Combinatorial commutative algebra

Combinatorial commutative algebra is a relatively new, rapidly developing mathematical discipline. As the name implies, it lies at the intersection of two more established fields, commutative algebra and combinatorics, and frequently uses methods of one to address problems arising in the other. Less obviously, polyhedral geometry plays a significant role.

One of the milestones in the development of the subject was Richard Stanley's 1975 proof of the Upper Bound Conjecture for simplicial spheres, which was based on earlier work of Melvin Hochster and Gerald Reisner. While the problem can be formulated purely in geometric terms, the methods of the proof drew on commutative algebra techniques.

A signature theorem in combinatorial commutative algebra is the characterization of h-vectors of simplicial polytopes conjectured in 1970 by Peter McMullen. Known as the g-theorem, it was proved in 1979 by Stanley (necessity of the conditions, algebraic argument) and by Louis Billera and Carl W. Lee (sufficiency, combinatorial and geometric construction). A major open question was the extension of this characterization from simplicial polytopes to simplicial spheres, the g-conjecture, which was resolved in 2018 by Karim Adiprasito.

Important notions of combinatorial commutative algebra

• Square-free monomial ideal in a polynomial ring and Stanley–Reisner ring of a simplicial complex.
• Cohen–Macaulay ring.
• Monomial ring, closely related to an affine semigroup ring and to the coordinate ring of an affine toric variety.
• Algebra with a straightening law. There are several versions of those, including Hodge algebras of Corrado de Concini, David Eisenbud, and Claudio Procesi.

See also

• Algebraic combinatorics
• Polyhedral combinatorics
• Zero-divisor graph

References

A foundational paper on Stanley–Reisner complexes by one of the pioneers of the theory:

• Hochster, Melvin (1977). "Cohen–Macaulay rings, combinatorics, and simplicial complexes". Ring Theory II: Proceedings of the Second Oklahoma Conference. Lecture Notes in Pure and Applied Mathematics. Vol. 26. Dekker. pp. 171–223. ISBN 0-8247-6575-3. OCLC 610144046. Zbl 0351.13009.

The first book is a classic (first edition published in 1983):

• Stanley, Richard (1996). Combinatorics and commutative algebra. Progress in Mathematics. Vol. 41 (2nd ed.). Birkhäuser. ISBN 0-8176-3836-9. Zbl 0838.13008.

Very influential, and well written, textbook-monograph:

• Bruns, Winfried; Herzog, Jürgen (1993). Cohen–Macaulay rings. Vol. 39. Cambridge Studies in Advanced Mathematics: Cambridge University Press. ISBN 0-521-41068-1. OCLC 802912314. Zbl 0788.13005.

Additional reading:

• Villarreal, Rafael H. (2001). Monomial algebras. Monographs and Textbooks in Pure and Applied Mathematics. Vol. 238. Marcel Dekker. ISBN 0-8247-0524-6. Zbl 1002.13010.
• Hibi, Takayuki (1992). Algebraic combinatorics on convex polytopes. Glebe, Australia: Carslaw Publications. ISBN 1875399046. OCLC 29023080.
• Sturmfels, Bernd (1996). Gröbner bases and convex polytopes. University Lecture Series. Vol. 8. American Mathematical Society. ISBN 0-8218-0487-1. OCLC 907364245. Zbl 0856.13020.
• Bruns, Winfried; Gubeladze, Joseph (2009). Polytopes, Rings, and K-Theory. Springer Monographs in Mathematics. Springer. doi:10.1007/b105283. ISBN 978-0-387-76355-2. Zbl 1168.13001.

A recent addition to the growing literature in the field, contains exposition of current research topics:

• Miller, Ezra; Sturmfels, Bernd (2005). Combinatorial commutative algebra. Graduate Texts in Mathematics. Vol. 227. Springer. ISBN 0-387-22356-8. Zbl 1066.13001.
• Herzog, Jürgen; Hibi, Takayuki (2011). Monomial Ideals. Graduate Texts in Mathematics. Vol. 260. Springer. ISBN 978-0-85729-106-6. Zbl 1206.13001.