Contents Chapter 1. Introduction 1 Chapter 2. Algebraic Discrete Morse Theory 7 Chapter 3. Resolution of the Residue Field in the Commutative Case 11 1. Gr¨obner Bases and Discrete Morse Theory 11 2. An Anick Resolution for the Commutative Polynomial Ring 14 3. Two Special Cases 17 Chapter 4. Resolution of the Residue Field in the Non-Commutative Case 21 1. Non-commutative Gr¨ obner Bases and Discrete Morse Theory 21 2. The Anick Resolution 23 3. The Poincar´ e-Betti Series of k 24 4. Examples 25 Chapter 5. Application to the Acyclic Hochschild Complex 29 1. Hochschild Homology and Discrete Morse Theory 29 2. Explicit Calculations of Hochschild Homology 31 Chapter 6. Minimal (Cellular) Resolutions for (p-)Borel Fixed Ideals 35 1. Cellular Resolutions 35 2. Cellular Minimal Resolution for Principal Borel Fixed Ideals 37 3. Cellular Minimal Resolution for a Class of p-Borel Fixed Ideals 40 Appendix A. The Bar and the Hochschild Complex 57 Appendix B. Proofs for Algebraic Discrete Morse Theory 61 Bibliography 71 Index 73 v
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