Combinatorial commutative algebra is a relatively new, rapidly developing

mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

discipline. As the name implies, it lies at the intersection of two more established fields,

commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Promi ...

and

combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many a ...

, 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 In mathematics, the upper bound theorem states that cyclic polytopes have the largest possible number of faces among all convex polytopes with a given dimension and number of vertices. It is one of the central results of polyhedral combinatorics. ...

for

simplicial sphere In geometry and combinatorics, a simplicial (or combinatorial) ''d''-sphere is a simplicial complex homeomorphic to the ''d''-dimensional sphere. Some simplicial spheres arise as the boundaries of convex polytopes, however, in higher dimensions m ...

s, which was based on earlier work of

Melvin Hochster Melvin Hochster (born August 2, 1943) is an American mathematician working in commutative algebra. He is currently the Jack E. McLaughlin Distinguished University Professor of Mathematics at the University of Michigan. Education Hochster attend ...

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 polytope In geometry, a simplicial polytope is a polytope whose facets are all simplices. For example, a ''simplicial polyhedron'' in three dimensions contains only triangular facesPolyhedra, Peter R. Cromwell, 1997. (p.341) and corresponds via Steinitz ...

s conjectured in 1970 by

Peter McMullen Peter McMullen (born 11 May 1942) is a British mathematician, a professor emeritus of mathematics at University College London. Education and career McMullen earned bachelor's and master's degrees from Trinity College, Cambridge, and studied at ...

. Known as the ''g''-theorem, it was proved in 1979 by Stanley (

necessity Necessary or necessity may refer to: * Need ** An action somebody may feel they must do ** An important task or essential thing to do at a particular time or by a particular moment * Necessary and sufficient condition, in logic, something that is ...

of the conditions, algebraic argument) and by

Louis Billera Louis Joseph Billera is a Professor of Mathematics at Cornell University. Career Billera completed his B.S. at the Rensselaer Polytechnic Institute in 1964. He earned his Ph.D. from the City University of New York in 1968, under the joint superv ...

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 Karim Alexander Adiprasito (born 1988) is a German mathematician working at the University of Copenhagen and the Hebrew University of Jerusalem who works in combinatorics. He completed his Ph.D. in 2013 at Free University Berlin under the supe ...

Important notions of combinatorial commutative algebra

* Square-free

monomial ideal In abstract algebra, a monomial ideal is an ideal generated by monomials in a multivariate polynomial ring over a field. A toric ideal is an ideal generated by differences of monomials (provided the ideal is a prime ideal). An affine or projective ...

in a

polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variable ...

and

Stanley–Reisner ring In mathematics, a Stanley–Reisner ring, or face ring, is a quotient of a polynomial algebra over a field by a square-free monomial ideal. Such ideals are described more geometrically in terms of finite simplicial complexes. The Stanley–Reisn ...

of a

simplicial complex In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial ...

. *

Cohen–Macaulay ring In mathematics, a Cohen–Macaulay ring is a commutative ring with some of the algebro-geometric properties of a smooth variety, such as local equidimensionality. Under mild assumptions, a local ring is Cohen–Macaulay exactly when it is a fini ...

. * Monomial ring, closely related to an affine semigroup ring and to the

coordinate ring In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field is the zero-locus in the affine space of some finite family of polynomials of variables with coefficients in that generate a prime ide ...

of an

affine Affine may describe any of various topics concerned with connections or affinities. It may refer to: * Affine, a Affinity_(law)#Terminology, relative by marriage in law and anthropology * Affine cipher, a special case of the more general substi ...

toric variety In algebraic geometry, a toric variety or torus embedding is an algebraic variety containing an algebraic torus as an open dense subset, such that the action of the torus on itself extends to the whole variety. Some authors also require it to be n ...

. * Algebra with a straightening law. There are several version of those, including Hodge algebras of

Corrado de Concini Corrado de Concini (born 28 July 1949 in Rome) is an Italian mathematician and professor at the Sapienza University of Rome. He studies algebraic geometry, quantum groups, invariant theory, and mathematical physics. Life and work He was born ...

David Eisenbud David Eisenbud (born 8 April 1947 in New York City) is an American mathematician. He is a professor of mathematics at the University of California, Berkeley and Director of the Mathematical Sciences Research Institute (MSRI); he previously se ...

, and

Claudio Procesi Claudio Procesi (born 31 March 1941 in Rome) is an Italian mathematician, known for works in algebra and representation theory. Career Procesi studied at the Sapienza University of Rome, where he received his degree (Laurea) in 1963. In 1966 he ...

References

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

, ''Cohen–Macaulay rings, combinatorics, and simplicial complexes''. Ring theory, II (Proc. Second Conf., Univ. Oklahoma, Norman, Okla., 1975), pp. 171–223. Lecture Notes in Pure and Appl. Math., Vol. 26, Dekker, New York, 1977. The first book is a classic (first edition published in 1983): * Richard Stanley, ''Combinatorics and commutative algebra''. Second edition. Progress in Mathematics, 41. Birkhäuser Boston, Inc., Boston, MA, 1996. x+164 pp. Very influential, and well written, textbook-monograph: * Winfried Bruns; Jürgen Herzog, ''Cohen–Macaulay rings''. Cambridge Studies in Advanced Mathematics, 39. Cambridge University Press, Cambridge, 1993. xii+403 pp. Additional reading: * Rafael Villarreal, ''Monomial algebras''. Monographs and Textbooks in Pure and Applied Mathematics, 238. Marcel Dekker, Inc., New York, 2001. x+455 pp. * Takayuki Hibi, ''Algebraic combinatorics on convex polytopes'', Carslaw Publications, Glebe, Australia, 1992 *

Bernd Sturmfels Bernd Sturmfels (born March 28, 1962 in Kassel, West Germany) is a Professor of Mathematics and Computer Science at the University of California, Berkeley and is a director of the Max Planck Institute for Mathematics in the Sciences in Leipzig si ...

, ''Gröbner bases and convex polytopes''. University Lecture Series, 8. American Mathematical Society, Providence, RI, 1996. xii+162 pp. * Winfried Bruns, Joseph Gubeladze, ''Polytopes, Rings, and K-Theory'', Springer Monographs in Mathematics, Springer, 2009. 461 pp. A recent addition to the growing literature in the field, contains exposition of current research topics: * Ezra Miller, Bernd Sturmfels, ''Combinatorial commutative algebra''.

Graduate Texts in Mathematics Graduate Texts in Mathematics (GTM) ( ISSN 0072-5285) is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standa ...

, 227. Springer-Verlag, New York, 2005. xiv+417 pp. {{ISBN, 0-387-22356-8 * Jürgen Herzog and Takayuki Hibi, ''Monomial Ideals''. Graduate Texts in Mathematics, 260. Springer-Verlag, New York, 2011. 304 pp. Commutative algebra Algebraic geometry Algebraic combinatorics

Important notions of combinatorial commutative algebra

See also

References