4

DAVID B. MASSEY

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

x

for the stalk of a in Ox x.

Definition 1.1. Let S be the multiplicatively closed set Ox

x

— [jp where the union is over

all p e Ass(Ox

x

/ a

x

) with \V(p)\ £ \W\. Then, we define a^W to equal

S'1^

0 GXx.

Thus, a^-^W is the ideal in Ox

x

consisting of the intersection of those (possibly embedded)

primary ideals, q, associated to a

x

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

oo

k=0

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

VigW

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;