View source for Combinatorial commutative algebra

'''Combinatorial commutative algebra''' is a relatively new, rapidly developing [[mathematics|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, [[convex polytope|polyhedral geometry]] plays a significant role.

One of the milestones in the development of the subject was [[Richard P. Stanley|Richard Stanley]]'s 1975 [[mathematical proof|proof]] of the '''[[Upper Bound Conjecture]]''' for [[simplicial sphere]]s, 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-vector|''h''-vectors]] of [[simplicial polytope]]s [[conjecture]]d in 1970 by [[Peter McMullen]]. Known as the [[g-theorem|''g''-theorem]], it was proved in 1979 by Stanley ([[Necessity and sufficiency|necessity]] of the conditions, algebraic argument) and by [[Louis Billera]] and Carl W. Lee ([[Necessity and sufficiency|sufficiency]], combinatorial and geometric construction). A major [[open problem|open question]] was the extension of this characterization from simplicial polytopes to simplicial spheres, the [[g-conjecture|''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]]s.
* '''[[Monomial ring]]''', closely related to an '''affine semigroup ring''' and to the [[coordinate ring]] of an [[affine variety|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:
*{{cite conference |author-link=Melvin Hochster |first=Melvin |last=Hochster |title=Cohen–Macaulay rings, combinatorics, and simplicial complexes |book-title=Ring Theory II: Proceedings of the Second Oklahoma Conference |publisher=Dekker |series=Lecture Notes in Pure and Applied Mathematics |volume=26 |date=1977 |isbn=0-8247-6575-3 |pages=171–223 |oclc=610144046 |url=https://archive.org/details/ringtheoryiiproc0026ring |zbl=0351.13009}}

The first book is a classic (first edition published in 1983):
* {{cite book |author-link=Richard P. Stanley |edition=2nd |first=Richard |last=Stanley |title=Combinatorics and commutative algebra |publisher=Birkhäuser |series=Progress in Mathematics |volume=41 |date=1996 |isbn=0-8176-3836-9 |zbl=0838.13008 |url={{GBurl|tZVCAAAAQBAJ|pg=PR5}}}}

Very influential, and well written, textbook-monograph:
* {{cite book |first1=Winfried |last1=Bruns |first2=Jürgen |last2=Herzog |title=Cohen–Macaulay rings |publisher=Cambridge University Press |location=Cambridge Studies in Advanced Mathematics |volume=39 |date=1993 |isbn=0-521-41068-1 |oclc=802912314 |zbl=0788.13005}}

Additional reading:
* {{cite book |first=Rafael H. |last=Villarreal |title=Monomial algebras |publisher=Marcel Dekker |series=Monographs and Textbooks in Pure and Applied Mathematics |volume=238 |date=2001 |isbn=0-8247-0524-6 |zbl=1002.13010 |url={{GBurl|rLfKlDJjLewC|pg=PP1}}}}
* {{cite book |first=Takayuki |last=Hibi |title=Algebraic combinatorics on convex polytopes |publisher=Carslaw Publications |location=Glebe, Australia |date=1992 |isbn=1875399046 |oclc=29023080}}
* {{cite book |author-link=Bernd Sturmfels |first=Bernd |last=Sturmfels |title=Gröbner bases and convex polytopes |publisher=American Mathematical Society |series=University Lecture Series |volume=8 |date=1996 |isbn=0-8218-0487-1 |oclc=907364245 |zbl=0856.13020 |url={{GBurl|K-bxBwAAQBAJ|pg=PR7}}}}
* {{cite book |first1=Winfried |last1=Bruns |first2=Joseph |last2=Gubeladze |series=Springer Monographs in Mathematics |title=Polytopes, Rings, and K-Theory |publisher=Springer |doi=10.1007/b105283 |date=2009 |isbn=978-0-387-76355-2 |zbl=1168.13001|url={{GBurl|pbgg1pFxW8YC|pg=PP9}}}}

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

* {{cite book |first1=Ezra |last1=Miller |first2=Bernd |last2=Sturmfels |title=Combinatorial commutative algebra |publisher=Springer |series=[[Graduate Texts in Mathematics]] |volume=227 |date=2005 |isbn=0-387-22356-8 |zbl=1066.13001|url={{GBurl|OYBCAAAAQBAJ|pg=PR11}}}}
* {{cite book |first1=Jürgen |last1=Herzog |first2=Takayuki |last2=Hibi |title=Monomial Ideals |publisher=Springer |series=Graduate Texts in Mathematics |volume=260 |date=2011 |isbn=978-0-85729-106-6 |zbl=1206.13001|url={{GBurl|3PxzYIBTibsC|pg=PR11}}}}
* {{cite book |first1=Jürgen |last1=Herzog |first2=Takayuki |last2=Hibi|first3=Hidefumi |last3=Oshugi |title=Binomial Ideals |publisher=Springer |series=Graduate Texts in Mathematics |volume=279 |date=2018 |isbn=978-3-319-95349-6 |zbl=1403.13004|url={{GBurl|https://books.google.com.tr/books?id=HsoCuAEACAAJ}}}}

[[Category:Commutative algebra]]
[[Category:Algebraic geometry]]
[[Category:Algebraic combinatorics]]