Combinatorial commutative algebra explained

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 rings.
• 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:

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

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

• Book: Stanley, Richard . Richard P. Stanley . 2nd . [{{GBurl|tZVCAAAAQBAJ|pg=PR5}} Combinatorics and commutative algebra ]. Birkhäuser . Progress in Mathematics . 41 . 1996 . 0-8176-3836-9 . 0838.13008 .

Very influential, and well written, textbook-monograph:

• Book: Winfried . Bruns . Jürgen . Herzog . Cohen–Macaulay rings . Cambridge University Press . Cambridge Studies in Advanced Mathematics . 39 . 1993 . 0-521-41068-1 . 802912314 . 0788.13005.

Additional reading:

• Book: Villarreal, Rafael H. . [{{GBurl|rLfKlDJjLewC|pg=PP1}} Monomial algebras ]. Marcel Dekker . Monographs and Textbooks in Pure and Applied Mathematics . 238 . 2001 . 0-8247-0524-6 . 1002.13010 .
• Book: Hibi, Takayuki . Algebraic combinatorics on convex polytopes . Carslaw Publications . Glebe, Australia . 1992 . 1875399046 . 29023080.
• Book: Sturmfels, Bernd . Bernd Sturmfels . [{{GBurl|K-bxBwAAQBAJ|pg=PR7}} Gröbner bases and convex polytopes ]. American Mathematical Society . University Lecture Series . 8 . 1996 . 0-8218-0487-1 . 907364245 . 0856.13020 .
• Book: Winfried . Bruns . Joseph . Gubeladze . Springer Monographs in Mathematics . [{{GBurl|pbgg1pFxW8YC|pg=PP9}} Polytopes, Rings, and K-Theory ]. Springer . 10.1007/b105283 . 2009 . 978-0-387-76355-2 . 1168.13001.

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

• Book: Ezra . Miller . Bernd . Sturmfels . [{{GBurl|OYBCAAAAQBAJ|pg=PR11}} Combinatorial commutative algebra ]. Springer . . 227 . 2005 . 0-387-22356-8 . 1066.13001.
• Book: Jürgen . Herzog . Takayuki . Hibi . [{{GBurl|3PxzYIBTibsC|pg=PR11}} Monomial Ideals ]. Springer . Graduate Texts in Mathematics . 260 . 2011 . 978-0-85729-106-6 . 1206.13001.