Preface

Summary

It all began with Serre's beautiful 1957/58 course at the Collège de France on Algèbre locale – Multiplicites [Se]. Here he introduced a local algebraic version of the geometric notion of intersection multiplicity as follows. Let A be a d-dimensional regular local ring and M and N finitely generated A-modules such that M⊕AN has finite length. Then he defined their

culminated in the three statements

(M_o) dim M + dim N ≤ d;

(M₁) In case of an inequality above, x(M,N) = 0;

(M₂) In case there is equality, x(M,N) > 0.

Serre proved (M_o) in general and (M₁) and (M₂ except for ramified local rings in mixed characteristic. He then cagìly added: “II est naturel de conjecturer que ces resultats sont vrais pour tous les anneaux reguliers.”, thus raising the first of the homological questions of the type this book addresses. Almost 30 years later, (M₁) was proved by P. Roberts [Ro 85], [Ro 87b] and by Gillet-Soulè [GS] using different methods, but remains open in the ramified case.

That the (M₁) are not true for an arbitrary noetherian local ring is seen in the example of a 3-dimensional local domain A = k[[X,Y,U,V]]/(XY-UV), k a field, and its 2-dimensional modules

It was then surmised that the operative fact for regular rings – which in those days had just been established by Auslander-Buchsbaum and by Serre himself – is that every module has finite projective dimension.