%0 Book Section
%T A law of conservation of number for local Euler characteristics
publisher American Mathematical Society
%D 2002
%U 0-8218-2957-2
%@ https://hdl.handle.net/20.500.14352/60709
%X Let T  and V  be complex analytic spaces, with T  reduced and locally irreducible and π:T×V→T  the projection map to the first factor. Let K ∗   be a complex 0→K n → X n  K n−1 → X n−1  …→ X 2  K 1 → X 1  K 0 →0  of O T×V   coherent sheaves, where all X j   are O T×V  -linear, all sheaves K j   are O T  -flat and such that the support of the homology sheaves H j (K ∗ )  is π -finite. For every t∈T , V t   denotes {t}×V≅V ; K ∗ t   denotes the complex obtained by tensoring K ∗   with O V t   ; K ∗ t;p   denotes the complex formed by the germs at (t,p)  of K ∗ t   and H j (K ∗ t;p )  denotes the j th homology group of the complex K ∗ t;p  . The Euler characteristic of the complex of sheaves K ∗ t   at a point (t,p)∈V t   is defined as χ(K ∗ t;p )=∑ j=0 n (−1) j dim C H j (K ∗ t;p ).  The authors show that, for every (t 0 ,p 0 )∈T×V  there are neighbourhoods T ′   and V ′   of t 0   and p 0  , respectively, such that for every t∈T ′  , χ(K ∗ t 0 ;p 0  )=∑ q∈V ′  χ(K ∗ t;p ) .
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