Abstract
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.
Highlights
Let S = {Sα : α ∈ I } be a family of non-zero, pairwise non-isomorphic objects in an Abelian k-category A, where k is a field
We shall use the category Ext(S) of iterated extensions of S to study the length category A(S) when S is a family of orthogonal k-rational points
Gabriel’s criterion for A(S) to be uniserial is a condition on the Gabriel quiver, and we use it to classify and explicitly construct all indecomposable objects in A(S) when the length category is uniserial
Summary
In this case, A(S) ⊆ A is a length category with S as its simple objects. We shall use the category Ext(S) of iterated extensions of S to study the length category A(S) when S is a family of orthogonal k-rational points. The category Ext(S) of iterated extensions has some interesting invariants, in addition to the length n, the simple factors {K1, . Gabriel’s criterion for A(S) to be uniserial is a condition on the Gabriel quiver, and we use it to classify and explicitly construct all indecomposable objects in A(S) when the length category is uniserial. There is a bijective correspondence between indecomposable objects of length n in A(S) and paths of length n − 1 in the Gabriel quiver of A(S) This result is well-known; see for instance Chen and Krause [3]. We show that in this case, the obstructions do not vanish, and there are no indecomposable modules of length n ≥ 3
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