Stable local cohomology

Thompson, Peder

doi:10.1080/00927872.2016.1175582

Cited by 2 publications

(1 citation statement)

References 36 publications

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“…Observe that A is a self-injective algebra with a unique simple module. The injective Leavitt complex I • is isomorphic to a complete resolution of the simple A-module; see [5, Definition 3.1.1] and compare [18,Proposition 2.20].…”

Section: The Injective Leavitt Complex Of a Finite Quiver Without Sinksmentioning

confidence: 99%

The injective Leavitt complex

2015

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For a finite quiver Q without sinks, we consider the corresponding finite dimensional algebra A with radical square zero. We construct an explicit compact generator for the homotopy category of acyclic complexes of injective A-modules. We call such a generator the injective Leavitt complex of Q. This terminology is justified by the following result: the differential graded endomorphism algebra of the injective Leavitt complex of Q is quasiisomorphic to the Leavitt path algebra of Q. Here, the Leavitt path algebra is naturally Z-graded and viewed as a differential graded algebra with trivial differential.Let A be a finite dimensional algebra over a field k. The homotopy category K ac (A-Inj) of acyclic complexes of injective A-modules is called the stable derived category of A in [12]. This category is a compactly generated triangulated category such that its subcategory of compact objects is triangle equivalent to the singularity category [5,15] of A.In general, it seems very difficult to give an explicit compact generator for the stable derived category of an algebra. In this paper, we construct an explicit compact generator for the homotopy category K ac (A-Inj), in the case that the algebra A is with radical square zero. The compact generator is called the injective Leavitt complex. This terminology is justified by the following result: the differential graded endomorphism algebra of the injective Leavitt complex is quasi-isomorphic to the Leavitt path algebra, which was introduced in [2, 3] as an algebraisation of graph C * -algebras [13,16] and in particular Cuntz-Krieger algebras [10]. Here, the Leavitt path algebra is naturally Z-graded and viewed as a differential graded algebra with trivial differential.Let Q be a finite quiver without sinks. Set A = kQ/J 2 to be the corresponding finite dimensional algebra with radical square zero. We introduce the injective Leavitt complex I • in Definition 1.4, which is an acyclic complex of injective A-modules; see Proposition 1.9.We denote by L k (Q) the Leavitt path algebra of Q over k. Recall that L k (Q) is a naturally Z-graded algebra, which is viewed as a differential graded algebra with trivial differential.The main result of this paper is as follows, which combines Theorem 2.13 with Theorem 4.2.Theorem I. Let Q be a finite quiver without sinks and A = kQ/J 2 the corresponding algebra with radical square zero.(1) The injective Leavitt complex I • of Q is a compact generator for the homotopy category K ac (A-Inj).(2) The differential graded endomorphism algebra of the injective Leavitt complexThis result is inspired by [9, Theorem 6.1], which describes the homotopy category K ac (A-Inj) in terms of Leavitt path algebras. We mention that [9, Theorem 6.1] extends [17, Theorem 7.2]; compare [7, Theorem 3.8]. The construction of the injective Leavitt complex is inspired by the basis of the Leavitt path algebra given by [1, Theorem 1].For the proof of (1) in Theorem I, we construct an explicit filtration of subcomplexes of the injective Leavitt complex I • . For (2)...

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Section: The Injective Leavitt Complex Of a Finite Quiver Without Sinksmentioning

confidence: 99%

The injective Leavitt complex

2015

Preprint

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The Injective Leavitt Complex

2017

Algebr Represent Theor

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Abstract. For a finite quiver Q without sinks, we consider the corresponding finite dimensional algebra A with radical square zero. We construct an explicit compact generator for the homotopy category of acyclic complexes of injective A-modules. We call such a generator the injective Leavitt complex of Q. This terminology is justified by the following result: the differential graded endomorphism algebra of the injective Leavitt complex of Q is quasiisomorphic to the Leavitt path algebra of Q. Here, the Leavitt path algebra is naturally Z-graded and viewed as a differential graded algebra with trivial differential.

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Stable local cohomology

Cited by 2 publications

References 36 publications

The injective Leavitt complex

The injective Leavitt complex

The Injective Leavitt Complex

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