标题： 计算直接和分解 显示英文标题

标题： Computing Direct Sum Decompositions

摘要： 我们描述并证明了两种实际算法，用于找到有限生成的k-代数R上的有限生成模的不可约直和项。第一个算法适用于（多）分次情况，这使得能够计算塔希克流形子簇上凝聚层的不可约直和项（特别是对于嵌入在射影空间中的流形）；第二个算法适用于R是局部代数且k是有限域的情况，这为奇点理论中的分解计算打开了大门。 我们还提供了多个示例，包括一些关于Frobenius提升映射的直和项行为（包括非分次情况）以及Artin环上的环量式的新现象。 显示英文摘要

摘要： We describe and prove correctness of two practical algorithms for finding indecomposable summands of finitely generated modules over a finitely generated k-algebra R. The first algorithm applies in the (multi)graded case, which enables the computation of indecomposable summands of coherent sheaves on subvarieties of toric varieties (in particular, for varieties embedded in projective space); the second algorithm applies when R is local and k is a finite field, opening the door to computing decompositions in singularity theory. We also present multiple examples, including some which present previously unknown phenomena regarding the behavior of summands of Frobenius pushforwards (including in the non-graded case) and syzygies over Artinian rings.