On linear complete intersections (Q1104988)

This note develops three essentially unrelated variations on the theme of characterizing the ring properties of the symmetric algebra S(E) of an R- module M in terms of data from a presentation \(R^ m\to^{\phi}R^ n\to E\to 0\), or of a longer piece of a free resolution of E. Thus, when R is a noetherian normal domain, Serre's condition \(R_ 1\) is expressed in terms of ranks of matrices constructed from \(\phi\), and this is used to establish the (non-)normality of certain symmetric and Rees algebras. When Ker(\(\phi)\) is free of rank 1, and R is a Cohen-Macaulay domain, then it is shown that S(E) has the same property, provided E is a first syzygy, its dual \(E^*\) is a third syzygy, and the ideal generated by the entries of the inclusion Ker(\(\phi)\to R^ m\) has certain Cohen- Macaulay and syzygetic properties. Finally, it is proved that if \(S(E)\otimes_ RS(E)\) is an integral domain, then \(pd_ R(E)\leq 1\), in case R is a regular local ring.