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Projective resolutions of simple modules and Hochschild cohomology for incidence algebras

Viktor Bekkert, John William MacQuarrie, Júlio Marques

Abstract

We give a practical, algorithmic method to calculate minimal projective resolutions of simple modules for a finite dimensional incidence $k$-algebra $Λ$, where $k$ is a field. We apply the method to the calculation of Ext groups between simple $Λ$-modules, Hochschild cohomology groups $\HH^i(Λ, Λ)$, and singular cohomology groups of finite $T_0$ topological spaces with coefficients in $k$.

Projective resolutions of simple modules and Hochschild cohomology for incidence algebras

Abstract

We give a practical, algorithmic method to calculate minimal projective resolutions of simple modules for a finite dimensional incidence -algebra , where is a field. We apply the method to the calculation of Ext groups between simple -modules, Hochschild cohomology groups , and singular cohomology groups of finite topological spaces with coefficients in .

Paper Structure

This paper contains 7 sections, 6 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Construction of the resolution
  3. Definition of $i$-cycles
  4. Definition of the resolution
  5. Application to Ext groups
  6. Hochschild cohomology
  7. Acknowledgments

Key Result

Lemma 1

Let $X$ be a poset and $x\in X$. With the definitions as above, the sequence is a chain complex.

Theorems & Definitions (11)

  • Remark 1
  • Remark 2
  • Example 3
  • Lemma 1
  • Lemma 2
  • Theorem 3
  • Proposition 4
  • Theorem 5: C89
  • Theorem 6
  • Remark 4
  • ...and 1 more