In this paper, we study length categories using iterated extensions. We fix a field k, and for any family S of orthogonal k-rational points in an Abelian k-category \(\mathcal {A}\), we consider the category Ext(S) of iterated extensions of S in \(\mathcal {A}\), equipped with the natural forgetful functor \(\mathbf {Ext}(\mathsf {S}) \to \mathbf {\mathcal {A}}(\mathsf {S})\) into the length category \(\mathbf {\mathcal {A}}(\mathsf {S})\). There is a necessary and sufficient condition for a length category to be uniserial, due to Gabriel, expressed in terms of the Gabriel quiver (or Ext-quiver) of the length category. Using Gabriel’s criterion, we give a complete classification of the indecomposable objects in \(\mathbf {\mathcal {A}}(\mathsf {S})\) when it is a uniserial length category. In particular, we prove that there is an obstruction for a path in the Gabriel quiver to give rise to an indecomposable object. The obstruction vanishes in the hereditary case, and can in general be expressed using matric Massey products. We discuss the close connection between this obstruction, and the noncommutative deformations of the family S in \(\mathcal {A}\). As an application, we classify all graded holonomic D-modules on a monomial curve over the complex numbers, obtaining the most explicit results over the affine line, when D is the first Weyl algebra. We also give a non-hereditary example, where we compute the obstructions and show that they do not vanish.

在本文中，我们使用迭代扩展研究长度类别。我们修复领域ķ，和任何家庭Ş正交ķ在阿贝尔-rational点ķ -category \（\ mathcal {A} \），我们考虑的范畴Ë X牛逼（小号）的迭代扩展的小号在\ （\ mathcal {A} \），安装有自然健忘算符\（\ mathbf {分机}（\ mathsf {S}）\到\ mathbf {\ mathcal {A}}（\ mathsf {S}）\）成长度类别\（\ mathbf {\ mathcal {A}}（\ mathsf {S}）\）。由于加百利，存在一个充分必要的条件，长度类别是无序的，用长度类别的加百利颤动（或Ext-quiver）表示。使用加百列准则，当它是一个无序长度类别时，我们给出\（\ mathbf {\ mathcal {A}}（\ mathsf {S}）\）中不可分解对象的完整分类。特别是，我们证明了加百利箭袋中的一条路径阻塞了无法分解的物体。遗传性情况下梗阻消失，通常可以使用基质Massey产品表示。我们讨论这个障碍物之间和家庭的非对易变形的紧密联系，小号在\（\ mathcal {A} \）。作为应用，我们将复数的单项曲线上的所有渐变完整D模块进行分类，当D是第一个Weyl代数时，在仿射线上获得最明确的结果。我们还给出了一个非世袭的示例，在此示例中，我们计算出障碍物并表明障碍物不会消失。