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Algebraic Discrete Morse Theory and Applications to Commutative Algebra Algebraic Discrete Morse Theory and Applications to Commutative Algebra (Algebraische Diskrete Morse-Theorie und Anwendungen in der Kommutativen Algebra) Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften (Dr. rer. nat.) vorgelegt dem Fachbereich Mathematik und Informatik der Philipps-Universit¨at Marburg von Michael J¨ollenbeck aus Marburg/Lahn Marburg/Lahn, Januar 2005 Vom Fachbereich Mathematik und Informatik der Philipps-Universit¨at Marburg als Dissertation angenommen am: 25.04.2005 Erstgutachter: Professor Dr. Volkmar Welker Zweitgutachter: Professor Dr. Tim R¨omer Tag der mundlic¨ hen Prufung:¨ 09.05.2005 Contents Part 1. Algebraic Discrete Morse Theory and Applications to Commutative Algebra Chapter 1. Introduction 5 Chapter 2. Basics from Commutative Algebra 11 x1. Free Resolutions of R-Modules 12 1.1. Cellular Resolutions 13 1.2. Hilbert and Poincar´e-Betti Series 14 x2. Examples for Chain Complexes in Commutative Algebra 15 2.1. Taylor and Scarf Complex 15 2.2. Poset Resolution for a Monomial Ordered Family 16 2.3. Koszul Complex 17 2.4. Bar Resolution 18 2.5. Acyclic Hochschild Complex 19 x3. Eagon Complex and the Golod Property 21 3.1. The Eagon Resolution 21 3.2. The Massey Operations and the Golod Property 23 Chapter 3. Algebraic Discrete Morse Theory 29 x1. Algebraic Discrete Morse Theory 29 x2. Proof of Theorem 1.2 32 x3. Normalized Bar and Hochschild Resolution via ADMT 41 Chapter 4. Free Resolutions of Monomial Ideals 43 x1. Algebraic Discrete Morse Theory on the Taylor Resolution 43 1.1. Standard Matching on the Taylor Resolution 43 1.2. Resolutions of Monomial Ideals Generated in Degree Two 45 1.3. Resolution of Stanley Reisner Ideals of a Partially Ordered Set 46 1.4. The gcd-Condition 49 x2. Algebraic Discrete Morse Theory for the Poset Resolution 50 2.1. ADMT for the Poset Resolution 51 2.2. What is a \good" underlying partially ordered set P ? 59 i ii Contents x3. Minimal Resolution and Regularity of Principal (p-)Borel Fixed Ideals 62 3.1. Cellular Minimal Resolution for Principal Borel Fixed Ideals 62 3.2. Cellular Minimal Resolution for a Class of p-Borel Fixed Ideals 65 Chapter 5. Free Resolution of the Residue Class Field k 81 x1. Resolution of the Residue Field in the Commutative Case 82 1.1. An Anick Resolution for the Commutative Polynomial Ring 84 1.2. Two Special Cases 88 x2. Resolution of the Residue Field in the Non-Commutative Case 90 2.1. The Anick Resolution 92 2.2. The Poincar´e-Betti Series of k 93 2.3. Examples 94 x3. Application to the Acyclic Hochschild Complex 96 Chapter 6. The Multigraded Hilbert and Poincar´e-Betti Series and the Golod Property 101 x1. The Multigraded Hilbert and Poincar´e-Betti Series 103 x2. The Homology of the Koszul Complex K A 106 x3. Hilbert and Poincar´e-Betti Series of the Algebra A = k[∆] 108 x4. Proof of Conjecture 1.2 for Several Classes of Algebras A 113 A 4.1. Proof for Algebras A for which H•(K ) is an M-ring 113 4.2. Proof for Koszul Algebras 115 4.3. Idea for a Proof in the General Case 121 x5. Applications to the Golod Property of Monomial Rings 125 Part 2. Two Problems in Algebraic Combinatorics Chapter 1. Introduction 131 Chapter 2. Homology of Nilpotent Lie Algebras of Finite Type 133 x1. General Theory 134 1.1. Root Space Decomposition 135 1.2. Root Systems and Reflection Groups 136 1.3. Homology of Lie Algebras 138 1.4. Conjectures and Open Questions 140 x2. New Invariance Theorem for Nilpotent Lie Algebras of Finite Type141 x3. Applications to Lie Algebras of Root Systems 146 3.1. Homology of Lie Algebras Associated to An 148 3.2. Homology of Lie Algebras Associated to other Root Systems 151 Chapter 3. The Neggers-Stanley Conjecture 153 x1. The Poset Conjecture 153 x2. The Naturally Labeled Case for Graded Posets 156 2.1. Proof of Theorem 2.6 157 x3. The Naturally Labeled Case for General Posets 160 3.1. W -Polynomial in Graph Theory 160 3.2. Unimodality for Naturally Labeled Posets 164 Contents iii Bibliography 171 Part 3. Appendix Appendix A. German Abstract (Deutsche Zusammenfassung) 175 x1. Struktur der Arbeit 175 x2. Algebraische Diskrete Morse-Theorie und Anwendungen 175 2.1. Einfuhrung¨ 175 2.2. Bisherige L¨osungsans¨atze 176 2.3. Die Algebraische Diskrete Morse-Theorie 177 2.4. Anwendungen in der Kommutativen Algebra 180 2.5. Struktur des ersten Teils 186 x3. Zwei Probleme aus der Algebraischen Kombinatorik 187 3.1. Einfuhrung¨ 187 3.2. Homologie von nilpotenten Lie-Algebren endlichen Typs 187 3.3. Neggers-Stanley-Vermutung 191 Appendix B. 195 x1. Danksagung / Acknowledgments 195 x2. Erkl¨arung 197 x3. Curriculum Vitae 199 The following text is a PhD thesis in Algebraic Combinatorics. It consists of two parts and an appendix. In the first part, "Algebraic Discrete Morse Theory and