Rings with monomial relations having linear resolutions (Q1063072)

By a ring with monomial relations is meant an algebra with presentation \(R=k[X_ 1,...,X_ n]/(M_ 1,...,M_ r)\), where k is a field and the \(M_ i's\) are monomials in the \(X_ j's\). The algebra R has a linear resolution if all syzygies of dimension at least two in a \(k[X_ 1,...,X_ n]\)-resolution of R are linear. This note gives a necessary and sufficient condition for a ring with monomial relations to have a linear resolution. (It is shown that this depends on the characteristic of k.) The crucial points are a reduction to the case of rings with squarefree monomial relations, and then a connection between the homology of the Koszul complex for such a ring and the homology for some associated simplicial complexes. This connection also gives some results of combinatorial nature for simplicial complexes. It also yields a precise condition for the ring to be Cohen-Macaulay.