Monomial Ideals

• Part I Gröbner bases: Monomial Ideals.- A short introduction to Gröbner bases.- Monomial orders and weights.- Generic initial ideals.- The exterior algebra.- Part II: Hilbert functions and resolutions.- Hilbert functions and the theorems of Macaulay and Kruskal-Katona.- Resolutions of monomial ideals and the Eliahou-Kervaire formula.- Alexander duality and resolutions.- Part III Combinatorics: Alexander duality and finite graphs.- Powers of monomial ideals.- Shifting theory.- Discrete Polymatroids.- Some homological algebra.- Geometry

From the reviews: "The authors ... who themselves have played an important, often crucial, role in recent developments of the subject, make an ideal co-author pairing for composing a book with such a comprehensive choice of material, including the latest achievements. It should serve as a useful resource for researchers in both commutative algebra and combinatorics. ... The presentation of the material is distinctive for its clarity and elegant style. ... The text can serve nicely as a basis for a couple of graduate courses in two semesters." (Rahim Zaare-Nahandi, Mathematical Reviews, Issue 2011 k)