Local algebra. Transl. from the French by CheeWhye Chin.

*(English)*Zbl 0959.13010
Springer Monographs in Mathematics. Berlin: Springer. xiii, 128 p. (2000).

This is an English translation of J.-P. Serre’s famous lecture notes “Algèbre locale. Multiplicités”, first appeared as Cours professé au Collège de France 1957/58 (written by P. Gabriel; Zbl 0091.03701) and then as Lect. Notes Math. 11 (1965; Zbl 0142.28603). This book has served generations of mathematicians who wanted to study homological methods in commutative algebra and, above all, the theory of intersection multiplicity.

As Serre wrote in the introduction, the main purpose of this course is to prove that the intersection multiplicities of algebraic geometry are equal to some Euler-Poincaré characteristics constructed by means of the Tor functor. To do this he had to recall almost all basic results of local algebra including many homological concepts which became standard now in commutative algebra.

Let \(A\) be a regular local ring. For every couple of finitely generated \(A\)-modules \(M\) and \(N\) such that \(M \otimes_k N\) has finite length, the intersection multiplicity of \(M\) and \(N\) is defined as \(\chi(M,N) := \sum_{i=0}^d(-1)^i\text{length(Tor}_i^A(M,N))\). The link to intersection multiplicities in algebraic geometry has led to the conjectures:

(1) \(\chi(M,N) = 0\) if \(\dim M + \dim N < \dim A\) (vanishing).

(2) \(\chi(M,N) > 0\) if \(\dim M + \dim N = \dim A\) (non-vanishing).

In this course, Serre proved these conjectures for regular local rings containing a field. However, his proof broke down in the ramified case. The vanishing conjecture was solved for the general case 30 years later independently by P. Roberts [Bull. Am. Soc., New Ser. 13, 127-130 (1985; Zbl 0585.13004)] and H. Gillet and C. Soulé [C. R. Acad. Sci., Paris, Sér. I 300, 71-74 (1985; Zbl 0587.13007)]. But the non-vanishing conjecture has remained open until today. Recently, O. Gabber proved \(\chi(M,N) \geq 0\) by using a major result on resolution of singularities [see P. Berthelot in: Séminaire Bourbaki, Vol. 1995/96, Astérisque 241, 273-311, exposé 815 (1997; Zbl 0924.14007)]. This development demonstrates that Serre’s course is still actual.

The translation differs from the previous editions in the following aspects: The terminology and references have been brought up to date; some proofs have been rewritten or clarified; some new sections have been added such as chapter II, A.8 (adic filtrations), chapter IV, D.5 (regularity in ring extensions), appendix III with 5 sections (graded algebras), index, index of notations.

Serre’s course is a book which can be profitably read and reread by beginner and expert alike.

As Serre wrote in the introduction, the main purpose of this course is to prove that the intersection multiplicities of algebraic geometry are equal to some Euler-Poincaré characteristics constructed by means of the Tor functor. To do this he had to recall almost all basic results of local algebra including many homological concepts which became standard now in commutative algebra.

Let \(A\) be a regular local ring. For every couple of finitely generated \(A\)-modules \(M\) and \(N\) such that \(M \otimes_k N\) has finite length, the intersection multiplicity of \(M\) and \(N\) is defined as \(\chi(M,N) := \sum_{i=0}^d(-1)^i\text{length(Tor}_i^A(M,N))\). The link to intersection multiplicities in algebraic geometry has led to the conjectures:

(1) \(\chi(M,N) = 0\) if \(\dim M + \dim N < \dim A\) (vanishing).

(2) \(\chi(M,N) > 0\) if \(\dim M + \dim N = \dim A\) (non-vanishing).

In this course, Serre proved these conjectures for regular local rings containing a field. However, his proof broke down in the ramified case. The vanishing conjecture was solved for the general case 30 years later independently by P. Roberts [Bull. Am. Soc., New Ser. 13, 127-130 (1985; Zbl 0585.13004)] and H. Gillet and C. Soulé [C. R. Acad. Sci., Paris, Sér. I 300, 71-74 (1985; Zbl 0587.13007)]. But the non-vanishing conjecture has remained open until today. Recently, O. Gabber proved \(\chi(M,N) \geq 0\) by using a major result on resolution of singularities [see P. Berthelot in: Séminaire Bourbaki, Vol. 1995/96, Astérisque 241, 273-311, exposé 815 (1997; Zbl 0924.14007)]. This development demonstrates that Serre’s course is still actual.

The translation differs from the previous editions in the following aspects: The terminology and references have been brought up to date; some proofs have been rewritten or clarified; some new sections have been added such as chapter II, A.8 (adic filtrations), chapter IV, D.5 (regularity in ring extensions), appendix III with 5 sections (graded algebras), index, index of notations.

Serre’s course is a book which can be profitably read and reread by beginner and expert alike.

Reviewer: Ngo Viet Trung (Hanoi)