Preface

The present text is an extended and updated version of my lecture notes

Residuen und Dualit¨ at auf projektiven algebraischen Variet¨ aten (Der Regensburger

Trichter 19 (1986)), based on a course I taught in the winter term 1985/86 at

the University of Regensburg. I am grateful to David Cox for helping me with the

translation and transforming the manuscript into the appropriate LTEX

A

2ε style, to

Alicia Dickenstein and David Cox for encouragement and critical comments and for

enriching the book by adding two sections, one on applications of algebraic residue

theory and the other explaining toric residues and relating them to the earlier text.

The main objective of my old lectures, which were strongly influenced by Lip-

man’s monograph [71], was to describe local and global duality in the special case of

irreducible algebraic varieties over an algebraically closed base field k in terms of dif-

ferential forms and their residues. Although the dualizing sheaf of a d-dimensional

algebraic variety V is only unique up to isomorphism, there is a canonical choice,

the sheaf ωV/k of regular d-forms. This sheaf is an intrinsically defined subsheaf of

the constant sheaf ΩR(V

d

)/k

, where R(V ) is the field of rational functions on V . We

construct ωV/k in § 9 after the necessary preparation. Similarly, for a closed point

x ∈ V , the stalk (ωV/k)x is a canonical choice for the dualizing (canonical) module

ωOV,x/k studied in local algebra. We have the residue map

Resx : Hx

d(ωV/k)

−→ k

defined on the d-th local cohomology of ωV/k. The local cohomology classes can be

written as generalized fractions

ω

f1, . . . , fd

where ω ∈ ωOV,x/k and f1, . . . , fd is a system of parameters of OV,x. Using the

residue map, we get the Grothendieck residue symbol

Resx

ω

f1, . . . , fd

.

For a projective variety V the residue map at the vertex of the aﬃne cone C(V )

induces a linear operator on global cohomology

V

:

Hd(V,

ωV/k) −→ k

called the integral. The local and global duality theorems are formulated in terms

of Resx and

V

. There is also the residue theorem stating that “the integral is

the sum of all of the residues.” Specializing to projective algebraic curves gives

the usual residue theorem for curves plus a version of the Serre duality theorem

expressed in terms of differentials and their residues. Basic rules of the residue

calculus are formulated and proved, and later generalized to toric residues by David

vii