Combinatorial commutative algebra
Encyclopedia
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 polytope
s 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 Lou Billera and Carl W. Lee (sufficiency, combinatorial and geometric construction). A major open question is the extension of this characterization from simplicial polytopes to simplicial spheres, the g-conjecture.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
discipline. As the name implies, it lies at the intersection of two more established fields, commutative algebra
Commutative algebra
Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...
and combinatorics
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
, and frequently uses methods of one to address problems arising in the other. Less obviously, polyhedral geometry
Convex polytope
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn...
plays a significant role.
One of the milestones in the development of the subject was Richard Stanley
Richard P. Stanley
Richard Peter Stanley is the Norman Levinson Professor of Applied Mathematics at the Massachusetts Institute of Technology, in Cambridge, Massachusetts. He received his Ph.D. at Harvard University in 1971 under the supervision of Gian-Carlo Rota...
's 1975 proof of the Upper Bound Conjecture for simplicial spheres, which was based on earlier work of Melvin Hochster
Melvin Hochster
Melvin Hochster is an eminent American mathematician, regarded as one of the leading commutative algebraists active today. He is currently the Jack E. McLaughlin Distinguished University Professor of Mathematics at the University of Michigan.Hochster attended Stuyvesant High School, where he was...
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
H-vector
In algebraic combinatorics, the h-vector of a simplicial polytope is a fundamental invariant of the polytope which encodes the number of faces of different dimensions and allows one to express the Dehn–Sommerville equations in a particularly simple form...
of simplicial polytope
Simplicial polytope
In geometry, a simplicial polytope is a d-polytope whose facets are all simplices.For example, a simplicial polyhedron contains only triangular faces and corresponds via Steinitz's theorem to a maximal planar graph....
s 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 Lou Billera and Carl W. Lee (sufficiency, combinatorial and geometric construction). A major open question is the extension of this characterization from simplicial polytopes to simplicial spheres, the g-conjecture.
Important notions of combinatorial commutative algebra
- Square-free monomial ideal in a polynomial ringPolynomial ringIn mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of...
and Stanley–Reisner ring of a simplicial complexSimplicial complexIn mathematics, a simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments, triangles, and their n-dimensional counterparts...
. - 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 version of those, including Hodge algebras of Corrado de Concini, David EisenbudDavid EisenbudDavid Eisenbud is an American mathematician. He is a professor of mathematics at the University of California, Berkeley and was Director of the Mathematical Sciences Research Institute from 1997 to 2007....
, and Claudio Procesi.