These are numerous, but in this final chapter we treat only two, of unequal importance. The first is a proof of the New Intersection Theorem 13.1.1. There exist other proofs, [PS 74, Th. 1], [Ho 75b, p. 171], [Ro 76], [Ro 80c, Th. 1], but the present one, due to Foxby [Fo 77b, 1], quickly and elegantly establishes this result in equal characteristic. In a major breakthrough, P. Roberts recently succeeded in deriving the unequal characteristic case from the result in equal characteristic p. His proof, which uses the intersection theory developed in algebraic geometry by Fulton and MacPherson [Fu], we can only reproduce in part, 13.1.2. An important consequence of the New Intersection is the Homological Height Theorem 8.4.3 and its derivates discussed in section 8.4. Two other results which follow are the Auslander Zerodivisor Theorem 13.1.11 and an answer to a question of Bass 13.1.7.

Our second application is more parochial and establishes the nonvanishing of Bass numbers in their allowed range, Theorem 13.2.5. Again there exist other proofs of this result [Ro 76, Th. 2], [Ro 80b, Ch. 2, Prop. 4.3] but ours, which is taken from [FFGR, Th. 1.1], has an amusing feature. A weak kind of Big Cohen-Macaulay module suffices, and this is shown to exist independently from the characteristic, Proposition 13.2.1.

The existence of Big Cohen-Macaulay modules allows one to prove a stronger version of New Intersection [EG, Th. 1.13] which is needed for the proof of the Syzygies Conjecture.