical algorithm for computing indecomposable summands of finitely generated modules over a finitely generated k-algebra, for k a field of positive characteristic. Our algorithm works over multigraded rings, which enables the computation of indecomposable summands of coherent sheaves on subvarieties of toric varieties (in particular, for varieties embedded in projective space). We also present multiple examples, including some which present previously unknown phenomena regarding the behavior of summands of Frobenius pushforwards and syzygies over Artinian rings.