Power of Monomial Ideals

1. Preliminaries 3 2. Gröbner Bases 3 2.1. Motivating Problems 3 2.2. Ordering of Monomials 3 2.3. The Division Algorithm in S = k[x1, . . . , xn] 5 2.4. Dickson’s Lemma 7 2.5. Gröbner Bases and the Hilbert Basis Theorem 10 2.6. Some Further Applications of Gröbner bases 18 3. Hilbert Functions 21 3.1. Macaulay’s Theorem 27 3.2. Hilbert Functions of Reduced Standard Graded k-algebras 37 3.3. Hilbert Functions of Points in Pn 42 3.4. Maximal Growth of Hilbert Functions and Consequences 45 3.5. The Eisenbud-Green-Harris Conjecture 49 3.6. A combinatorial approach to EGH 51 3.7. Some enumeration 54 3.8. Sum fun with points and regular sequences 57 4. Lex ideals and Betti numbers 58 4.1. Group actions on S = k[x1, . . . , xn] 59 4.2. Generic initial ideals 61 4.3. Some comments on Gröbner bases and modules 63 4.4. Comparing lex-segment ideals and Borel-fixed ideals 63 5. Squarefree monomial ideals 71 5.1. Hilbert series 73 5.2. Shellable simplicial complexes and H-vectors 75 6. A crash course on resolutions 76 6.1. The graded world 77 7. Group Presentations 80 7.1. Lifting monomial ideals: A. Croll, C. Gibbons, and B. Johnson 80 7.2. Representations of monomial orders: R. Brase, A. Denkert, and M. Janssen 83