C h a p t e r 1. GAP SHEAVES
Let W be analytic subset of an analytic space X and let a be a coherent sheaf of ideals
in Ox. At each point x of V(a), we wish to consider scheme-theoretically those components
of V(a) which are not contained in \W\. This leads one to the notion of a gap sheaf. Our
primary references for gap sheaves are [Si-TV] and [Fi].
Let j3 be a second coherent sheaf of ideals in Ox. We write a
for the stalk of a in Ox x.
Definition 1.1. Let S be the multiplicatively closed set Ox
[jp where the union is over
all p e Ass(Ox
/ a
) with \V(p)\ £ \W\. Then, we define a^W to equal
0 GXx.
Thus, a^-^W is the ideal in Ox
consisting of the intersection of those (possibly embedded)
primary ideals, q, associated to a
such that \V(q)\ £ \W\.
Now, we have defined a^W in each stalk. By [Si-TV], if we perform this operation
simultaneously at all points of V(a)1 then we obtain a coherent sheaf of ideals called a
gap sheaf ; we write this sheaf as a-^W. If /3 is any coherent sheaf of ideals such that
W = supp(C?x//?), then
If V = V(a), we let V-W denote the scheme V(a^W). It is important to note that the
scheme V~W does not depend on the structure of W as a scheme, but only as an analytic
set. The scheme V-W is sometimes referred to as the analytic closure ofV W [Fi, p.41];
this is certainly the correct, intuitive way to think of V-^W.
We find it convenient to extend this gap sheaf notation to the case of analytic sets (reduced
schemes) and analytic cycles.
Hence, if Z and W are analytic sets, then we let Z-W denote the union of the components
of Z which are not contained in W; if C = ^2
mi[Vi] *s a n
analytic cycle in a complex manifold
M and W is an analytic subset of M, then we define C~W by
If a is a coherent sheaf of ideals in OM, C is a cycle in M, and W is an analytic subset of
M, then clearly [V(a)-iW] - [V{a)]^W and |C-W| = \C\-W.
The following properties of gap sheaves are immediate from the definition.
Proposition 1.2.
i) ax = f3x for all x £ X W if and only if a-W (3-W;
ii) if on is a finite collection of coherent ideals in Ox, then D(a^VF) = (nai)-^W;
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