§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
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