This chapter is an introduction to ‘combinatorial commutative algebra’, a fascinating new branch of commutative algebra created by Hochster and Stanley in the mid-seventies. The combinatorial objects considered are simplicial complexes to which one assigns algebraic objects, the Stanley–Reisner rings. We study how the face numbers of a simplicial complex are related to the Hilbert series of the corresponding Stanley–Reisner ring. This is the basis of all further investigations which culminate in Stanley's proof of the upper bound theorem for simplicial spheres. It turns out that most of the important algebraic notions introduced in the earlier chapters, such as ‘Cohen–Macaulay’, ‘Gorenstein’, ‘local cohomology’, and ‘Hilbert series’, are the proper concepts in solving purely combinatorial problems. Other applications of commutative algebra to combinatorics will be given in the next chapter.

Simplicial complexes

The present section is devoted to introducing the Stanley–Reisner ring associated with a simplicial complex, and studying its Hilbert series. The most important invariant of a simplicial complex, its f-vector, can be easily transformed into the h-vector, an invariant encoded by the Hilbert function of the associated Stanley–Reisner ring. It is of interest to know when a Stanley–Reisner ring is Cohen–Macaulay, because then the results about Hilbert functions of Chapter 4 may be employed to get information about the f-vector. In concluding this section we show that the Stanley–Reisner ring of a shellable simplicial complex is Cohen–Macaulay, and study systems of parameters of such a ring.