Hammocks to visualize the support of finitely presented functors

TL;DR

This paper introduces a geometric visualization called 'hammocks' for understanding the support of finitely presented functors in module categories, linking algebraic properties to Auslander-Reiten structures.

Contribution

It provides a new hammock visualization method to analyze the support of finitely presented functors and relates it to Auslander-Reiten theory using the Cokernel Complex Lemma.

Findings

Hammocks typically span between sources and sinks in the Auslander-Reiten quiver.

Tangent regions in hammocks correspond to meshes with specific value inequalities.

The approach applies to quiver representations and invariant subspaces of nilpotent operators.

Abstract

Many properties of a module can be expressed in terms of the dimension of the vector space obtained by applying a finitely presented functor to that module. For example, the dimension of the kernel, image or cokernel of the multiplication map given by an algebra element; or the number of summands of a certain type when the module is considered a module over a subalgebra. When the indecomposable modules over the algebra are arranged in the Auslander-Reiten quiver, the support of the finitely presented functor typically has the shape of a hammock, spanned between sources and sinks. There may also be tangents which are meshes where the hammock function at the middle term exceeds the sum of the values at the start and end terms. We describe how sources, sinks and tangents of the hammock relate to the modules which define the projective resolution of the finitely presented functor. The key tool is the Cokernel Complex Lemma which links the values of the hammock function to the Auslander-Reiten structure of the category. We are also interested in exact subcategories of module categories which have Auslander-Reiten sequences. Our examples include quiver representations and invariant subspaces of nilpotent linear operators.