Algebraic Discrete Morse Theory and Applications to Commutative Algebra

205 Pages·2005·1.09 MB·English

by  Michael Jöllenbeck

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-Universita(cid:127)t Marburg
von
Michael Jo(cid:127)llenbeck
aus Marburg/Lahn
Marburg/Lahn, Januar 2005
Vom Fachbereich Mathematik und Informatik
der Philipps-Universita(cid:127)t Marburg als Dissertation
angenommen am: 25.04.2005
Erstgutachter: Professor Dr. Volkmar Welker
Zweitgutachter: Professor Dr. Tim Ro(cid:127)mer
Tag der mu(cid:127)ndlichen Pru(cid:127)fung: 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(cid:19)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(cid:19)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(cid:19)e-Betti Series and the
Golod Property 101
x1. The Multigraded Hilbert and Poincar(cid:19)e-Betti Series 103
x2. The Homology of the Koszul Complex KA 106
x3. Hilbert and Poincar(cid:19)e-Betti Series of the Algebra A = k[(cid:1)] 108
x4. Proof of Conjecture 1.2 for Several Classes of Algebras A 113
4.1. Proof for Algebras A for which H (KA) is an M-ring 113
(cid:15)
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 Re(cid:13)ection 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 A 148
n
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. Einfu(cid:127)hrung 175
2.2. Bisherige Lo(cid:127)sungsansa(cid:127)tze 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. Einfu(cid:127)hrung 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. Erkla(cid:127)rung 197
x3. Curriculum Vitae 199

ThefollowingtextisaPhDthesisinAlgebraicCombinatorics. Itconsistsof
two partsandanappendix. Inthe (cid:12)rstpart, "Algebraic Discrete MorseTheory
and Applications to Commutative Algebra", we generalize Forman’s Discrete
Morsetheoryandgiveseveralapplicationstoproblemsincommutative 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