Applications to Commutative Algebra", we generalize Forman's Discrete Morse theory and give several applications to problems in commutative algebra. In the second part we present results on two related problems in Algebraic Combinatorics, namely "Homology of Nilpotent Lie Algebras of Finite Type" and the "Neggers-Stanley Conjecture". The appendix consists of the German abstract, acknowledgments, curricu- lum vitae, and the declaration of authorship. Part 1 Algebraic Discrete Morse Theory and Applications to Commutative Algebra Chapter 1 Introduction In linear algebra there is the fundamental concept of linear independence. The situation turns out to be simple due to the fact that all vector spaces V over a field k are free k-modules. In particular, the two conditions \maximal inde- pendent" and \minimal generating" for a set of vectors are equivalent. The concept of dependence of polynomials p1; : : : ; pr 2 S := k[x1; : : : ; xn] is more complex. For example the conditions \maximal independent" and \min- imal generating" are not equivalent anymore. In order to measure dependence of polynomials, one considers free reso- lutions of the ideal hp1; : : : ; pri E S, especially minimal free resolutions. Even though minimal free resolutions always exist, even in the monomial case, it is still an open problem to explicitly construct a minimal free resolution in general. For monomial ideals there exist many explicit free resolutions, but they are mostly not minimal. One central idea of our work is to extract from a given free resolution of a monomial ideal a minimal resolution, by dividing out acyclic subcomplexes. In commutative algebra one considers, besides minimal resolutions of mono- mial ideals, many other invariants, for example regularity, Poincar´e-Betti series, Ext, Tor, which are calculated as well by the homology of chain complexes of free R-modules. Often we are in the situation that the homology of a given chain complex calculates an invariant, but the complex is very large in the sense that there exist homotopy-equivalent chain complexes such that the mod- ules have a smaller rank. For example, the homology of the Taylor resolution of a monomial ideal tensored with the field k calculates the Betti numbers, but if one had the minimal resolution at hand, this calculation would be more efficient. Theoretically, one can minimize each chain complex by dividing out acyclic subcomplexes, but in praxis there does not exist any efficient algorithm which minimizes a given chain complex. The concept of cellular resolutions is a good tool for minimizing free resolutions with topological arguments, namely Forman's Discrete Morse theory (see [21], [22]). The main idea is that a given cellular resolution can be made smaller by finding a so-called acyclic matching on the supporting CW-complex. This method was studied by Batzies in his PhD thesis (see [4]). But this concept is limited. First, the theory only works 5 6 1. Introduction if the given chain complex is supported by a regular CW-complex, which is not always the case in applications. Moreover, it is still an open problem if any monomial ideal admits a minimal cellular resolution. But even if the given chain complex is supported by a regular CW-complex, it can happen that, af- ter applying Discrete Morse theory once, the resulting Morse complex is not minimal and the CW-complex is not regular anymore. This obstructs a further application of this theory. In this thesis we define an algebraic version of Forman's Discrete Morse theory. We call this generalization "Algebraic Discrete Morse theory". It gen- eralizes the idea of \matching down" given chain complexes to smaller ones from the realm of cellular chain complexes to all algebraic chain complexes. Another advantage of this theory over the Discrete Morse theory for cellular resolutions is the fact that it can be applied iteratively. We apply our method to several problems in commutative algebra. Similar to Discrete Morse theory for cellular resolutions, one major field of application for Algebraic Discrete Morse theory is finding minimal resolutions of monomial ideals in the commutative polynomial ring. For example, in this thesis we are able to give new minimal resolutions for a subclass of p-Borel fixed ideals. Another field of applications are minimal resolutions of the residue class field k =∼ A=m over standard graded k-algebras A, where m is the unique graded maximal ideal in A. Once we have found a minimal resolution of k, it allows us to calculate the multigraded Hilbert and Poincar´e-Betti series. For example we were able to specify and prove a conjecture about the Poincar´e-Betti series made by Charalambous and Reeves (see [13]). We get new connections between Hilbert and Poincar´e-Betti series and find interesting corollaries for the Golod property.
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