§0.

INTRODUCTION

The residue symbol introduced by Grothendieck [RD, pp. 195-199] has been found

useful in various contexts: duality theory of algebraic varieties, Gysin homomorphisms

of manifolds with vector fields having isolated zeros, integral representations in several

complex variables,

just to mention a few ( cf. for example [L], [AC], [AY], and their

bibliographies).

However, in spite of its broad interest, the theory of the residue symbol does not

seem to have been written down in a really satisfactory manner. One difficulty is that

Grothendieck's approach depends on the global duality machinery developed in [RD];

and furthermore proofs are not given there.

(A

more detailed version is presented in

[Bv];

and for a complete treatment of the case of algebraic varieties, with a somewhat

different slant, cf.

[L].)

Grothendieck considers a smooth map f:

X--+- Y

of locally

noetherian schemes, with q-dimensional fibres, and a closed subscheme

Z

of

X

defined by an ideal I which is locally generated by q elements, and such that

Z

is

finite over

Y.

With i:

Z --+- X

the inclusion, and g

=

f

o

i:

Z --+- Y,

there is a residue

isomorphism i1f1

.::...

g1, or, more concretely, a sheaf isomorphism:

g*(Hom0z(Aq(I/!

2

),

i*O~/Y)).::...

Homoj_g*

Oz,

Oy)

(O~;y

=

relative differential q-forms) upon which the theory of the residue symbol is

built.

But in fact the residue symbol can be viewed as a formal algebraic construct,

which can be defined and studied directly with only the elements of ring theory and

homological algebra. Indeed, while duality theory may provide the primary motivation

for residues,(!) eliminating it from their theoretical foundation results not only in

greater simplicity, but also in greater generality, and ultimately, one hopes, in more

1980 Mathematics Subject Classification: 13099, 14F10, 16A61, 32A27.

Partially supported by NSF' Grant MCS-8200624 at Purdue University.

(I)

and that is why [L] appeared before this paper. (The relation of this paper to [L]

is

made explicit in Appendix A of §3 below.) My own interest in the subject was inspired

by p. 81 of [SJ, and by §§10 and 15 of [Z].

1

http://dx.doi.org/10.1090/conm/061