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Hammocks to visualize the support of finitely presented functors

Markus Schmidmeier

TL;DR

The paper introduces hammock visualizations for the support of finitely presented functors on module categories by placing the functor's projective presentation into the Auslander–Reiten quiver. Central to the method is the Cokernel Complex Lemma, which connects exact-cokernel data to AR-structure and yields a hammock function e_*(M)=dim E(M) whose sources, sinks, and tangents reflect the presentation components. The framework is developed for both covariant and contravariant finitely presented functors, with dual results, and is then applied to two broad domains: modules (via kernels, images, and cokernels of multiplication maps and quiver representations) and invariant subspaces (via nilpotent operator pairs and their summands). The approach provides concrete, computable relationships between module-theoretic invariants and AR-theoretic geometry, and it demonstrates how adjoint functors can translate hammock data between ambient and submodule categories. Overall, the hammock viewpoint offers a geometric, combinatorial tool to extract multiplicities and structural information from finitely presented functors, with explicit illustrated examples including E₆ quivers, kernel and image data, and invariant-subspace hierarchies.

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.

Hammocks to visualize the support of finitely presented functors

TL;DR

The paper introduces hammock visualizations for the support of finitely presented functors on module categories by placing the functor's projective presentation into the Auslander–Reiten quiver. Central to the method is the Cokernel Complex Lemma, which connects exact-cokernel data to AR-structure and yields a hammock function e_*(M)=dim E(M) whose sources, sinks, and tangents reflect the presentation components. The framework is developed for both covariant and contravariant finitely presented functors, with dual results, and is then applied to two broad domains: modules (via kernels, images, and cokernels of multiplication maps and quiver representations) and invariant subspaces (via nilpotent operator pairs and their summands). The approach provides concrete, computable relationships between module-theoretic invariants and AR-theoretic geometry, and it demonstrates how adjoint functors can translate hammock data between ambient and submodule categories. Overall, the hammock viewpoint offers a geometric, combinatorial tool to extract multiplicities and structural information from finitely presented functors, with explicit illustrated examples including E₆ quivers, kernel and image data, and invariant-subspace hierarchies.

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.

Paper Structure

This paper contains 14 sections, 11 theorems, 51 equations, 7 figures.

Table of Contents

  1. Introduction
  2. The Cokernel Complex Lemma
  3. Hammock Functions
  4. Applications I: Modules
  5. The multiplication map for modules
  6. Quiver representations
  7. The homology at two composable arrows or paths
  8. Applications II: Invariant subspaces
  9. Our tool: Adjoint functors
  10. The multiplicity of $P^m$ as a summand of $V$
  11. The multiplicity of $P^m$ as a summand of $U$
  12. The adjoint isomorphism between covers and cokernels
  13. The multiplicity of $P^m$ as a summand of $V/U$
  14. Example: A hammock with no isolated source or sink

Key Result

Theorem 1.2

Let $\mathcal{E}: X\stackrel u\to Y\to Z\to 0$ be a short right exact sequence and $e_*$ the hammock function for $\mathcal{E}$ on $\Gamma$.

Figures (7)

  • Figure 1: The hammock for the functor $E:V\mapsto \mathop{\mathrm{Cok}}\nolimits V_\alpha$
  • Figure 2: The hammock for the functor $E:M\mapsto \mathop{\mathrm{Ker}}\nolimits M_\beta$
  • Figure 3: The hammock for the functor $E: M\mapsto{\rm Im}\, M_\beta$
  • Figure 4: The Auslander-Reiten quiver for $\mathcal{S}(4)$
  • Figure 5: Hammocks for summands of $U$, $V$ and $V/U$
  • ...and 2 more figures

Theorems & Definitions (34)

  • Example 1.1
  • Definition
  • Theorem 1.2
  • Proposition 1.3
  • Proposition 1.4: Cokernel Complex Lemma
  • Remark
  • Lemma 2.1
  • Claim 2.2
  • proof
  • Claim 2.3
  • ...and 24 more