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
A signature theorem in combinatorial commutative algebra is the characterization of
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.
- 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
References
A foundational paper on Stanley–Reisner complexes by one of the pioneers of the theory:
- Zbl 0351.13009.
The first book is a classic (first edition published in 1983):
- 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. Zbl 0788.13005.
Additional reading:
- Villarreal, Rafael H. (2001). Monomial algebras. Monographs and Textbooks in Pure and Applied Mathematics. Vol. 238. Marcel Dekker. Zbl 1002.13010.
- Hibi, Takayuki (1992). Algebraic combinatorics on convex polytopes. Glebe, Australia: Carslaw Publications. OCLC 29023080.
- Zbl 0856.13020.
- Bruns, Winfried; Gubeladze, Joseph (2009). Polytopes, Rings, and K-Theory. Springer Monographs in Mathematics. Springer. 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. Zbl 1066.13001.
- Herzog, Jürgen; Hibi, Takayuki (2011). Monomial Ideals. Graduate Texts in Mathematics. Vol. 260. Springer. Zbl 1206.13001.